1SECTION 3.3.1.1.1 (JULY 1974) PAGE 1 AUTHORS: T.J.DEKKER AND W.HOFFMANN. CONTRIBUTORS: W.HOFFMANN, J.G.VERWER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730716. BRIEF DESCRIPTION: THIS SECTION CONTAINS FOUR PROCEDURES FOR CALCULATING EIGENVALUES OR EIGENVECTORS OF A SYMMETRIC TRIDIAGONAL MATRIX. VALSYMTRI CALCULATES ALL, OR SOME CONSECUTIVE, EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX BY MEANS OF LINEAR INTERPOLATION USING A STURM SEQUENCE; VECSYMTRI CALCULATES THE CORRESPONDING EIGENVECTORS BY MEANS OF INVERSE ITERATION. QRIVALSYMTRI CALCULATES ALL EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX BY MEANS OF QR ITERATION; QRISYMTRI CALCULATES THE EIGENVECTORS AS WELL. WHEN ALL EIGENVALUES HAVE TO BE CALCULATED, QRIVALSYMTRI IS PREFERABLE WITH RESPECT TO THE RUNNING TIME; WHEN THE EIGENVECTORS ALSO HAVE TO BE CALCULATED, INVERSE ITERATION IS PREFERABLE. KEYWORDS: EIGENVALUES, EIGENVECTORS, TRIDIAGONAL MATRIX, STURM-SEQUENCE, INVERSE ITERATION, QR ITERATION. 1SECTION 3.3.1.1.1 (DECEMBER 1975) PAGE 2 SUBSECTION: VALSYMTRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" VALSYMTRI(D, BB, N, N1, N2, VAL, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, BB, VAL, EM; "CODE" 34151; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; D: ; "ARRAY" D[1:N]; ENTRY: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX; BB: ; "ARRAY" BB[1:N-1]; ENTRY: THE SQUARES OF THE CODIAGONAL ELEMENTS OF THE SYMMETRIC TRIDIAGONAL MATRIX; N1,N2: ; THE SERIAL NUMBER OF THE FIRST AND LAST EIGENVALUE TO BE CALCULATED, RESPECTIVELY; VAL: ; "ARRAY" VAL[N1:N2]; EXIT: THE N2-N1+1 CALCULATED CONSECUTIVE EIGENVALUES IN NONINCREASING ORDER; EM: ; "ARRAY" EM[0:3]; ENTRY: EM[0], THE MACHINE PRECISION, EM[1], AN UPPERBOUND FOR THE MODULI OF THE EIGENVALUES OF THE GIVEN MATRIX, EM[2], A RELATIVE TOLERANCE FOR THE EIGENVALUES; EXIT: EM[3], THE TOTAL NUMBER OF ITERATIONS USED FOR CALCULATING THE EIGENVALUES. PROCEDURES USED: ZEROIN = CP34150. RUNNING TIME: DEPENDS STRONGLY ON THE DISTANCE OF SUCCESSIVE EIGENVALUES. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: LET T DENOTE THE GIVEN SYMMETRIC TRIDIAGONAL MATRIX OF ORDER N AND I THE IDENTITY MATRIX. THE EIGENVALUES OF T ARE THE ZEROES OF THE N-TH DEGREE POLYNOMIAL P(N,X) = DET(T - X*I). INSTEAD OF SEARCHING FOR THE ZEROES OF P(N,X) WE LOOK FOR THE ZEROES OF THE FUNCTION F(N,X) = P(N,X) / P(N-1,X). MAINTAINING A LOWER BOUND FOR ABS(P(N-1,X)) WE DO AVOID OVERFLOW OF THE REAL NUMBER CAPACITY IN THE COMPUTATION OF F(N,X). THIS FUNCTION CAN BE CALCULATED AS FOLLOWS: 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 3 F(1,X) = D[1] - X, F(K,X) = D[K] - X - BB[K-1] / ("IF" ABS(F(K-1,X)) > MACHTOL "THEN" F(K-1,X) "ELSE" "IF" F(K-1,X) <= 0 "THEN" -MACHTOL "ELSE" MACHTOL), K = 2, . . . ,N, WHERE MACHTOL EQUALS EM[0] * EM[1]. USING THE STURM SEQUENCE PROPERTY OF (F(K,X)), K=1,2,...,N, WE CAN LOCATE THE DESIRED EIGENVALUES BY MEANS OF THE PROCEDURE ZEROIN (SECTION 5.1.1.1). FOR FURTHER DETAILS SEE REF[1], REF[2]. SUBSECTION: VECSYMTRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" VECSYMTRI(D, B, N, N1, N2, VAL, VEC, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, B, VAL, VEC, EM; "CODE" 34152; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; D: ; "ARRAY" D[1:N], ENTRY: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX; B: ; "ARRAY" B[1:N]; ENTRY: THE CODIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX FOLLOWED BY AN ADDITIONAL ELEMENT 0; N1, N2: ; LOWER AND UPPER BOUND OF THE ARRAY VAL (SEE ALSO METHOD AND PERFORMANCE); VAL: ; "ARRAY" VAL[N1:N2]; ENTRY: A ROW OF NONINCREASING EIGENVALUES AS DELIVERED BY VALSYMTRI; VEC: ; "ARRAY" VEC[1:N,N1:N2]; EXIT: THE EIGENVECTORS CORRESPONDING WITH THE GIVEN EIGENVALUES (SEE ALSO METHOD AND PERFORMANCE); EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION, EM[1], A NORM OF THE GIVEN MATRIX, EM[4], THE ORTHOGONALISATION PARAMETER (SEE ALSO METHOD AND PERFORMANCE), EM[6], THE RELATIVE TOLERANCE FOR THE EIGENVECTORS, EM[8], THE MAXIMUM NUMBER OF ITERATIONS ALLOWED FOR THE CALCULATION OF EACH EIGENVECTOR; 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 4 EXIT: EM[5], THE NUMBER OF EIGENVECTORS INVOLVED IN THE LAST GRAM-SCHMIDT ORTHOGONALISATION (SEE METHOD AND PERFORMANCE), EM[7], THE MAXIMUM EUCLIDEAN NORM OF THE RESIDUES, EM[9], THE LARGEST NUMBER OF ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGENVECTOR (SEE METHOD AND PERFORMANCE). PROCEDURES USED: VECVEC = CP34010, TAMVEC = CP34012, ELMVECCOL= CP34021. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: FIVE AUXILIARY ONE-DIMENSIONAL REAL ARRAYS AND ONE BOOLEAN ARRAY, ALL OF LENGTH N, ARE USED. RUNNING TIME: THE PROCESS IS OF ORDER N FOR EACH EIGENVECTOR. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: AN EIGENVECTOR OF A SYMMETRIC TRIDIAGONAL MATRIX T, CORRESPONDING TO AN EIGENVALUE LAMBDA, IS CALCULATED BY MEANS OF INVERSE ITERATION; I.E. STARTING FROM SOME INITIAL VECTOR X, THE LINEAR SYSTEM (T - LAMBDA * I)Y = X IS SOLVED ITERATIVELY, THE SOLUTION Y, DIVIDED BY ITS EUCLIDEAN NORM REPLACING X EACH TIME. IF THE DISTANCE BETWEEN SOME APPROXIMATE EIGENVALUES IS SMALLER THAN MACHTOL (=EM[0] * EM[1]), THEN THEY ARE SLIGHTLY MODIFIED SUCH THAT THE DISTANCE BETWEEN THEM EQUALS MACHTOL. IF THE DISTANCE BETWEEN SOME EIGENVALUES IS SMALLER THAN THE ORTHOGONALISATION PARAMETER (=EM[4]) TIMES EM[1], THEN IN EACH ITERATION STEP GRAM- SCHMIDT ORTHOGONALISATION IS CARRIED OUT, SO THAT THE EIGENVECTORS OBTAINED ARE ORTHOGONAL WITHIN WORKING PRECISION. THE ITERATION ENDS AS SOON AS EITHER THE EUCLIDEAN NORM OF THE RESIDUE IS SMALLER THAN EM[1] * EM[6], OR THE MAXIMUM ALLOWED NUMBER OF ITERATIONS (=EM[8]) HAS BEEN PERFORMED. IN THE LATTER CASE EM[9]:= EM[8] + 1. IF N1 > 1, THEN VECSYMTRI SHOULD BE PRECEDED BY ONE OR MORE CALLS OF VECSYMTRI PRODUCING A NUMBER OF EIGENVECTORS CORRESPONDING TO THE PRECEDING EIGENVALUES. MOREOVER ONE MUST GIVE EM[5], AS PRODUCED BY THE LAST CALL OF VECSYMTRI; THE K-TH TO N2-TH EIGENVALUES, WHERE K = N1 - EM[5], MUST BE GIVEN IN ARRAY VAL[K:N2] IN MONOTONICALLY NONINCREASING ORDER (THE K-TH TO (N1-1)-TH EIGENVALUES BEING NEEDED FOR THE MODIFYING MENTIONED ABOVE), AND THE CORRESPONDING EIGENVECTORS UP TO THE (N1-1)-TH (WHICH ARE NEEDED FOR THE GRAM-SCHMIDT ORTHOGONALISATION) IN THE CORRESPONDING COLUMNS OF ARRAY VEC[1:N,K:N2]. THE TOLERANCES SHOULD SATISFY: EM[0] ( < EM[2]) < EM[6] AND EM[4] >= EM[0] / EM[6]. FOR FURTHER DETAILS SEE REF[1]. 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 5 SUBSECTION: QRIVALSYMTRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" QRIVALSYMTRI(D, BB, N, EM); "VALUE" N; "INTEGER" N; "ARRAY" D, BB, EM; "CODE" 34160; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; D: ; "ARRAY" D[1:N]; ENTRY: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX; EXIT: THE EIGENVALUES OF THE MATRIX IN SOME ARBITRARY ORDER; BB: ; "ARRAY" BB[1:N]; ENTRY: THE SQUARES OF THE CODIAGONAL ELEMENTS OF THE SYMMETRIC TRIDIAGONAL MATRIX FOLLOWED BY AN ADDITONAL ELEMENT O; EXIT: THE SQUARES OF THE CODIAGONAL ELEMENTS OF THE SYMMETRIC TRIDIAGONAL MATRIX RESULTING FROM THE QR ITERATION; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[2], A RELATIVE TOLERANCE FOR THE EIGENVALUES; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF ITERATIONS PERFORMED. MOREOVER: QRIVALSYMTRI:= THE NUMBER OF EIGENVALUES NOT CALCULATED. PROCEDURES USED: NONE. RUNNING TIME: THE PROCESS IS OF ORDER N SQUARED. LANGUAGE: ALGOL 60. 1SECTION 3.3.1.1.1 (DECEMBER 1975) PAGE 6 METHOD AND PERFORMANCE: IN QRIVALSYMTRI THE EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX ARE CALCULATED BY MEANS OF QR-ITERATION. FOR THIS PROCEDURE WE USED ESSENTIALLY THE SQUARE-ROOT-FREE VERSION OF THE QR ALGORITHM DUE TO REINSCH[3]. IN ADDITION TO THE RELATIVE ERROR, WHICH IS SUPPOSED TO BE BOUNDED BY EM[1] * EM[2] (I.E. MATRIX NORM TIMES RELATIVE TOLERANCE), THE CALCULATED EIGENVALUES HAVE AN ABSOLUTE ERROR WHICH IS BOUNDED BY BY EM[0] * EM[1] (I.E. MACHINE PRECISION TIMES MATRIX NORM). IN PARTICULAR, WHEN SOME EIGENVALUES ARE VERY SMALL COMPARED TO THE MATRIX NORM, THE ACCURACY OF THE CALCULATED EIGENVALUES CAN BE INCREASED BY GIVING EM[0] A (POSITIVE) VALUE WHICH IS LESS THAN THE MACHINE PRECISION. A PARTICULAR CHOICE OF EM[0] IS HARMLESS FOR THE PROCEDURE PROVIDED THAT FOR EACH I THE CALCULATION OF BB[I] / EM[0] ** 2 CAUSES NO OVERFLOW AND THE CALCULATION OF (EM[0] * EM[1]) ** 2 CAUSES NO UNDERFLOW. ONE SHOULD NOTICE THAT THE NUMBER OF QR ITERATIONS INCREASES BY A SMALLER CHOICE OF EM[0]. FOR FURTHER DETAILS SEE [2], [3]. SUBSECTION: QRISYMTRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" QRISYMTRI(A, N, D, B, BB, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, D, B, BB, EM; "CODE" 34161; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; D: ; "ARRAY" D[1:N]; ENTRY: THE MAIN DIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX; EXIT: THE EIGENVALUES OF THE MATRIX IN SOME ARBITRARY ORDER; B: ; "ARRAY" B[1:N]; ENTRY: THE CODIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX FOLLOWED BY AN ADDITIONAL ELEMENT 0; EXIT: THE CODIAGONAL OF THE SYMMETRIC TRIDIAGONAL MATRIX RESULTING FROM THE QR ITERATION, FOLLOWED BY AN ADDITIONAL ELEMENT 0; BB: ; "ARRAY" BB[1:N]; ENTRY: THE SQUARED CODIAGONAL ELEMENTS OF THE SYMMETRIC TRIDIAGONAL MATRIX, FOLLOWED BY AN ADDITIONAL ELEMENT O; EXIT: THE SQUARED CODIAGONAL ELEMENTS OF THE SYMMETRIC TRIDIAGONAL MATRIX RESULTING FROM THE QR ITERATION; 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 7 A: ; "ARRAY" A[1:N,1:N]; ENTRY: SOME MATRIX S, SAY, (POSSIBLY THE IDENTITY MATRIX); EXIT: THE EIGENVECTORS OF THE ORIGINAL SYMMETRIC TRIDIAGONAL MATRIX, PREMULTIPLIED BY S (SEE METHOD AND PERFORMANCE); EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[2], A RELATIVE TOLERANCE FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF ITERATIONS PERFORMED. MOREOVER: QRISYMTRI:= THE NUMBER OF EIGENVALUES AND -VECTORS NOT CALCULATED. PROCEDURES USED: ROTCOL = CP34040. RUNNING TIME: THE PROCESS IS OF ORDER N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: IN QRISYMTRI THE EIGENVALUES AND EIGENVECTORS OF A SYMMETRIC TRIDIAGONAL MATRIX ARE COMPUTED SIMULTANEOUSLY. IN MOST APPLICATIONS QRISYMTRI IS USED IN THE COMPUTATION OF EIGENVALUES AND -VECTORS OF A GENERAL SYMMETRIC MATRIX (SEE QRISYM SECTION 3.3.1.1.2); IN THAT CASE ARRAY A IS INITIALLY GIVEN THE VALUE OF THE TRANSFORMING MATRIX (TFMPREVEC SECTION 3.2.1.2.1.1). FOR THE COMPUTATION OF EIGENVALUES AND EIGENVECTORS OF A SYMMETRIC TRIDIAGONAL MATRIX, ARRAY A HAS TO BE INITIALIZED TO THE IDENTITY MATRIX. THE AVERAGE NUMBER OF ITERATIONS IS ABOUT 3N. WHEN THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS, THEN QRISYMTRI:= 0; OTHERWISE QRISYMTRI:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED, EM[5]:= EM[4] + 1 AND ONLY THE LAST N - K ELEMENTS OF D AND THE LAST N - K COLUMNS OF A ARE APPROXIMATE EIGENVALUES AND -VECTORS RESPECTIVELY, OF THE ORIGINAL MATRIX. FOR FURTHER DETAILS SEE REF[1], REF[2]. 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 8 REFERENCES: [1] DEKKER, T.J. AND HOFFMANN, W. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2, MATHEMATICAL CENTRE TRACTS 23, MATHEMATISCH CENTRUM, AMSTERDAM, 1968; [2] WILKINSON, J.H. THE ALGEBRAIC EIGENVALUE PROBLEM, CLARENDON PRESS, OXFORD 1965. [3] REINSCH, CHR.H. A STABLE, RATIONAL QR ALGORITHM FOR THE COMPUTATION OF THE EIGENVALUES OF AN HERMITIAN, TRIDIAGONAL MATRIX. MATH. OF COMP. VOL 25(1971) PP. 591-597. EXAMPLE OF USE: THE FIRST AND SECOND EIGENVALUE IN MONOTONICALLY NON-INCREASING ORDER AND THE CORRESPONDING EIGENVECTORS OF T, WITH N = 4 AND T[I,J] = "IF" I = J "THEN" 2 "ELSE" "IF" ABS(I - J) = 1 "THEN" - 1 "ELSE" 0, MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" J; "ARRAY" B, D[1:4], BB[1:3], VAL[1:2], EM[0:9], VEC[1:4,1:2]; EM[0]:= "-14; EM[1]:= 4; EM[2]:= "-12; EM[4]:= "-3; EM[6]:= "-10; EM[8]:= 5; "FOR" J:= 1, 2, 3, 4 "DO" D[J]:= 2; B[4]:= 0; "FOR" J:= 1, 2, 3 "DO" "BEGIN" BB[J]:= 1; B[J]:= -1 "END"; VALSYMTRI(D, BB, 4, 1, 2, VAL, EM); VECSYMTRI(D, B, 4, 1, 2, VAL, VEC, EM); OUTPUT(61, "("2(+.13D"+2D, 2B), 2/")", VAL[1], VAL[2]); "FOR" J:= 1, 2, 3, 4 "DO" OUTPUT(61, "("2(+.13D"+2D, 2B), /")", VEC[J,1], VEC[J,2]); OUTPUT(61, "("/, .2D"+2D, /, 3(2ZD, /)")", EM[7], EM[3], EM[5], EM[9]) "END" THE PROGRAM DELIVERS: THE EIGENVALUES: +.3618033988751"+01 +.2618033988750"+01 THE EIGENVECTORS: +.3717480344602"+00 +.6015009550075"+00 +.6015009550075"+00 +.3717480344602"+00 +.6015009550075"+00 -.3717480344602"+00 +.3717480344602"+00 -.6015009550075"+00 EM[7] = .15"-11 EM[3] = 24 EM[5] = 1 EM[9] = 1 . 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 9 SOURCE TEXT(S): 0"CODE" 34151; "COMMENT" MCA 2311; "PROCEDURE" VALSYMTRI(D, BB, N, N1, N2, VAL, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, BB, VAL, EM; "BEGIN" "INTEGER" K, COUNT; "REAL" MAX, X, Y, MACHEPS, NORM, RE, MACHTOL, UB, LB, LAMBDA; "REAL" "PROCEDURE" STURM; "BEGIN" "INTEGER" P, I; "REAL" F; COUNT:= COUNT + 1; P:= K; F:= D[1] - X; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" F <= 0 "THEN" "BEGIN" P:= P + 1; "IF" P > N "THEN" "GOTO" OUT "END" "ELSE" "IF" P < I - 1 "THEN" "BEGIN" LB:= X; "GOTO" OUT "END"; "IF" ABS(F) < MACHTOL "THEN" F:= "IF" F <= 0 "THEN" - MACHTOL "ELSE" MACHTOL; F:= D[I] - X - BB[I - 1] / F "END"; "IF" P = N "OR" F <= 0 "THEN" "BEGIN" "IF" X < UB "THEN" UB:= X "END" "ELSE" LB:= X; OUT: STURM:= "IF" P = N "THEN" F "ELSE" (N - P) * MAX "END" STURM; MACHEPS:= EM[0]; NORM:= EM[1]; RE:= EM[2]; MACHTOL:= NORM * MACHEPS; MAX:= NORM / MACHEPS; COUNT:= 0; UB:= 1.1 * NORM; LB:= - UB; LAMBDA:= UB; "FOR" K:= N1 "STEP" 1 "UNTIL" N2 "DO" "BEGIN" X:= LB; Y:= UB; LB:= -1.1 * NORM; ZEROIN(X, Y, STURM, ABS(X) * RE + MACHTOL); VAL[K]:= LAMBDA:= "IF" X > LAMBDA "THEN" LAMBDA "ELSE" X; "IF" UB > X "THEN" UB:= "IF" X > Y "THEN" X "ELSE" Y "END"; EM[3]:= COUNT "END" VALSYMTRI 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 10 ; "EOP" 0"CODE" 34152; "COMMENT" MCA 2312; "PROCEDURE" VECSYMTRI(D, B, N, N1, N2, VAL, VEC, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" D, B, VAL, VEC, EM; "BEGIN" "INTEGER" I, J, K, COUNT, MAXCOUNT, COUNTLIM, ORTH, IND; "REAL" BI, BI1, U, W, Y, MI1, LAMBDA, OLDLAMBDA, ORTHEPS, VALSPREAD, SPR, RES, MAXRES, OLDRES, NORM, NEWNORM, OLDNORM, MACHTOL, VECTOL; "ARRAY" M, P, Q, R, X[1:N]; "BOOLEAN" "ARRAY" INT[1:N]; NORM:= EM[1]; MACHTOL:= EM[0] * NORM; VALSPREAD:= EM[4] * NORM; VECTOL:= EM[6] * NORM; COUNTLIM:= EM[8]; ORTHEPS:= SQRT(EM[0]); MAXCOUNT:= IND:= 0; MAXRES:= 0; "IF" N1 > 1 "THEN" "BEGIN" ORTH:= EM[5]; OLDLAMBDA:= VAL[N1 - ORTH]; "FOR" K:= N1 - ORTH + 1 "STEP" 1 "UNTIL" N1 - 1 "DO" "BEGIN" LAMBDA:= VAL[K]; SPR:= OLDLAMBDA - LAMBDA; "IF" SPR < MACHTOL "THEN" LAMBDA:= OLDLAMBDA - MACHTOL; OLDLAMBDA:= LAMBDA "END" "END" "ELSE" ORTH:= 1; "FOR" K:= N1 "STEP" 1 "UNTIL" N2 "DO" "BEGIN" LAMBDA:= VAL[K]; "IF" K > 1 "THEN" "BEGIN" SPR:= OLDLAMBDA - LAMBDA; "IF" SPR < VALSPREAD "THEN" "BEGIN" "IF" SPR < MACHTOL "THEN" LAMBDA:= OLDLAMBDA - MACHTOL; ORTH:= ORTH +1 "END" "ELSE" ORTH:= 1 "END"; COUNT:= 0; U:= D[1] - LAMBDA; BI:= W:= B[1]; "IF" ABS(BI) < MACHTOL "THEN" BI:= MACHTOL; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" BI1:= B[I + 1]; "IF" ABS(BI1) < MACHTOL "THEN" BI1:= MACHTOL; "IF" ABS(BI) >= ABS(U) "THEN" "BEGIN" MI1:= M[I + 1]:= U / BI; P[I]:= BI; Y:= Q[I]:= D[I + 1] - LAMBDA; R[I]:= BI1; U:= W - MI1 * Y; W:= - MI1 * BI1; INT[I]:= "TRUE" "END" "ELSE" "BEGIN" MI1:= M[I + 1]:= BI / U; P[I]:= U; Q[I]:= W; R[I]:= 0; U:= D[I + 1] - LAMBDA - MI1 * W;W:= BI1; INT[I]:= "FALSE" "END"; X[I]:= 1; BI:= BI1 "END" TRANSFORM 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 11 ; P[N]:= "IF" ABS(U) < MACHTOL "THEN" MACHTOL "ELSE" U; Q[N]:= R[N]:= 0; X[N]:= 1; "GOTO" ENTRY; ITERATE: W:= X[1]; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" INT[I - 1] "THEN" "BEGIN" U:= W; W:= X[I - 1]:= X[I] "END" "ELSE" U:= X[I]; W:= X[I]:= U - M[I] * W "END" ALTERNATE; ENTRY: U:= W:= 0; "FOR" I:= N "STEP" -1 "UNTIL" 1 "DO" "BEGIN" Y:= U; U:= X[I]:= (X[I] - Q[I] * U - R[I] * W) / P[I]; W:= Y "END" NEXT ITERATION; NEWNORM:= SQRT(VECVEC(1, N, 0, X, X)); "IF" ORTH > 1"THEN" "BEGIN" OLDNORM:= NEWNORM; "FOR" J:= K - ORTH + 1 "STEP" 1 "UNTIL" K - 1 "DO" ELMVECCOL(1, N, J, X, VEC, -TAMVEC(1, N, J, VEC, X)); NEWNORM:= SQRT(VECVEC(1, N, 0, X, X)); "IF" NEWNORM < ORTHEPS * OLDNORM "THEN" "BEGIN" IND:= IND + 1; COUNT:= 1; "FOR" I:= 1 "STEP" 1 "UNTIL" IND - 1, IND + 1 "STEP" 1 "UNTIL" N "DO" X[I]:= 0; X[IND]:= 1; "IF" IND = N "THEN" IND:= 0; "GOTO" ITERATE "END" NEW START "END" ORTHOGONALISATION; RES:= 1 / NEWNORM; "IF" RES > VECTOL "OR" COUNT = 0 "THEN" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT <= COUNTLIM "THEN" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" X[I]:= X[I] * RES; "GOTO" ITERATE "END" "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" VEC[I,K]:= X[I] * RES; "IF" COUNT > MAXCOUNT "THEN" MAXCOUNT:= COUNT; "IF" RES > MAXRES "THEN" MAXRES:= RES; OLDLAMBDA:= LAMBDA "END"; EM[5]:= ORTH; EM[7]:= MAXRES; EM[9]:= MAXCOUNT "END" VECSYMTRI 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 12 ; "EOP" 0"CODE" 34160; "INTEGER" "PROCEDURE" QRIVALSYMTRI(D, BB, N, EM); "VALUE" N; "INTEGER" N; "ARRAY" D, BB, EM; "BEGIN" "INTEGER" I, I1, LOW, OLDLOW, N1, COUNT, MAX; "REAL" BBTOL, BBMAX, BBI, BBN1, MACHTOL, DN, DELTA, F, NUM, SHIFT, G, H, T, P, R, S, C, OLDG; BBTOL:= (EM[2] * EM[1]) ** 2; MACHTOL:= EM[0] * EM[1]; MAX:= EM[4]; BBMAX:= 0; COUNT:= 0; OLDLOW:= N; "FOR" N1:= N - 1 "WHILE" N > 0 "DO" "BEGIN" "FOR" I:= N, I - 1 "WHILE" ("IF" I >= 1 "THEN" BB[I] > BBTOL "ELSE" "FALSE") "DO" LOW:= I; "IF" LOW > 1 "THEN" "BEGIN" "IF" BB[LOW-1] > BBMAX "THEN" BBMAX:= BB[LOW-1] "END"; "IF" LOW = N "THEN" N:= N1 "ELSE" "BEGIN" DN:= D[N]; DELTA:= D[N1] - DN; BBN1:= BB[N1]; "IF" ABS(DELTA) < MACHTOL "THEN" R:= SQRT(BBN1) "ELSE" "BEGIN" F:= 2 / DELTA; NUM:= BBN1 * F; R:= -NUM / (SQRT(NUM * F + 1) + 1) "END"; "IF" LOW = N1 "THEN" "BEGIN" D[N]:= DN + R; D[N1]:= D[N1] - R; N:= N - 2 "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" END; "IF" LOW < OLDLOW "THEN" "BEGIN" SHIFT:= 0; OLDLOW:= LOW "END" "ELSE" SHIFT:= DN + R; H:= D[LOW] - SHIFT; "IF" ABS(H) < MACHTOL "THEN" H:= "IF" H <= 0 "THEN" -MACHTOL "ELSE" MACHTOL; G:= H; T:= G * H; BBI:= BB[LOW]; P:= T + BBI; I1:= LOW; "FOR" I:= LOW + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" S:= BBI / P; C:= T / P; H:= D[I] - SHIFT - BBI / H; "IF" ABS(H) < MACHTOL "THEN" H:= "IF" H <= 0 "THEN" -MACHTOL "ELSE" MACHTOL; OLDG:= G; G:= H * C; T:= G * H; D[I1]:= OLDG - G + D[I]; BBI:= "IF" I = N "THEN" 0 "ELSE" BB[I]; P:= T + BBI; BB[I1]:= S * P; I1:= I "END"; D[N]:= G + SHIFT "END" QRSTEP "END" "END"; END: EM[3]:= SQRT(BBMAX); EM[5]:= COUNT; QRIVALSYMTRI:= N "END" QRIVALSYMTRI 1SECTION 3.3.1.1.1 (JULY 1974) PAGE 13 ; "EOP" 0"CODE" 34161; "COMMENT" MCA 2321; "INTEGER" "PROCEDURE" QRISYMTRI(A, N, D, B, BB, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, D, B, BB, EM; "BEGIN" "INTEGER" I, J, J1, K, M, M1, COUNT, MAX; "REAL" BBMAX, R, S, SIN, T, C, COS, OLDCOS, G, P, W, TOL, TOL2, LAMBDA, DK1, A0, A1; TOL:= EM[2] * EM[1]; TOL2:= TOL * TOL; COUNT:= 0; BBMAX:= 0; MAX:= EM[4]; M:= N; IN: K:= M; M1:= M - 1; NEXT: K:= K - 1; "IF" K > 0 "THEN" "BEGIN" "IF" BB[K] >= TOL2 "THEN" "GOTO" NEXT; "IF" BB[K] > BBMAX "THEN" BBMAX:= BB[K] "END"; "IF" K = M1 "THEN" M:= M1 "ELSE" "BEGIN" T:= D[M] - D[M1]; R:= BB[M1]; "IF" ABS(T) < TOL "THEN" S:= SQRT(R) "ELSE" "BEGIN" W:= 2 / T; S:= W * R / (SQRT(W * W * R + 1) + 1) "END"; "IF" K = M - 2 "THEN" "BEGIN" D[M]:= D[M] + S; D[M1]:= D[M1] - S; T:= - S / B[M1]; R:= SQRT(T * T + 1); COS:= 1 / R; SIN:= T / R; ROTCOL(1,N,M1,M,A,COS,SIN); M:= M - 2 "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" END; LAMBDA:= D[M] + S; "IF" ABS(T) < TOL "THEN" "BEGIN" W:= D[M1] - S; "IF" ABS(W) < ABS(LAMBDA) "THEN" LAMBDA:= W "END"; K:= K + 1; T:= D[K] - LAMBDA; COS:= 1; W:= B[K]; P:= SQRT(T * T + W * W); J1:= K; "FOR" J:= K + 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" OLDCOS:= COS; COS:= T / P; SIN:= W / P; DK1:= D[J] - LAMBDA; T:= OLDCOS * T; D[J1]:= (T + DK1) * SIN * SIN + LAMBDA + T; T:= COS * DK1 - SIN * W * OLDCOS; W:= B[J]; P:= SQRT(T * T + W * W); G:= B[J1]:= SIN * P; BB[J1]:= G * G; ROTCOL(1, N, J1, J, A, COS, SIN); J1:= J "END"; D[M]:= COS * T + LAMBDA; "IF" T < 0 "THEN" B[M1]:= - G "END" QRSTEP "END"; "IF" M > 0 "THEN" "GOTO" IN; END: EM[3]:= SQRT(BBMAX); EM[5]:= COUNT; QRISYMTRI:= M "END" QRISYMTRI; "EOP" 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 1 AUTHORS: T.J.DEKKER AND W.HOFFMANN. CONTRIBUTORS: W.HOFFMANN, J.G.VERWER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730924. BRIEF DESCRIPTION: THIS SECTION CONTAINS SEVEN PROCEDURES. A) EIGVALSYM1 AND EIGVALSYM2 CALCULATE ALL EIGENVALUES, OR SOME CONSECUTIVE EIGENVALUES INCLUDING THE LARGEST, OF A SYMMETRIC MATRIX USING LINEAR INTERPOLATION ON A FUNCTION DERIVED FROM A STURM SEQUENCE, B) EIGSYM1 AND EIGSYM2 CALCULATE THE CORRESPONDING EIGENVECTORS AS WELL, BY MEANS OF INVERSE ITERATION, C) QRIVALSYM1 AND QRIVALSYM2 CALCULATE ALL EIGENVALUES OF A SYMMETRIC MATRIX BY MEANS OF QR ITERATION, D) QRISYM CALCULATES ALL EIGENVECTORS AS WELL IN THE SAME ITERATION PROCESS. EIGVALSYM1, EIGSYM1 AND QRIVALSYM1 USE IONAL ARRAY FOR THE GIVEN SYMMETRIC MATRIX; THE OTHER PROCEDURES EXPECT THE MATRIX TO BE STORED IN "ARRAY". QRISYM DELIVERS THE EIGENVECTORS IN THE ARRAY THAT WAS USED FOR THE ORIGINAL MATRIX IN CONTRAST WITH EIGSYM1 AND EIGSYM2 WHICH DELIVER THE EIGENVECTORS IN AN EXTRA ARRAY. WHEN ALL EIGENVALUES HAVE TO BE CALCULATED, THE PROCEDURES USING QR ITERATION ARE PREFERABLE WITH RESPECT TO THEIR RUNNING TIME. WHEN ALSO THE EIGENVECTORS HAVE TO BE CALCULATED THE PROCEDURES USING INVERSE ITERATION ARE FASTER; HOWEVER, THE ONE USING QR ITERATION USES LESS MEMORY SPACE. KEYWORDS: EIGENVALUES, EIGENVECTORS, SYMMETRIC MATRIX, STURM-SEQUENCE, INVERSE ITERATION, QR ITERATION. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 2 SUBSECTION: EIGVALSYM2. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" EIGVALSYM2(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "CODE" 34153; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; THE SERIAL NUMBER OF THE LAST EIGENVALUE TO BE CALCULATED; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<= J); EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION (WHICH ISN'T USED BY THIS PROCEDURE) IS DELIVERED IN THE UPPER TRIANGULAR PART OF A; THE ELEMENTS A[I,J] FOR I > J ARE NEITHER USED NOR CHANGED; VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: THE NUMVAL LARGEST EIGENVALUES IN MONOTONICALLY NON-INCREASING ORDER; EM: ; "ARRAY" EM[0:3]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES; EXIT: EM[1], THE INFINITY NORM OF THE ORIGINAL MATRIX, EM[3], THE NUMBER OF ITERATIONS USED FOR CALCULATING THE NUMVAL EIGENVALUES. PROCEDURES USED: TFMSYMTRI2 = CP34140, VALSYMTRI = CP34151. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: THREE ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE USED. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF EIGVALSYM2 CONSISTS OF TWO PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI2 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S TRANSFORMATION; THE SECOND IS A CALL OF VALSYMTRI TO CALCULATE THE DESIRED EIGENVALUES. OPERATION DETAILS OF BOTH PROCEDURES ARE GIVEN IN THEIR DESCRIPTION. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 3 SUBSECTION: EIGVALSYM1. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" EIGVALSYM1(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "CODE" 34155; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; THE SERIAL NUMBER OF THE LAST EIGENVALUE TO BE CALCULATED; A: ; "ARRAY" A[1:(N+1)*N//2]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN SUCH A WAY THAT THE (I,J)-TH ELEMENT OF THE MATRIX IS A[(J-1)*J//2+I], 1 <= I <= J <= N; EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION (WHICH ISN'T USED BY THIS PROCEDURE). VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: THE NUMVAL LARGEST EIGENVALUES IN MONOTONICALLY NON-INCREASING ORDER; EM: ; "ARRAY" EM[0:3]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES; EXIT: EM[1], THE INFINITY NORM OF THE ORIGINAL MATRIX, EM[3], THE NUMBER OF ITERATIONS USED FOR CALCULATING THE NUMVAL EIGENVALUES. PROCEDURES USED: TFMSYMTRI1 = CP34143, VALSYMTRI = CP34151. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: THREE ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE USED. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF EIGVALSYM1 CONSISTS OF TWO PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI1 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S TRANSFORMATION; THE SECOND IS A CALL OF VALSYMTRI TO CALCULATE THE DESIRED EIGENVALUES. OPERATION DETAILS OF BOTH PROCEDURES ARE GIVEN IN THEIR DESCRIPTION. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 4 SUBSECTION: EIGSYM2. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" EIGSYM2(A, N, NUMVAL, VAL, VEC, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VEC, EM; "CODE" 34154; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; THE SERIAL NUMBER OF THE LAST EIGENVALUE TO BE CALCULATED; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<= J); EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION IS DELIVERED IN THE UPPER TRIANGULAR PART OF A; THE ELEMENTS A[I,J] FOR I > J ARE NEITHER USED NOR CHANGED; VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: THE NUMVAL LARGEST EIGENVALUES IN MONOTONICALLY NON-INCREASING ORDER; VEC: ; "ARRAY" VEC[1:N,1:NUMVAL]; EXIT: THE NUMVAL CALCULATED EIGENVECTORS, STORED COLUMN- WISE, CORRESPONDING TO THE CALCULATED EIGENVALUES; EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES, EM[4], THE ORTHOGONALISATION PARAMETER (SEE METHOD AND PERFORMANCE), EM[6], THE TOLERANCE FOR THE EIGENVECTORS, EM[8], THE MAXIMUM NUMBER OF INVERSE ITERATIONS ALLOWED FOR THE CALCULATION OF EACH EIGEN- VECTOR; EXIT: EM[1], THE INFINITY NORM OF THE MATRIX, EM[3], THE NUMBER OF ITERATIONS USED FOR CALCULATING THE NUMVAL EIGENVALUES, EM[5], THE NUMBER OF EIGENVECTORS INVOLVED IN THE LAST GRAM-SCHMIDT ORTHOGONALISATION, EM[7], THE MAXIMUM EUCLIDEAN NORM OF THE RESIDUES OF THE CALCULATED EIGENVECTORS, EM[9], THE LARGEST NUMBER OF INVERSE ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGEN- VECTOR. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 5 PROCEDURES USED: TFMSYMTRI2 = CP34140, VALSYMTRI = CP34151, VECSYMTRI = CP34152, BAKSYMTRI2 = CP34141. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: THREE ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE DECLARED; MOREOVER, VECSYMTRI USES FIVE ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N AND ONE BOOLEAN ARRAY OF LENGTH N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF EIGSYM2 CONSISTS OF FOUR PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI2 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDERS TRANSFORMATION, THE SECOND IS A CALL OF VALSYMTRI TO CALCULATE THE DESIRED EIGENVALUES, THE THIRD IS A CALL OF VECSYMTRI TO CALCULATE THE CORRESPONDING EIGENVECTORS AND THE FOURTH IS A CALL OF BAKSYMTRI2 TO PERFORM THE BACK TRANSFORMATION. THE PARAMETERS EM[5], EM[7] AND EM[9] ARE GIVEN ITS VALUE IN THE PROCEDURE VECSYMTRI. FOR A POSSIBLY SUBSEQUENT CALL OF VECSYMTRI THE VALUE OF EM[5] IS NEEDED. WHEN CONSECUTIVE EIGENVALUES ARE TOO CLOSE TOGETHER, THE CORRESPONDING EIGENVECTORS ARE NOT NECESSARILY DELIVERED ORTHOGONAL BY INVERSE ITERATION (THE METHOD WHICH IS USED IN VECSYMTRI). THEREFORE GRAM-SCHMIDT ORTHOGONALISATION IS APPLIED ON THE EIGENVECTORS WHEN THE DISTANCE BETWEEN TWO CONSECUTIVE EIGENVALUES IS SMALLER THAN EM[4]. FOR FURTHER DETAILS ONE IS REFERRED TO THE SPECIFIC PROCEDURE DESCRIPTIONS. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 6 SUBSECTION: EIGSYM1. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" EIGSYM1(A, N, NUMVAL, VAL, VEC, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VEC, EM; "CODE" 34156; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; THE SERIAL NUMBER OF THE LAST EIGENVALUE TO BE CALCULATED; A: ; "ARRAY" A[1:(N+1)*N//2]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN SUCH A WAY THAT THE (I,J)-TH ELEMENT OF THE MATRIX IS A[(J-1)*J//2+I], 1 <= I <= J <= N; EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION; VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: THE NUMVAL LARGEST EIGENVALUES IN MONOTONICALLY NON-INCREASING ORDER; VEC: ; "ARRAY" VEC[1:N,1:NUMVAL]; EXIT: THE NUMVAL CALCULATED EIGENVECTORS, STORED COLUMN- WISE, CORRESPONDING TO THE CALCULATED EIGENVALUES; EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES, EM[4], THE ORTHOGONALISATION PARAMETER (SEE METHOD AND PERFORMANCE), EM[6], THE TOLERANCE FOR THE EIGENVECTORS, EM[8], THE MAXIMUM NUMBER OF INVERSE ITERATIONS ALLOWED FOR THE CALCULATION OF EACH EIGEN- VECTOR; EXIT: EM[1], THE INFINITY NORM OF THE MATRIX, EM[3], THE NUMBER OF ITERATIONS USED FOR CALCULATING THE NUMVAL EIGENVALUES, EM[5], THE NUMBER OF EIGENVECTORS INVOLVED IN THE LAST GRAM-SCHMIDT ORTHOGONALISATION, EM[7], THE MAXIMUM EUCLIDEAN NORM OF THE RESIDUES OF THE CALCULATED EIGENVECTORS, EM[9], THE LARGEST NUMBER OF INVERSE ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGEN- VECTOR. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 7 PROCEDURES USED: TFMSYMTRI1 = CP34143, VALSYMTRI = CP34151, VECSYMTRI = CP34152, BAKSYMTRI1 = CP34144. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: THREE ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE DECLARED; MOREOVER, VECSYMTRI AND BAKSYMTRI1 USE A TOTAL AMOUNT OF SIX ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N AND ONE BOOLEAN ARRAY OF LENGTH N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF EIGSYM1 CONSISTS OF FOUR PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI1 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDERS TRANSFORMATION, THE SECOND IS A CALL OF VALSYMTRI TO CALCULATE THE DESIRED EIGENVALUES, THE THIRD IS A CALL OF VECSYMTRI TO CALCULATE THE CORRESPONDING EIGENVECTORS AND THE FOURTH IS A CALL OF BAKSYMTRI1 TO PERFORM THE BACK TRANSFORMATION. FOR DETAILS ONE IS REFERRED TO EIGSYM2 OR TO THE DESCRIPTIONS OF THE FOUR PROCEDURES USED. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 8 SUBSECTION: QRIVALSYM2. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" QRIVALSYM2(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "CODE" 34162; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<= J); EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION (WHICH ISN'T USED BY THIS PROCEDURE) IS DELIVERED IN THE UPPER TRIANGULAR PART OF A; THE ELEMENTS A[I,J] FOR I > J ARE NEITHER USED NOR CHANGED; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE MATRIX IN SOME ARBITRARY ORDER; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES, EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[1], THE INFINITY NORM OF THE MATRIX, EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED, EM[5], THE NUMBER OF ITERATIONS PERFORMED; MOREOVER: QRIVALSYM2:= THE NUMBER OF EIGENVALUES NOT CALCULATED. PROCEDURES USED: TFMSYMTRI2 = CP34140, QRIVALSYMTRI = CP34160. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: IONAL REAL ARRAYS OF LENGTH N ARE USED. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 9 METHOD AND PERFORMANCE: THE BODY OF QRIVALSYM2 CONSISTS OF TWO PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI2 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S TRANSFORMATION; THE SECOND IS A CALL OF QRIVALSYMTRI TO CALCULATE THE EIGENVALUES. WHEN THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS THEN QRIVALSYM2:= 0; OTHERWISE QRIVALSYM2:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED, EM[5]:= EM[4] + 1 AND ONLY THE LAST N - K ELEMENTS OF VAL ARE APPROXIMATE EIGENVALUES OF THE GIVEN MATRIX. OPERATION DETAILS OF BOTH PROCEDURES USED ARE GIVEN IN THEIR DESCRIPTION. SUBSECTION: QRIVALSYM1. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" QRIVALSYM1(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "CODE" 34164; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; A: ; "ARRAY" A[1:(N+1)*N//2]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN SUCH A WAY THAT THE (I,J)-TH ELEMENT OF THE MATRIX IS A[(J-1)*J//2+I], 1 <= I <= J <= N; EXIT: THE DATA FOR HOUSEHOLDER'S BACK TRANSFORMATION (WHICH ISN'T USED BY THIS PROCEDURE). VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE MATRIX IN SOME ARBITRARY ORDER; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE EIGENVALUES, EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[1], THE INFINITY NORM OF THE MATRIX, EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED, EM[5], THE NUMBER OF ITERATIONS PERFORMED; MOREOVER: QRIVALSYM1:= THE NUMBER OF EIGENVALUES NOT CALCULATED. PROCEDURES USED: TFMSYMTRI1 = CP34143, QRIVALSYMTRI = CP34160. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: TWO ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE USED. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 10 RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF QRIVALSYM1 CONSISTS OF TWO PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI1 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S TRANSFORMATION; THE SECOND IS A CALL OF QRIVALSYMTRI TO CALCULATE THE EIGENVALUES. WHEN THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS THEN QRIVALSYM1:= 0; OTHERWISE QRIVALSYM1:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED, EM[5]:= EM[4] + 1 AND ONLY THE LAST N - K ELEMENTS OF VAL ARE APPROXIMATE EIGENVALUES OF THE GIVEN MATRIX. OPERATION DETAILS OF BOTH PROCEDURES USED ARE GIVEN IN THEIR DESCRIPTION. SUBSECTION: QRISYM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" QRISYM(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "CODE" 34163; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRIX; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE UPPER TRIANGLE OF THE SYMMETRIC MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<= J); EXIT: THE EIGENVECTORS OF THE SYMMETRIC MATRIX, STORED COLUMNWISE; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE MATRIX CORRESPONDING TO THE CALCULATED EIGENVECTORS; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION, EM[2], THE RELATIVE TOLERANCE FOR THE QR ITERATION, EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[1], THE INFINITY NORM OF THE MATRIX, EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED, EM[5], THE NUMBER OF ITERATIONS PERFORMED; MOREOVER: QRISYM := THE NUMBER OF EIGENVALUES AND -VECTORS NOT CALCULATED. 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 11 PROCEDURES USED: TFMSYMTRI2 = CP34140, TFMPREVEC = CP34142, QRISYMTRI = CP34161. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: TWO ONE-DIMENSIONAL REAL ARRAYS OF LENGTH N ARE USED. RUNNING TIME: PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE BODY OF QRISYM CONSISTS OF THREE PROCEDURE STATEMENTS; THE FIRST IS A CALL OF TFMSYMTRI2 TO TRANSFORM THE SYMMETRIC MATRIX INTO A SIMILAR TRIDIAGONAL MATRIX BY MEANS OF HOUSEHOLDER'S TRANSFORMATION, THE SECOND IS A CALL OF TFMPREVEC TO PERFORM THE DESIRED BACK TRANSFORMATION ON THE EIGENVECTORS IN ADVANCE AND THE THIRD IS A CALL OF QRISYMTRI TO CALCULATE THE EIGENVALUES AND THE EIGENVECTORS. WHEN THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS THEN QRISYM:= 0; OTHERWISE QRISYM:= THE NUMBER, K, OF EIGENVALUES AND -VECTORS NOT CALCULATED, EM[5]:= EM[4] + 1 AND ONLY THE LAST N - K ELEMENTS OF VAL AND THE LAST N - K COLUMNS OF A ARE APPROXIMATE EIGENVALUES AND EIGENVECTORS RESPECTIVELY OF THE GIVEN MATRIX. OPERATION DETAILS OF THE PROCEDURES USED ARE GIVEN IN THEIR DESCRIPTION. EXAMPLES OF USE: THE TWO LARGEST EIGENVALUES IN MONOTONICALLY NON INCREASING ORDER AND THE CORRESPONDING EIGENVECTORS OF M, WITH N = 4 AND M[I,J] = 1 / (I + J - 1), MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I, J; "ARRAY" A[1:10], VAL[1:2], EM[0:9], VEC[1:4,1:2]; EM[0]:= "-14; EM[2]:= "-12; EM[4]:= "-3; EM[6]:= 10"-10; EM[8]:= 5; "FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" "FOR" J:= I "STEP" 1 "UNTIL" 4 "DO" A[(J * J - J) / 2 + I]:= 1 / (I + J - 1); EIGSYM1(A, 4, 2, VAL, VEC, EM); OUTPUT(61, "("2(+.13D"+2D, 2B), 2/")", VAL[1], VAL[2]); "FOR" I:= 1, 2, 3, 4 "DO" OUTPUT(61, "("2(+.13D"+2D, 2B), /")", VEC[I,1], VEC[I,2]); OUTPUT(61, "("2(.2D"+2D, /), 3(2ZD, /)")", EM[1], EM[7], EM[3], EM[5], EM[9]) "END" 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 12 THE PROGRAM DELIVERS(THE RESULTS ARE CORRECT UP TO TWELVE DIGITS): THE EIGENVALUES: +.1500214280059"+01 +.1691412202214"+00 THE EIGENVECTORS: -.7926082911638"+00 +.5820756994972"+00 -.4519231209016"+00 -.3705021850671"+00 -.3224163985818"+00 -.5095786345018"+00 -.2521611696882"+00 -.5140482722222"+00 EM[1] = .21"+01 EM[7] = .92"-14 EM[3] = 32 EM[5] = 1 EM[9] = 1 . THE TWO LARGEST EIGENVALUES OF M, WITH N = 4 AND M[I,J] = 1 / (I + J - 1), MAY BE OBTAINED IN MONOTONICALLY NON INCREASING ORDER BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I, J; "ARRAY" A[1:4,1:4], VAL[1:2], EM[0:3]; EM[0]:= "-14; EM[2]:= "-12; "FOR" I:= 1 "STEP" 1 "UNTIL" 4 "DO" "FOR" J:= I "STEP" 1 "UNTIL" 4 "DO" A[I,J]:= 1 / (I + J -1); EIGVALSYM2(A, 4, 2, VAL, EM); OUTPUT(61, "("2(+.13D"+2D, 2B)")", VAL[1], VAL[2]); OUTPUT(61, "("2/, .2D"+2D, /, 2ZD")", EM[1], EM[3]) "END" THE PROGRAM DELIVERS(THE RESULTS ARE CORRECT UP TO TWELVE DIGITS): THE EIGENVALUES: +.1500214280059"+01 +.1691412202214"+00 EM[3] = .21"+01 EM[1] = 32 . 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 13 SOURCE TEXT(S) : 0"CODE" 34153; "COMMENT" MCA 2313; "PROCEDURE" EIGVALSYM2(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B, BB, D[1:N]; TFMSYMTRI2(A, N, D, B, BB, EM); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM) "END" EIGVALSYM2; "EOP" 0"CODE" 34154; "COMMENT" MCA 2314; "PROCEDURE" EIGSYM2(A, N, NUMVAL, VAL, VEC, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VEC, EM; "BEGIN" "ARRAY" B, BB, D[1:N]; TFMSYMTRI2(A, N, D, B, BB, EM); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM); VECSYMTRI(D, B, N, 1, NUMVAL, VAL, VEC, EM); BAKSYMTRI2(A, N, 1, NUMVAL, VEC) "END" EIGSYM2; "EOP" 0"CODE" 34155; "COMMENT" MCA 2318; "PROCEDURE" EIGVALSYM1(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B, BB, D[1:N]; TFMSYMTRI1(A, N, D, B, BB, EM); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM) "END" EIGVALSYM1 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 14 ; "EOP" 0"CODE" 34156; "COMMENT" MCA 2319; "PROCEDURE" EIGSYM1(A, N, NUMVAL, VAL, VEC, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VEC, EM; "BEGIN" "ARRAY" B, BB, D[1:N]; TFMSYMTRI1(A, N, D, B, BB, EM); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM); VECSYMTRI(D, B, N, 1, NUMVAL, VAL, VEC, EM); BAKSYMTRI1(A, N, 1, NUMVAL, VEC) "END" EIGSYM1; "EOP" 0"CODE" 34162; "COMMENT" MCA 2322; "INTEGER" "PROCEDURE" QRIVALSYM2(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B, BB[1:N]; TFMSYMTRI2(A, N, VAL, B, BB, EM); QRIVALSYM2:= QRIVALSYMTRI(VAL, BB, N, EM) "END" QRIVALSYM2 1SECTION 3.3.1.1.2 (JULY 1974) PAGE 15 ; "EOP" 0"CODE" 34163; "COMMENT" MCA 2323; "INTEGER" "PROCEDURE" QRISYM(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B, BB[1:N]; TFMSYMTRI2(A, N, VAL, B, BB, EM); TFMPREVEC(A, N); QRISYM:= QRISYMTRI(A, N, VAL, B, BB, EM) "END" QRISYM; "EOP" 0"CODE" 34164; "COMMENT" MCA 2327; "INTEGER" "PROCEDURE" QRIVALSYM1(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B, BB[1 : N]; TFMSYMTRI1(A, N, VAL, B, BB, EM); QRIVALSYM1:= QRIVALSYMTRI(VAL, BB, N, EM) "END" QRIVALSYM1; "EOP" 1SECTION : 3.3.1.1.3.1 (NOVEMBER 1976) PAGE 1 AUTHORS: J.J.G. ADMIRAAL, A.C. YSSELSTEIN. CONTRIBUTOR: J.J.G. ADMIRAAL. INSTITUTE: UNIVERSITY OF AMSTERDAM. RECEIVED: 761101. BRIEF DESCRIPTION: THIS SECTION CONTAINS THREE PROCEDURES FOR SORTING THE ELEMENTS OF A VECTOR AND CORRESPONDINGLY PERMUTING THE ELEMENTS OF A VECTOR OR A MATRIX ROW. A) MERGESORT DELIVERS A PERMUTATION OF INDICES CORRESPONDING TO SORTING THE ELEMENTS OF A GIVEN VECTOR INTO NON-DECREASING ORDER. B) VECPERM PERMUTES THE ELEMENTS OF A GIVEN VECTOR CORRESPONDING TO A GIVEN PERMUTATION OF INDICES. C) ROWPERM PERMUTES THE ELEMENTS OF A GIVEN ROW OF A MATRIX CORRESPONDING TO A GIVEN PERMUTATION OF INDICES. KEYWORDS: SORTING, PERMUTING. SUBSECTION: MERGESORT. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" MERGESORT(VEC1,VEC2,LOW,UPP); "VALUE" LOW,UPP;"INTEGER" LOW,UPP;"ARRAY" VEC1; "INTEGER""ARRAY" VEC2; "CODE" 36405; THE MEANING OF THE FORMAL PARAMETERS IS: VEC1 :; "ARRAY" VEC1[LOW:UPP]; ENTRY: THE VECTOR TO BE SORTED INTO NONDECREASING ORDER; EXIT: THE CONTENTS OF VEC1 ARE LEFT INVARIANT; VEC2 :; "INTEGER""ARRAY" VEC2[LOW:UPP]; EXIT: THE PERMUTATION OF INDICES CORRESPONDING TO SORTING THE ELEMENTS OF VEC1 INTO NON-DECREASING ORDER; LOW : ; THE LOWER INDEX OF THE ARRAYS VEC1 AND VEC2; UPP : ; THE UPPER INDEX OF THE ARRAYS VEC1 AND VEC2; 1SECTION : 3.3.1.1.3.1 (NOVEMBER 1976) PAGE 2 PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: ONE LOCAL INTEGER ARRAY OF LENGTH N,WHERE N = UPP - LOW + 1. RUNNING TIME: AVERAGE PROPORTIONAL TO N * LN(N). LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SORTING BY MERGING. ([1],[2]) EXAMPLE OF USE: THE PROCEDURE MERGESORT IS USED IN SYMEIGIMP (SECTION 3.3.1.1.3.3). SUBSECTION: VECPERM. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" VECPERM(PERM,LOW,UPP,VECTOR); "VALUE" LOW,UPP; "INTEGER" LOW,UPP; "INTEGER" "ARRAY" PERM;"ARRAY" VECTOR; "CODE" 36404; THE MEANING OF THE FORMAL PARAMETERS IS: PERM :; "INTEGER" "ARRAY" PERM[LOW:UPP]; ENTRY: A GIVEN PERMUTATION (E.G.AS PRODUCED BY MERGESORT) OF THE NUMBERS IN THE ARRAY VECTOR; LOW :; THE LOWER INDEX OF THE ARRAYS PERM AND VECTOR; UPP :; THE UPPER INDEX OF THE ARRAYS PERM AND VECTOR; VECTOR :; "ARRAY" VECTOR[LOW:UPP]; ENTRY: THE REAL VECTOR TO BE PERMUTED; EXIT: THE PERMUTED VECTOR ELEMENTS. PROCEDURE USED: NONE. REQUIRED CENTRAL MEMORY : ONE LOCAL BOOLEAN ARRAY OF LENGTH N IS DECLARED. RUNNING TIME: PROPORTIONAL TO N. LANGUAGE: ALGOL 60. EXAMPLE OF USE: THE PROCEDURE VECPERM IS USED IN SYMEIGIMP; (SECTION 3.3.1.1.3.3). 1SECTION : 3.3.1.1.3.1 (NOVEMBER 1976) PAGE 3 SUBSECTION: ROWPERM. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" ROWPERM(PERM,LOW,UPP,I,MATRIX); "VALUE" LOW,UPP,I;"INTEGER" LOW,UPP,I; "INTEGER" "ARRAY" PERM;"ARRAY" MATRIX; "CODE" 36403; THE MEANING OF THE FORMAL PARAMETERS IS: PERM :; "INTEGER" "ARRAY" PERM[LOW:UPP]; ENTRY: A GIVEN PERMUTATION (E.G. AS PRODUCED BY MERGESORT) OF THE NUMBERS IN THE ARRAY VECTOR; LOW :; THE LOWER INDEX OF THE ARRAY PERM; UPP :; THE UPPER INDEX OF THE ARRAY PERM; I :; THE ROW INDEX OF THE MATRIX ELEMENTS; MATRIX :: "ARRAY" MATRIX [I :I, LOW : UP]; ENTRY:MATRIX [I, LOW : UP] SHOULD CONTAIN THE ELEMENTS TO BE PERMUTED; EXIT: MATRIX[I, LOW : UP] CONTAINS THE ROW OF PERMUTED ELEMENTS; PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: ONE LOCAL BOOLEAN ARRAY OF LENGTH N IS DECLARED. RUNNING TIME: PROPORTIONAL TO N, WHERE N = UPP - LOW + 1. LANGUAGE: ALGOL 60. EXAMPLE OF USE: THE PROCEDURE ROWPERM IS USED IN SYMEIGIMP. (SECTION 3.3.1.1.3.3). REFERENCES: [1] D.E. KNUTH , THE ART OF COMPUTER PROGRAMMING, VOL. 3/ SORTING AND SEARCHING,ADDISON-WESLEY 1973. (SECTION 5.2.4 P.159-173). [2] A.V. AHO, J.E. HOPCROFT & J.D. ULLMAN, THE DESIGN AND ANALYSIS OF COMPUTER ALGORITHMS, ADDISON-WESLEY 1974. (SECTION I.M, P65-67). 1SECTION : 3.3.1.1.3.1 (NOVEMBER 1976) PAGE 4 SOURCE TEXTS: 0"CODE" 36405; "PROCEDURE" MERGESORT(A,P,LOW,UP);"VALUE" LOW,UP; "INTEGER" LOW,UP;"ARRAY" A;"INTEGER" "ARRAY" P; "BEGIN" "INTEGER" I,L,R,PL,PR,LO,STEP,STAP,UMLP1,UMSP1,REST,RESTV; "BOOLEAN" ROUT,LOUT; "INTEGER" "ARRAY" HP[LOW:UP]; "PROCEDURE" MERGE(LO,LS,RS);"VALUE" LO,LS,RS;"INTEGER" LO,LS,RS; "BEGIN" L:=LO; R:=LO+LS; LOUT:=ROUT:="FALSE"; "FOR" I:=LO,I+1 "WHILE" ^(LOUT "OR" ROUT) "DO" "BEGIN" PL:=P[L];PR:=P[R];"IF" A[PL]>A[PR] "THEN" "BEGIN" HP[I]:=PR;R:=R+1;ROUT:=R=LO+LS+RS "END" "ELSE" "BEGIN" HP[I]:=PL;L:=L+1;LOUT:=L=LO+LS "END" "END" FOR I; "IF" ROUT "THEN" "BEGIN" "FOR" I:=LO+LS-1 "STEP" -1 "UNTIL" L "DO" P[I+RS]:=P[I];R:=L+RS "END"; "FOR" I:=R-1 "STEP" -1 "UNTIL" LO "DO" P[I]:=HP[I]; "END" MERGE; "FOR" I:=LOW "STEP" 1 "UNTIL" UP "DO" P[I]:=I;RESTV:=0; UMLP1:=UP-LOW+1; "FOR" STEP:=1, STEP*2 "WHILE" STEP < UMLP1 "DO" "BEGIN" STAP:=2*STEP;UMSP1:=UP-STAP+1; "FOR" LO:=LOW "STEP" STAP "UNTIL" UMSP1 "DO" MERGE(LO,STEP,STEP); REST:=UP-LO+1; "IF" REST>RESTV & RESTV>0 "THEN" MERGE(LO,REST-RESTV,RESTV); RESTV:=REST "END" FOR STEP "END" MERGESORT 1SECTION : 3.3.1.1.3.1 (NOVEMBER 1976) PAGE 5 ; "EOP" "CODE" 36404; "PROCEDURE" VECPERM(PERM,LOW,UPP,VECTOR);"VALUE" LOW,UPP; "INTEGER" LOW,UPP;"INTEGER" "ARRAY" PERM;"REAL" "ARRAY" VECTOR; "BEGIN" "INTEGER" T,J,K;"REAL" A;"BOOLEAN" "ARRAY" TODO[LOW:UPP]; "FOR" T:=LOW "STEP" 1 "UNTIL" UPP "DO" TODO[T]:="TRUE"; "FOR" T:=LOW "STEP" 1 "UNTIL" UPP "DO" "BEGIN" "IF" TODO[T] "THEN" "BEGIN" K:=T;A:=VECTOR[K]; "FOR" J:=PERM[K] "WHILE" J^=T "DO" "BEGIN" VECTOR[K]:=VECTOR[J];TODO[K]:="FALSE";K:=J "END";VECTOR[K]:=A;TODO[K]:="FALSE" "END" CYCLE; "END" FOR T; "END" VECPERM; "EOP" "CODE" 36403; "PROCEDURE" ROWPERM(PERM,LOW,UPP,I,MAT);"VALUE" LOW,UPP,I; "INTEGER" LOW,UPP,I;"INTEGER" "ARRAY" PERM;"REAL" "ARRAY" MAT; "BEGIN" "INTEGER" T,J,K;"REAL" A;"BOOLEAN" "ARRAY" TODO[LOW:UPP]; "FOR" T:=LOW "STEP" 1 "UNTIL" UPP "DO" TODO[T]:="TRUE"; "FOR" T:=LOW "STEP" 1 "UNTIL" UPP "DO" "BEGIN" "IF" TODO[T] "THEN" "BEGIN" K:=T;A:=MAT[I,K]; "FOR" J:=PERM[K] "WHILE" J^=T "DO" "BEGIN" MAT[I,K]:=MAT[I,J];TODO[K]:="FALSE";K:=J "END";MAT[I,K]:=A;TODO[K]:="FALSE" "END" CYCLE; "END" FOR T; "END" ROWPERM; "EOP" 1SECTION : 3.3.1.1.3.2 (NOVEMBER 1976) PAGE 1 AUTHOR/CONTRIBUTOR: J.J.G. ADMIRAAL. INSTITUTE: UNIVERSITY OF AMSTERDAM. RECEIVED: 761101. BRIEF DESCRIPTION: THE PROCEDURE ORTHOG ORTHOGONALIZES SOME ADJACENT MATRIX COLUMNS ACCORDING TO THE MODIFIED GRAM SCHMIDT METHOD (SEE [1]). KEYWORDS: MATRIX COLUMNS, MODIFIED GRAM SCHMIDT ORTHOGONALIZATION. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" ORTHOG(N,LC,UC,X); "VALUE" N,LC,UC; "INTEGER" N,LC,UC;"ARRAY" X; "CODE" 36402; THE MEANING OF THE FORMAL PARAMETERS IS: N :; THE ORDER OF THE MATRIX X; LC :; THE LOWER COLUMN INDEX OF THE MATRIX COLUMNS; UC :; THE UPPER COLUMN INDEX OF THE MATRIX COLUMNS; X :; "ARRAY" X[1:N,LC:UC]; ENTRY: THE MATRIX COLUMNS,TO BE ORTHOGONALIZED; EXIT: THE ORTHOGONALIZED MATRIX COLUMNS. PROCEDURES USED: TAMMAT = CP34014, ELMCOL = CP34023. REQUIRED CENTRAL MEMORY: NO LOCAL ARRAYS ARE DECLARED. RUNNING TIME: PROPORTIONAL TO N**3. 1SECTION : 3.3.1.1.3.2 (NOVEMBER 1976) PAGE 2 LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE MODIFIED GRAM SCHMIDT METHOD (SEE [1],CHAPTER 4.54). EXAMPLE OF USE: THE PROCEDURE ORTHOG IS USED IN SYMEIGIMP. (SECTION 3.3.1.1.3.3). REFERENCES: [1] J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS,OXFORD,1965. SOURCE TEXT: 0"CODE" 36402; "PROCEDURE" ORTHOG(N,LC,UC,X);"VALUE" N,LC,UC; "INTEGER" N,LC,UC;"ARRAY" X; "BEGIN" "INTEGER" I,J,K; "REAL" NORMX; "FOR" J:=LC "STEP" 1 "UNTIL" UC "DO" "BEGIN" NORMX:=SQRT(TAMMAT(1,N,J,J,X,X)); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" X[I,J]:=X[I,J]/NORMX; "FOR" K:=J+1 "STEP" 1 "UNTIL" UC "DO" ELMCOL(1,N,K,J,X,X,-TAMMAT(1,N,K,J,X,X)) "END" "END" ORTHOG; "EOP" 1SECTION : 3.3.1.1.3.3 (DECEMBER 1979) PAGE 1 AUTHOR/CONTRIBUTOR: J.J.G. ADMIRAAL. INSTITUTE: UNIVERSITY OF AMSTERDAM. RECEIVED: 761101. BRIEF DESCRIPTION: THE PROCEDURE SYMEIGIMP IMPROVES A GIVEN APPROXIMATION OF A REAL SYMMETRIC EIGENSYSTEM AND CALCULATES ERROR BOUNDS FOR THE EIGENVALUES. KEYWORDS: EIGENVALUES. EIGENVECTORS. SYMMETRIC MATRIX. RAYLEIGH QUOTIENTS. ERROR BOUNDS. IMPROVED EIGENSYSTEM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS : "PROCEDURE" SYMEIGIMP(N,A,VEC,VAL1,VAL2,LBOUND,UBOUND,AUX); "VALUE" N;"INTEGER" N; "ARRAY" A,VEC,VAL1,VAL2,LBOUND,UBOUND,AUX; "CODE" 36401; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF MATRIX A; A: ; "ARRAY" A[1:N,1:N] CONTAINS A REAL SYMMETRIC MATRIX WHOSE EIGENSYSTEM HAS TO BE IMPROVED; VEC: ; "ARRAY" VEC[1:N,1:N] CONTAINS A MATRIX WHOSE COLUMNS ARE A SYSTEM OF APPROXIMATE EIGENVECTORS OF MATRIX A; ENTRY: INITIAL APPROXIMATIONS; EXIT: IMPROVED APPROXIMATIONS; VAL1: ; "ARRAY" VAL1[1:N]; ENTRY: INITIAL APPROXIMATIONS OF THE EIGENVALUES OF A; EXIT: THE HEAD PARTS OF THE DOUBLE PRECISION IMPROVED APPROXIMATIONS OF THE EIGENVALUES OF A; VAL2: ; "ARRAY" VAL2[1:N]; EXIT: THE TAIL PARTS OF THE DOUBLE PRECISION IMPROVED EIGENVALUES OF A; 1SECTION : 3.3.1.1.3.3 (DECEMBER 1979) PAGE 2 LBOUND, UBOUND: ; EXIT: "ARRAY" LBOUND, UBOUND [1:N] CONTAIN THE LOWER AND UPPER ERRORBOUNDS RESPECTIVELY FOR THE EIGENVALUE APPROXIMATIONS IN VAL1,VAL2[1:N] SUCH THAT THE I-TH EXACT EIGENVALUE LIES BETWEEN VAL1[I]+VAL2[I] -LBOUND[I] AND VAL1[I]+VAL2[I]+UBOUND[I]; AUX: ; "ARRAY" AUX[0:5]; ENTRY: AUX[0]= THE RELATIVE PRECISION OF THE ELEMENTS OF A; AUX[2]= THE RELATIVE TOLERANCE FOR THE RESIDUAL MATRIX; THE ITERATION ENDS WHEN THE MAXIMUM ABSOLUTE VALUE OF THE RESIDUAL ELEMENTS IS SMALLER THAN AUX[2]*AUX[1]. AUX[4]= THE MAXIMUM NUMBER OF ITERATIONS ALLOWED; EXIT: AUX[1]= INFINITY NORM OF THE MATRIX A; AUX[3]= MAXIMUM ABSOLUTE ELT. OF THE RESIDUAL MATRIX AUX[5]= NUMBER OF ITERATIONS; PROCEDURES USED: LNGMATVEC = CP34411, LNGMATMAT = CP34413, LNGTAMMAT = CP34414, VECVEC = CP34010, MATMAT = CP34013, TAMMAT = CP34014. MERGESORT = CP36405, VECPERM = CP36404, ROWPERM = CP36403, ORTHOG = CP36402, QRISYM = CP34163, INFNRMMAT = CP31064. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. REQUIRED CENTRAL MEMORY: AUXILIARY ARRAYS ARE DECLARED TO A TOTAL OF 3*N*N + 6*N REALS AND N INTEGERS; MOREOVER, N INTEGERS OR N BOOLEANS ARE USED BY MERGESORT, VECPERM AND ROWPERM. METHOD AND PERFORMANCE: SEE[1]. 1SECTION : 3.3.1.1.3.3 (DECEMBER 1979) PAGE 3 REFERENCES: [1]. J.J.G. ADMIRAAL. ITERATIEF VERBETEREN VAN REEEL SYMMETRISCH EIGENSYSTEEM EN BEREKENEN VAN FOUTGRENZEN VOOR DE VERKREGEN EIGENWAARDEN. DOCTORAL SCRIPTION,MARCH 1976, UNIVERSITEIT VAN AMSTERDAM. [2]. R.T. GREGORY AND D.L. KARNEY. A COLLECTION OF MATRICES FOR TESTING COMPUTATIONAL ALGORITHMS, WILEY-INTERSCIENCE, 1969. EXAMPLE OF USE. "BEGIN" "INTEGER" I,J;"REAL" S; "ARRAY" A,X[1:4,1:4],VAL1,VAL2,LBOUND,UBOUND[1:4],EM,AUX[0:5]; A[1,1]:=A[2,2]:=A[3,3]:=A[4,4]:=6; A[1,2]:=A[2,1]:=A[3,1]:=A[1,3]:=4; A[4,2]:=A[2,4]:=A[3,4]:=A[4,3]:=4; A[1,4]:=A[4,1]:=A[3,2]:=A[2,3]:=1; "FOR" I:=1 "STEP" 1 "UNTIL" 4 "DO" "FOR" J:=I "STEP" 1 "UNTIL" 4 "DO" X[I,J]:=X[J,I]:=A[I,J]; OUTPUT(61,"(""("A")",/,4(4(+DB),/)")",A); EM[0]:="-14;EM[4]:=100;EM[2]:="-5; QRISYM(X,4,VAL1,EM); AUX[0]:=0;AUX[4]:=10;AUX[2]:="-14; SYMEIGIMP(4,A,X,VAL1,VAL2,LBOUND,UBOUND,AUX); OUTPUT(61,"("/,"("THE EXACT EIGENVALUES ARE: -1 , +5 , +5 , +15")", //,"("THE DIFFERENCES BETWEEN THE CALCULATED AND THE EXACT ")", "("EIGENVALUES")", //,4(N,/)")",(VAL1[1]+1)+VAL2[1],(VAL1[2]-5)+VAL2[2],(VAL1[3]- 5)+VAL2[3],(VAL1[4]-15)+VAL2[4]); OUTPUT(61,"("/,"("LOWERBOUNDS UPPERBOUNDS")",//")"); "FOR" I:=1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61,"("2(+D.D"+DD5B),/")",LBOUND[I],UBOUND[I]); OUTPUT(61,"("/,"("NUMBER OF ITERATIONS = ")",ZD//, "("INFINITY NORM OF A = ")",ZD//, "("MAXIMUM ABSOLUTE ELEMENT OF RESIDU = ")",D.D"+DD")", AUX[5],AUX[1],AUX[3]) "END" EXAMPLE OF USE 1SECTION : 3.3.1.1.3.3 (NOVEMBER 1976) PAGE 4 DELIVERS: A +6 +4 +4 +1 +4 +6 +1 +4 +4 +1 +6 +4 +1 +4 +4 +6 THE EXACT EIGENVALUES ARE: -1 , +5 , +5 , +15 THE DIFFERENCES BETWEEN THE CALCULATED AND THE EXACT EIGENVALUES -6.3423147029256"-022 +5.5934784498910"-018 +4.0389678347316"-028 -5.5947317864427"-018 LOWERBOUNDS UPPERBOUNDS +1.2"-23 +1.2"-23 +7.5"-09 +7.5"-09 +1.0"-13 +1.0"-13 +5.6"-18 +5.6"-18 NUMBER OF ITERATIONS = 2 INFINITY NORM OF A = 15 MAXIMUM ABSOLUTE ELEMENT OF RESIDU = 2.8"-14 1SECTION : 3.3.1.1.3.3 (NOVEMBER 1976) PAGE 5 SOURCETEXT: 0"CODE" 36401; "PROCEDURE" SYMEIGIMP(N,A,VEC,VAL1,VAL2,LBOUND,UBOUND,AUX); "VALUE" N;"INTEGER" N;"ARRAY" A,VEC,VAL1,VAL2,LBOUND,UBOUND,AUX; "BEGIN" "INTEGER" K,I,J,I0,I1,ITER,MAXITP1;"REAL" S,HEAD,TAIL,MAX,TOL, MATEPS,RELERRA,RELTOLR,NORMA;"INTEGER" "ARRAY" PERM[1:N]; "ARRAY" R,P,Y[1:N,1:N],RQ,RQT,EPS,Z,VAL3,ETA[1:N]; "PROCEDURE" BOUNDS(I0,I1,N,LBOUND,UBOUND);"VALUE" I0,I1,N; "INTEGER" I0,I1,N;"ARRAY" LBOUND,UBOUND; "BEGIN" "INTEGER" K,I,J,I01;"REAL" EPS2,DL,DR; "FOR" I:=I0,I01 "WHILE" I<=I1 "DO" "BEGIN" J:=I01:=I; "FOR" J:=J+1 "WHILE" "IF" J>I1 "THEN" "FALSE" "ELSE" RQ[J]-RQ[J-1]<=EPS[J]+EPS[J-1] "DO" I01:=J; "IF" I = I01 "THEN" "BEGIN" "IF" IMAX "THEN" MAX:=ABS(R[I,J]) "END";"IF" MAX > TOL "THEN" STOP:="FALSE"; "IF" "NOT" STOP "AND" ITERN "THEN" "FALSE" "ELSE" RQ[J]-RQ[J-1]<=SQRT((EPS[J]+EPS[J-1])*NORMA) "DO" I1:=J; "IF" STOP "OR" ITER=MAXITP1 "THEN" BOUNDS(I0,I1,N,LBOUND,UBOUND) "ELSE" "BEGIN" "IF" I0=I1 "THEN" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" N "DO" "IF" K=I0 "THEN" Y[K,I0]:=1 "ELSE" R[K,I0]:=P[K,I0]; VAL1[I0]:=RQ[I0];VAL2[I0]:=RQT[PERM[I0]] "END" "ELSE" "BEGIN""INTEGER" N1,I0M1,I1P1;"REAL" M1;"ARRAY"EM[0:5]; N1:=I1-I0+1;EM[0]:=EM[2]:="-14;EM[4]:=10*N1; "BEGIN" "ARRAY" PP[1:N1,1:N1],VAL4[1:N1];M1:=0; "FOR" K:=I0 "STEP" 1 "UNTIL" I1 "DO" M1:=M1+VAL3[K];M1:=M1/N1; "COMMENT" 1SECTION : 3.3.1.1.3.3 (DECEMBER 1979) PAGE 7 ; "FOR" I:=1 "STEP" 1 "UNTIL" N1 "DO" "FOR" J:=1 "STEP" 1 "UNTIL" N1 "DO" "BEGIN" PP[I,J]:=P[I+I0-1,J+I0-1]; "IF" I=J "THEN" PP[I,J]:=PP[I,J]+VAL3[J+I0-1]-M1 "END";"FOR" I:=I0 "STEP" 1 "UNTIL" I1 "DO" "BEGIN" VAL3[I]:=M1;VAL1[I]:=RQ[I]; VAL2[I]:=RQT[PERM[I]] "END"; QRISYM(PP,N1,VAL4,EM); MERGESORT(VAL4,PERM,1,N1); "FOR" I:=1 "STEP" 1 "UNTIL" N1 "DO" "FOR" J:=1 "STEP" 1 "UNTIL" N1 "DO" P[I+I0-1,J+I0-1]:=PP[I,PERM[J]]; I0M1:=I0-1;I1P1:=I1+1; "FOR" J:=I0 "STEP" 1 "UNTIL" I1 "DO" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" I0M1, I1P1 "STEP" 1 "UNTIL" N "DO" "BEGIN" S:=0; "FOR" K:=I0 "STEP" 1 "UNTIL" I1 "DO" S:=S+P[I,K]*P[K,J]; R[I,J]:=S "END";"FOR" I:=I0 "STEP" 1 "UNTIL" I1 "DO" Y[I,J]:=P[I,J] "END" FOR J "END" INNERBLOCK "END" I1>I0 "END" NOT STOP "END" FOR I0; "IF" "NOT" STOP "AND" ITER; "ARRAY" A[1:N,1:N]; ENTRY: THE ELEMENTS OF THE REAL UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF ARRAY A; EXIT: THE HESSENBERG PART OF ARRAY A IS ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; IF THE ABSOLUTE VALUE OF SOME SUBDIAGONAL ELEMENT IS SMALLER THAN EM[1] * EM[2], THEN THIS ELEMENT IS NEGLECTED AND THE MATRIX IS PARTITIONED; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF VAL ARE APPROXIMATE EIGEN- VALUES OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN REAVALQRI; VAL: ; "ARRAY" VAL[1:N]; THE EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN VAL. MOREOVER: REAVALQRI DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE REAVALQRI DELIVERS THE NUMBER OF EIGEN- VALUES NOT CALCULATED. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 3 PROCEDURES USED: ROTCOL = CP34040, ROTROW = CP34041. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE METHOD USED IN THE PROCEDURE REAVALQRI IS THE SINGLE QR ITERATION OF FRANCIS (SEE REF[1], P. 54, REF[2] P. 515 - 543 AND REF[3]). THE EIGENVALUES OF A REAL UPPER-HESSENBERG MATRIX ARE CALCULATED, PROVIDED THAT THE MATRIX HAS REAL EIGENVALUES ONLY. REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. [2]. J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS, OXFORD, 1965. [3]. J.G. FRANCIS. THE QR TRANSFORMATION, PARTS 1 AND 2. COMP. J. 4 (1961), 265 - 271 AND 332 - 345. EXAMPLE OF USE: THE PROCEDURE REAVALQRI IS USED IN REAEIGVAL, SECTION 3.3.1.2.2. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 4 SUBSECTION: REAVECHES. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" REAVECHES(A, N, LAMBDA, EM, V); "VALUE" N, LAMBDA; "INTEGER" N; "REAL" LAMBDA; "ARRAY" A, EM, V; "CODE" 34181; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE ELEMENTS OF THE REAL UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF ARRAY A; EXIT: THE HESSENBERG PART OF ARRAY A IS ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; LAMBDA: ; THE GIVEN REAL EIGENVALUE OF THE UPPER-HESSENBERG MATRIX; EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[6], THE TOLERANCE USED FOR THE EIGENVECTOR; THE INVERSE ITERATION ENDS IF THE EUCLIDIAN NORM OF THE RESIDUE VECTOR IS SMALLER THAN EM[1] * EM[6]; EM[8], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[6] = "-10, EM[8] = 5; EXIT: EM[7], THE EUCLIDIAN NORM OF THE RESIDUE VECTOR OF THE CALCULATED EIGENVECTOR; EM[9], THE NUMBER OF INVERSE ITERATIONS PERFORMED; IF EM[7] REMAINS LARGER THAN EM[1] * EM[6] DURING EM[8] ITERATIONS, THE VALUE EM[8] + 1 IS DELIVERED; V: ; "ARRAY" V[1:N]; THE CALCULATED EIGENVECTOR IS DELIVERED IN V. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 5 PROCEDURES USED: VECVEC = CP34010, MATVEC = CP34011. RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE PROCEDURE REAVECHES CALCULATES AN EIGENVECTOR CORRESPONDING TO A GIVEN APPROXIMATE REAL EIGENVALUE OF A REAL UPPER-HESSENBERG MATRIX, BY MEANS OF INVERSE ITERATION (SEE REF[1], P. 55, REF[2], P. 619 - 629 AND REF[3]). REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. [2]. J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS, OXFORD, 1965. [3]. J.M. VARAH. EIGENVECTORS OF A REAL MATRIX BY INVERSE ITERATION. STANFORD UNIVERSITY, TECH. REP. NO. CS 34, 1966. EXAMPLE OF USE: THE PROCEDURE REAVECHES IS USED IN REAEIG1, SECTION 3.3.1.2.2. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 6 SUBSECTION: REAQRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" REAQRI(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "CODE" 34186; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE ELEMENTS OF THE REAL UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF ARRAY A; EXIT: THE HESSENBERG PART OF ARRAY A IS ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; IF THE ABSOLUTE VALUE OF SOME SUBDIAGONAL ELEMENT IS SMALLER THAN EM[1] * EM[2], THEN THIS ELEMENT IS NEGLECTED AND THE MATRIX IS PARTITIONED; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED; IN THIS CASE ONLY THE LAST N - K ELEMENTS OF VAL AND THE LAST N - K COLUMNS OF VEC ARE APPROXIMATED EIGENVALUES AND EIGENVECTORS OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN REAQRI; VAL: ; "ARRAY" VAL[1:N]; THE EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN VAL; VEC: ; "ARRAY" VEC[1:N,1:N]; THE CALCULATED EIGENVECTORS, CORRESPONDING TO THE EIGEN- VALUES IN ARRAY VAL[1:N], ARE DELIVERED IN THE COLUMNS OF ARRAY VEC. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 7 MOREOVER: REAQRI DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE REAQRI DELIVERS THE NUMBER OF EIGEN- VALUES AND EIGENVECTORS NOT CALCULATED. PROCEDURES USED: MATVEC = CP34011, ROTCOL = CP34040, ROTROW = CP34041. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE PROCEDURE REAQRI CALCULATES THE EIGENVALUES OF AN UPPER- HESSENBERG MATRIX BY MEANS OF SINGLE QR ITERATION (SEE METHOD AND PERFORMANCE OF REAVALQRI, THIS SECTION). THE EIGENVECTORS ARE CALCULATED BY A DIRECT METHOD (SEE REF[1], P. 55-56), IN CONTRAST WITH REAVECHES WHICH USES INVERSE ITERATION. IF THE HESSENBERG MATRIX IS NOT TOO ILL-CONDITIONED WITH RESPECT TO ITS EIGENVALUE PROBLEM, THEN THIS METHOD YIELDS NUMERICALLY INDEPENDENT EIGENVECTORS AND IS COMPETITIVE WITH INVERSE ITERATION AS TO ACCURACY AND COMPUTATION TIME. IF THE QR ITERATION PROCESS IS NOT COMPLETED WITHIN THE GIVEN NUMBER OF ITERATIONS, NOT ALL EIGENVALUES AND EIGENVECTORS ARE DELIVERED. THE PROCEDURE REAQRI SHOULD BE USED ONLY IF ALL EIGENVALUES ARE REAL. REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. EXAMPLE OF USE: THE PROCEDURE REAQRI IS USED IN REAEIG3, SECTION 3.3.1.2.2. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 8 SUBSECTION: COMVALQRI. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" COMVALQRI(A, N, EM, RE, IM); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM; "CODE" 34190; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE ELEMENTS OF THE REAL UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF ARRAY A; EXIT: THE HESSENBERG PART OF ARRAY A IS ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; IF THE ABSOLUTE VALUE OF SOME SUBDIAGONAL ELEMENT IS SMALLER THAN EM[1] * EM[2], THEN THIS ELEMENT IS NEGLECTED AND THE MATRIX IS PARTITIONED; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF RE AND IM ARE APPROXIMATE EIGENVALUES OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN COMVALQRI; RE,IM: ; "ARRAY" RE, IM[1:N]; THE REAL AND IMAGINARY PARTS OF THE CALCULATED EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN ARRAY RE, IM[1:N], THE MEMBERS OF EACH NONREAL COMPLEX CONJUGATE PAIR BEING CONSECUTIVE. MOREOVER: COMVALQRI DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE COMVALQRI DELIVERS THE NUMBER OF EIGEN- VALUES NOT CALCULATED. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 9 PROCEDURES USED: NONE. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE METHOD USED IN THE PROCEDURE COMVALQRI FOR CALCULATING THE REAL AND COMPLEX EIGENVALUES OF A REAL UPPER-HESSENBERG MATRIX IS THE DOUBLE QR ITERATION OF FRANCIS (SEE REF[1], P. 74, REF[2] P. 528 - 537 AND REF[3]). REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. [2]. J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS, OXFORD, 1965. [3]. J.G. FRANCIS. THE QR TRANSFORMATION, PARTS 1 AND 2. COMP. J. 4 (1961), 265 - 271 AND 332 - 345. EXAMPLE OF USE: THE COMPLEX EIGENVALUES AND -VECTORS OF H, WITH N = 4 AND H[I,J] = "IF" I = 1 "THEN" -1 "ELSE" "IF" I - J = 1 "THEN" 1 "ELSE" 0, MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I, J, M; "ARRAY" A[1:4,1:4], RE, IM[1:4], EM[0:9]; EM[0]:= "-14; EM[2]:= "-13; EM[1]:= 4; EM[4]:= 40; EM[6]:= "-10; EM[8]:= 5; "FOR" I:= 1, 2, 3, 4 "DO" "FOR" J:= 1, 2, 3, 4 "DO" A[I,J]:= "IF" I = 1 "THEN" -1 "ELSE" "IF" I - J = 1 "THEN" 1 "ELSE" 0; M:= COMVALQRI(A, 4, EM, RE, IM); OUTPUT(61, "("D, /")", M); 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 9 "FOR" J:= M + 1 "STEP" 1 "UNTIL" 4 "DO" "BEGIN" "INTEGER" K; "ARRAY" U, V[1:4]; "FOR" I:= 1, 2, 3, 4 "DO" "FOR" K:= 1, 2, 3, 4 "DO" A[I,K]:= "IF" I = 1 "THEN" -1 "ELSE" "IF" I - K = 1 "THEN" 1 "ELSE" 0; COMVECHES(A, 4, RE[J], IM[J], EM, U, V); OUTPUT(61, "("/, 2(+.13D"+2D, 2B), 2/")", RE[J], IM[J]); "FOR" I:= 1, 2, 3, 4 "DO" OUTPUT(61, "("21B, 2(+.13D"+2D, 2B), /")", U[I], V[I]) "END"; OUTPUT(61, "("/, 2(.2D"+2D, /), 2(ZD, /)")", EM[3], EM[7], EM[5], EM[9]) "END" THE PROGRAM DELIVERS (THE RESULTS ARE CORRECT UP TO TWELVE DIGITS): THE NUMBER OF NOT CALCULATED EIGENVALUES: 0 THE EIGENVALUES AND -VECTORS: +.3090169943750"+00 +.9510565162952"+00 -.2527643931136"+00 -.4314048696688"+00 -.4883989055049"+00 +.1070817869743"+00 -.4908273055667"-01 +.4975850535950"+00 +.4580641097602"+00 +.2004426884413"+00 +.3090169943750"+00 -.9510565162952"+00 -.2527643931136"+00 +.4314048696688"+00 -.4883989055049"+00 -.1070817869743"+00 -.4908273055667"-01 -.4975850535950"+00 +.4580641097602"+00 -.2004426884413"+00 -.8090169943749"+00 +.5877852522924"+00 +.1095191711534"+00 -.4878581260468"+00 -.3753586823743"+00 +.3303117611685"+00 +.4978239349006"+00 -.4659753040772"-01 -.4301373647081"+00 -.2549153731770"+00 -.8090169943749"+00 -.5877852522924"+00 +.1095191711534"+00 +.4878581260468"+00 -.3753586823743"+00 -.3303117611685"+00 +.4978239349006"+00 +.4659753040772"-01 -.4301373647081"+00 +.2549153731770"+00 THE ARRAY EM: EM[3] = .67"-22 EM[7] = .17"-13 EM[5] = 9 EM[9] = 1 . 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 11 SUBSECTION: COMVECHES. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" COMVECHES(A, N, LAMBDA, MU, EM, U, V); "VALUE" N, LAMBDA, MU; "INTEGER" N; "REAL" LAMBDA, MU; "ARRAY" A, EM, U, V; "CODE" 34191; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE ELEMENTS OF THE REAL UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE UPPER TRIANGLE AND THE FIRST SUBDIAGONAL OF ARRAY A; EXIT: THE HESSENBERG PART OF ARRAY A IS ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; LAMBDA, MU: ; THE REAL AND IMAGINARY PART OF THE GIVEN EIGENVALUE; EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION; EM[1], A NORM OF THE GIVEN MATRIX; EM[6], THE TOLERANCE USED FOR THE EIGENVECTOR; THE INVERSE ITERATION ENDS IF THE EUCLIDIAN NORM OF THE RESIDUE VECTOR IS SMALLER THAN EM[1] * EM[6]; EM[8], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[6] = "-10, EM[8] = 5; EXIT: EM[7], THE EUCLIDIAN NORM OF THE RESIDUE VECTOR OF THE CALCULATED EIGENVECTOR; EM[9], THE NUMBER OF INVERSE ITERATIONS PERFORMED; IF EM[7] REMAINS LARGER THAN EM[1] * EM[6] DURING EM[8] ITERATIONS, THE VALUE EM[8] + 1 IS DELIVERED; U, V: ; "ARRAY" U, V[1:N]; THE REAL AND IMAGINARY PARTS OF THE CALCULATED EIGENVECTOR ARE DELIVERED IN THE ARRAYS U, V[1:N]. 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 12 PROCEDURES USED: VECVEC = CP34010, MATVEC = CP34011, TAMVEC = CP34012. RUNNING TIME: ROUGHLY PROPORTIONAL TO N SQUARED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE PROCEDURE COMVECHES CALCULATES AN EIGENVECTOR CORRESPONDING TO A GIVEN APPROXIMATE EIGENVALUE OF A REAL UPPER-HESSENBERG MATRIX, BY MEANS OF INVERSE ITERATION (SEE REF[1], P. 75, REF[2], P. 629 - 633 AND REF[3]). REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. [2]. J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS, OXFORD, 1965. [3]. J.M. VARAH. EIGENVECTORS OF A REAL MATRIX BY INVERSE ITERATION. STANFORD UNIVERSITY, TECH. REP. NO. CS 34, 1966. EXAMPLE OF USE: SEE EXAMPLE OF USE OF COMVALQRI, THIS SECTION. SOURCE TEXT(S) : 0"CODE" 34180; "COMMENT" MCA 2410; "INTEGER" "PROCEDURE" REAVALQRI(A, N, EM, VAL); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL; "BEGIN" "INTEGER" N1, I, I1, J, Q, MAX, COUNT; "REAL" DET, W, SHIFT, KAPPA, NU, MU, R, TOL, DELTA, MACHTOL, S; "COMMENT" 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 13 ; MACHTOL:= EM[0] * EM[1]; TOL:= EM[1] * EM[2]; MAX:= EM[4]; COUNT:= 0; R:= 0; IN: N1:= N - 1; "FOR" I:= N, I - 1 "WHILE" ("IF" I >= 1 "THEN" ABS(A[I + 1,I]) > TOL "ELSE" "FALSE") "DO" Q:= I; "IF" Q > 1 "THEN" "BEGIN" "IF" ABS(A[Q,Q - 1]) > R "THEN" R:= ABS(A[Q,Q - 1]) "END"; "IF" Q = N "THEN" "BEGIN" VAL[N]:= A[N,N]; N:= N1 "END" "ELSE" "BEGIN" DELTA:= A[N,N] - A[N1,N1]; DET:= A[N,N1] * A[N1,N]; "IF" ABS(DELTA) < MACHTOL "THEN" S:= SQRT(DET) "ELSE" "BEGIN" W:= 2 / DELTA; S:= W * W * DET + 1; S:= "IF" S <= 0 "THEN" -DELTA * .5 "ELSE" W * DET / (SQRT(S) + 1) "END"; "IF" Q = N1 "THEN" "BEGIN" VAL[N]:= A[N,N] + S; VAL[N1]:= A[N1,N1] - S; N:= N - 2 "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" OUT; SHIFT:= A[N,N] + S; "IF" ABS(DELTA) < TOL "THEN" "BEGIN" W:= A[N1,N1] - S; "IF" ABS(W) < ABS(SHIFT) "THEN" SHIFT:= W "END"; A[Q,Q]:= A[Q,Q] - SHIFT; "FOR" I:= Q "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" I1:= I + 1; A[I1,I1]:= A[I1,I1] - SHIFT; KAPPA:= SQRT(A[I,I] ** 2 + A[I1,I] ** 2); "IF" I > Q "THEN" "BEGIN" A[I,I - 1]:= KAPPA * NU; W:= KAPPA * MU "END" "ELSE" W:= KAPPA; MU:= A[I,I] / KAPPA; NU:= A[I1,I] / KAPPA; A[I,I]:= W; ROTROW(I1, N, I, I1, A, MU, NU); ROTCOL(Q, I, I, I1, A, MU, NU); A[I,I]:= A[I,I] + SHIFT "END"; A[N,N - 1]:= A[N,N] * NU; A[N,N]:= A[N,N] * MU + SHIFT "END" "END"; "IF" N > 0 "THEN" "GOTO" IN; OUT: EM[3]:= R; EM[5]:= COUNT; REAVALQRI:= N "END" REAVALQRI 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 14 ; "EOP" 0"CODE" 34181; "COMMENT" MCA 2411; "PROCEDURE" REAVECHES(A, N, LAMBDA, EM, V); "VALUE" N, LAMBDA; "INTEGER" N; "REAL" LAMBDA; "ARRAY" A, EM, V; "BEGIN" "INTEGER" I, I1, J, COUNT, MAX; "REAL" M, R, NORM, MACHTOL, TOL; "BOOLEAN" "ARRAY" P[1:N]; NORM:= EM[1]; MACHTOL:= EM[0] * NORM; TOL:= EM[6] * NORM; MAX:= EM[8]; A[1,1]:= A[1,1] - LAMBDA; GAUSS: "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" I1:= I + 1; R:= A[I,I]; M:= A[I1,I]; "IF" ABS(M) < MACHTOL "THEN" M:= MACHTOL; P[I]:= ABS(M) <= ABS(R); "IF" P[I] "THEN" "BEGIN" A[I1,I]:= M:= M / R; "FOR" J:= I1 "STEP" 1 "UNTIL" N "DO" A[I1,J]:= ("IF" J > I1 "THEN" A[I1,J] "ELSE" A[I1,J] - LAMBDA) - M * A[I,J] "END" "ELSE" "BEGIN" A[I,I]:= M; A[I1,I]:= M:= R / M; "FOR" J:= I1 "STEP" 1 "UNTIL" N "DO" "BEGIN" R:= ("IF" J > I1 "THEN" A[I1,J] "ELSE" A[I1,J] - LAMBDA); A[I1,J]:= A[I,J] - M * R; A[I,J]:= R "END" "END" "END" GAUSS; "IF" ABS(A[N,N]) < MACHTOL "THEN" A[N,N]:= MACHTOL; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" V[J]:= 1; COUNT:= 0; FORWARD: COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" OUT; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" I1:= I + 1; "IF" P[I] "THEN" V[I1]:= V[I1] - A[I1,I] * V[I] "ELSE" "BEGIN" R:= V[I1]; V[I1]:= V[I] - A[I1,I] * R; V[I]:=R "END" "END" FORWARD; BACKWARD: "FOR" I:= N "STEP" -1 "UNTIL" 1 "DO" V[I]:= (V[I] - MATVEC(I + 1, N, I, A, V)) / A[I,I]; R:= 1 / SQRT(VECVEC(1, N, 0, V, V)); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" V[J]:= V[J] * R; "IF" R > TOL "THEN" "GOTO" FORWARD; OUT: EM[7]:= R; EM[9]:= COUNT "END" REAVECHES 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 15 ; "EOP" 0"CODE" 34186; "COMMENT" MCA 2416; "INTEGER" "PROCEDURE" REAQRI(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "BEGIN" "INTEGER" M1, I, I1, M, J, Q, MAX, COUNT; "REAL" W, SHIFT, KAPPA, NU, MU, R, TOL, S, MACHTOL, ELMAX, T, DELTA, DET; "ARRAY" TF[1:N]; MACHTOL:= EM[0] * EM[1]; TOL:= EM[1] * EM[2]; MAX:= EM[4]; COUNT:= 0; ELMAX:= 0; M:= N; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" VEC[I,I]:= 1; "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" VEC[I,J]:= VEC[J,I]:= 0 "END"; IN: M1:= M - 1; "FOR" I:= M, I - 1 "WHILE" ("IF" I >= 1 "THEN" ABS(A[I + 1,I]) > TOL "ELSE" "FALSE") "DO" Q:= I; "IF" Q > 1 "THEN" "BEGIN" "IF" ABS(A[Q,Q - 1]) > ELMAX "THEN" ELMAX:= ABS(A[Q, Q - 1]) "END"; "IF" Q = M "THEN" "BEGIN" VAL[M]:= A[M,M]; M:= M1 "END" "ELSE" "BEGIN" DELTA:= A[M,M] - A[M1,M1]; DET:= A[M,M1] * A[M1,M]; "IF" ABS(DELTA) < MACHTOL "THEN" S:= SQRT(DET) "ELSE" "BEGIN" W:= 2 / DELTA; S:= W * W * DET + 1; S:= "IF" S <= 0 "THEN" -DELTA * .5 "ELSE" W * DET / (SQRT(S) + 1) "END"; "IF" Q = M1 "THEN" "BEGIN" A[M,M]:= VAL[M]:= A[M,M] + S; A[Q,Q]:= VAL[Q]:= A[Q,Q] - S; T:= "IF" ABS(S) < MACHTOL "THEN" (S + DELTA) / A[M,Q] "ELSE" A[Q,M] / S; R:= SQRT(T * T + 1); NU:= 1 / R; MU:= -T * NU; A[Q,M]:= A[Q,M] - A[M,Q]; ROTROW(Q + 2, N, Q, M, A, MU, NU); ROTCOL(1, Q - 1, Q, M, A, MU, NU); ROTCOL(1, N, Q, M, VEC, MU, NU); M:= M - 2 "END" 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 16 "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" END; SHIFT:= A[M,M] + S; "IF" ABS(DELTA) < TOL "THEN" "BEGIN" W:= A[M1,M1] - S; "IF" ABS(W) < ABS(SHIFT) "THEN" SHIFT:= W "END"; A[Q,Q]:= A[Q,Q] - SHIFT; "FOR" I:= Q "STEP" 1 "UNTIL" M1 "DO" "BEGIN" I1:= I + 1; A[I1,I1]:= A[I1,I1] - SHIFT; KAPPA:= SQRT(A[I,I] ** 2 + A[I1,I] ** 2); "IF" I > Q "THEN" "BEGIN" A[I,I - 1]:= KAPPA * NU; W:= KAPPA * MU "END" "ELSE" W:= KAPPA; MU:= A[I,I] / KAPPA; NU:= A[I1,I] / KAPPA; A[I,I]:= W; ROTROW(I1, N, I, I1, A, MU, NU); ROTCOL(1, I, I, I1, A, MU, NU); A[I,I]:= A[I,I] + SHIFT; ROTCOL(1, N, I, I1, VEC, MU, NU) "END"; A[M,M1]:= A[M,M] * NU; A[M,M]:= A[M,M] * MU + SHIFT "END" "END"; "IF" M > 0 "THEN" "GOTO" IN; "FOR" J:= N "STEP" -1 "UNTIL" 2 "DO" "BEGIN" TF[J]:= 1; T:= A[J,J]; "FOR" I:= J - 1 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" DELTA:= T - A[I,I]; TF[I]:= MATVEC(I + 1, J, I, A, TF) / ("IF" ABS(DELTA) < MACHTOL "THEN" MACHTOL "ELSE" DELTA) "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" VEC[I,J]:= MATVEC(1, J, I, VEC, TF) "END"; END: EM[3]:= ELMAX; EM[5]:= COUNT; REAQRI:= M "END" REAQRI 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 17 ; "EOP" 0"CODE" 34190; "COMMENT" MCA 2420; "INTEGER" "PROCEDURE" COMVALQRI(A, N, EM, RE, IM); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM; "BEGIN" "INTEGER" I, J, P, Q, MAX, COUNT, N1, P1, P2, IMIN1, I1, I2, I3; "REAL" DISC, SIGMA, RHO, G1, G2, G3, PSI1, PSI2, AA, E, K, S, NORM, MACHTOL2, TOL, W; "BOOLEAN" B; NORM:= EM[1]; MACHTOL2:= (EM[0] * NORM) ** 2; TOL:= EM[2] * NORM; MAX:= EM[4]; COUNT:= 0; W:= 0; IN: "FOR" I:= N, I - 1 "WHILE" ("IF" I >= 1 "THEN" ABS(A[I + 1,I]) > TOL "ELSE" "FALSE") "DO" Q:= I; "IF" Q > 1 "THEN" "BEGIN" "IF" ABS(A[Q,Q - 1]) > W "THEN" W:= ABS(A[Q,Q - 1]) "END"; "IF" Q >= N - 1 "THEN" "BEGIN" N1:= N - 1; "IF" Q = N "THEN" "BEGIN" RE[N]:= A[N,N]; IM[N]:= 0; N:= N1 "END" "ELSE" "BEGIN" SIGMA:= A[N,N] - A[N1,N1]; RHO:= -A[N,N1] * A[N1,N]; DISC:= SIGMA ** 2 - 4 * RHO; "IF" DISC > 0 "THEN" "BEGIN" DISC:= SQRT(DISC); S:= -2 * RHO / (SIGMA + ("IF" SIGMA >= 0 "THEN" DISC "ELSE" -DISC)); RE[N]:= A[N,N] + S; RE[N1]:= A[N1,N1] - S; IM[N]:= IM[N1]:= 0 "END" "ELSE" "BEGIN" RE[N]:= RE[N1]:= (A[N1,N1] + A[N,N]) / 2; IM[N1]:= SQRT( -DISC) / 2; IM[N]:= -IM[N1] "END"; N:= N - 2 "END" "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" OUT; N1:= N - 1; SIGMA:= A[N,N] + A[N1,N1] + SQRT(ABS(A[N1,N - 2] * A[N,N1]) * EM[0]); RHO:= A[N,N] * A[N1,N1] - A[N,N1] * A[N1,N]; "FOR" I:= N - 1, I - 1 "WHILE" ("IF" I - 1 >= Q "THEN" ABS(A[I,I - 1] * A[I1,I] * (ABS(A[I,I] + A[I1,I1] - SIGMA) + ABS(A[I + 2,I1]))) > ABS(A[I,I] * ((A[I,I] - SIGMA) + A[I,I1] * A[I1,I] + RHO)) * TOL "ELSE" "FALSE") "DO" P1:= I1:= I; P:= P1 - 1; P2:= P + 2; "COMMENT" 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 18 ; "FOR" I:= P "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" IMIN1:= I - 1; I1:= I + 1; I2:= I + 2; "IF" I = P "THEN" "BEGIN" G1:= A[P,P] * (A[P,P] - SIGMA) + A[P,P1] * A[P1,P] + RHO; G2:= A[P1,P] * (A[P,P] + A[P1,P1] - SIGMA); "IF" P1 <= N1 "THEN" "BEGIN" G3:= A[P1,P] * A[P2,P1]; A[P2,P]:= 0 "END" "ELSE" G3:= 0 "END" "ELSE" "BEGIN" G1:= A[I,IMIN1]; G2:= A[I1,IMIN1]; G3:= "IF" I2 <= N "THEN" A[I2,IMIN1] "ELSE" 0 "END"; K:= "IF" G1 >= 0 "THEN" SQRT(G1 ** 2 + G2 ** 2 + G3 ** 2) "ELSE" -SQRT(G1 ** 2 + G2 ** 2 + G3 ** 2); B:= ABS(K) > MACHTOL2; AA:= "IF" B "THEN" G1 / K + 1 "ELSE" 2; PSI1:= "IF" B "THEN" G2 / (G1 + K) "ELSE" 0; PSI2:= "IF" B "THEN" G3 / (G1 + K) "ELSE" 0; "IF" I ^= Q "THEN" A[I,IMIN1]:= "IF" I = P "THEN" -A[I,IMIN1] "ELSE" -K; "FOR" J:= I "STEP" 1 "UNTIL" N "DO" "BEGIN" E:= AA * (A[I,J] + PSI1 * A[I1,J] + ("IF" I2 <= N "THEN" PSI2 * A[I2,J] "ELSE" 0)); A[I,J]:= A[I,J] - E; A[I1,J]:= A[I1,J] - PSI1 * E; "IF" I2 <= N "THEN" A[I2,J]:= A[I2,J] - PSI2 * E "END"; "FOR" J:= Q "STEP" 1 "UNTIL" ("IF" I2 <= N "THEN" I2 "ELSE" N) "DO" "BEGIN" E:= AA * (A[J,I] + PSI1 * A[J,I1] + ("IF" I2 <= N "THEN" PSI2 * A[J,I2] "ELSE" 0)); A[J,I]:= A[J,I] - E; A[J,I1]:= A[J,I1] - PSI1 * E; "IF" I2 <= N "THEN" A[J,I2]:= A[J,I2] - PSI2 * E "END"; "IF" I2 <= N1 "THEN" "BEGIN" I3:= I + 3; E:= AA * PSI2 * A[I3,I2]; A[I3,I]:= -E; A[I3,I1]:= -PSI1 * E; A[I3,I2]:= A[I3,I2] - PSI2 * E "END" "END" "END"; "IF" N > 0 "THEN" "GOTO" IN; OUT: EM[3]:= W; EM[5]:= COUNT; COMVALQRI:= N "END" COMVALQRI 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 19 ; "EOP" 0"CODE" 34191; "COMMENT" MCA 2421; "PROCEDURE" COMVECHES(A, N, LAMBDA, MU, EM, U, V); "VALUE" N, LAMBDA, MU; "INTEGER" N; "REAL" LAMBDA, MU; "ARRAY" A, EM, U, V; "BEGIN" "INTEGER" I, I1, J, COUNT, MAX; "REAL" AA, BB, D, M, R, S, W, X, Y, NORM, MACHTOL, TOL; "ARRAY" G, F[1:N]; "BOOLEAN" "ARRAY" P[1:N]; NORM:= EM[1]; MACHTOL:= EM[0] * NORM; TOL:= EM[6] * NORM; MAX:= EM[8]; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" "BEGIN" F[I - 1]:= A[I,I - 1]; A[I,1]:= 0 "END"; AA:= A[1,1] - LAMBDA; BB:= -MU; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" I1:= I + 1; M:= F[I]; "IF" ABS(M) < MACHTOL "THEN" M:= MACHTOL; A[I,I]:= M; D:= AA ** 2 + BB ** 2; P[I]:= ABS(M) < SQRT(D); "IF" P[I] "THEN" "BEGIN" "COMMENT" A[I,J] * FACTOR AND A[I1,J] - A[I,J]; F[I]:= R:= M * AA / D; G[I]:= S:= -M * BB / D; W:= A[I1,I]; X:= A[I,I1]; A[I1,I]:= Y:= X * S + W * R; A[I,I1]:= X:= X * R - W * S; AA:= A[I1,I1] - LAMBDA - X; BB:= -(MU + Y); "FOR" J:= I + 2 "STEP" 1 "UNTIL" N "DO" "BEGIN" W:= A[J,I]; X:= A[I,J]; A[J,I]:= Y:= X * S + W * R; A[I,J]:= X:= X * R - W * S; A[J,I1]:= -Y; A[I1,J]:= A[I1,J] - X "END" "END" "ELSE" "BEGIN" "COMMENT" INTERCHANGE A[I1,J] AND A[I,J] - A[I1,J] * FACTOR; F[I]:= R:= AA / M; G[I]:= S:= BB / M; W:= A[I1,I1] - LAMBDA; AA:= A[I,I1] - R * W - S * MU; A[I,I1]:= W; BB:= A[I1,I] - S * W + R * MU; A[I1,I]:= -MU; "FOR" J:= I + 2 "STEP" 1 "UNTIL" N "DO" "BEGIN" W:= A[I1,J]; A[I1,J]:= A[I,J] - R * W; A[I,J]:= W; A[J,I1]:= A[J,I] - S * W; A[J,I]:= 0 "END" "END" "END" 1SECTION : 3.3.1.2.1 (JULY 1974) PAGE 20 ; P[N]:= "TRUE"; D:= AA ** 2 + BB ** 2; "IF" D < MACHTOL ** 2 "THEN" "BEGIN" AA:= MACHTOL; BB:= 0; D:= MACHTOL ** 2 "END"; A[N,N]:= D; F[N]:= AA; G[N]:= -BB; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" U[I]:= 1; V[I]:= 0 "END"; COUNT:= 0; FORWARD: "IF" COUNT > MAX "THEN" "GOTO" OUTM; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" P[I] "THEN" "BEGIN" W:= V[I]; V[I]:= G[I] * U[I] + F[I] * W; U[I]:= F[I] * U[I] - G[I] * W; "IF" I < N "THEN" "BEGIN" V[I + 1]:= V[I + 1] - V[I]; U[I + 1]:= U[I + 1] - U[I] "END" "END" "ELSE" "BEGIN" AA:= U[I + 1]; BB:= V[I + 1]; U[I + 1]:= U[I] - (F[I] * AA - G[I] * BB); U[I]:= AA; V[I + 1]:= V[I] - (G[I] * AA + F[I] * BB); V[I]:= BB "END" "END" FORWARD; BACKWARD: "FOR" I:= N "STEP" -1 "UNTIL" 1 "DO" "BEGIN" I1:= I + 1; U[I]:= (U[I] - MATVEC(I1, N, I, A, U) + ("IF" P[I] "THEN" TAMVEC(I1, N, I, A, V) "ELSE" A[I1,I] * V[I1])) / A[I,I]; V[I]:= (V[I] - MATVEC(I1, N, I, A, V) - ("IF" P[I] "THEN" TAMVEC(I1, N, I, A, U) "ELSE" A[I1,I] * U[I1])) / A[I,I] "END" BACKWARD; NORMALISE: W:= 1 / SQRT(VECVEC(1, N, 0, U, U) + VECVEC(1, N, 0, V, V)); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" U[J]:= U[J] * W; V[J]:= V[J] * W "END"; COUNT:= COUNT + 1; "IF" W > TOL "THEN" "GOTO" FORWARD; OUTM: EM[7]:= W; EM[9]:= COUNT "END" COMVECHES; "EOP" 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 1 AUTHORS : T.J. DEKKER, W. HOFFMANN. CONTRIBUTORS: W. HOFFMANN, S.P.N. VAN KAMPEN, J.G. VERWER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 731205. BRIEF DESCRIPTION: THIS SECTION CONTAINS FIVE PROCEDURES FOR CALCULATING EIGENVALUES AND / OR EIGENVECTORS OF REAL MATRICES: A) REAEIGVAL CALCULATES THE EIGENVALUES OF A MATRIX, PROVIDED THAT ALL EIGENVALUES ARE REAL, B) REAEIG1 CALCULATES THE EIGENVALUES, PROVIDED THAT THEY ARE ALL REAL, AND THE EIGENVECTORS OF A MATRIX, C) REAEIG3 CALCULATES THE EIGENVALUES, PROVIDED THAT THEY ARE ALL REAL, AND THE EIGENVECTORS OF A MATRIX, D) COMEIGVAL CALCULATES THE EIGENVALUES OF A MATRIX, E) COMEIG1 CALCULATES THE EIGENVALUES AND EIGENVECTORS OF A MATRIX. KEYWORDS: EIGENVALUES, EIGENVECTORS. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 2 SUBSECTION: REAEIGVAL. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" REAEIGVAL(A, N, EM, VAL); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL; "CODE" 34182; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE MATRIX WHOSE EIGENVALUES ARE TO BE CALCULATED; EXIT: THE ARRAY ELEMENTS ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[1], THE INFINITY NORM OF THE EQUILIBRATED MATRIX; EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF VAL ARE APPROXIMATE EIGEN- VALUES OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN REAEIGVAL; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN MONOTONICALLY NONINCREASING ORDER; MOREOVER: REAEIGVAL DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE REAEIGVAL DELIVERS K, THE NUMBER OF EIGENVALUES NOT CALCULATED. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 3 PROCEDURES USED: EQILBR = CP34173, TFMREAHES = CP34170, REAVALQRI = CP34180. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: 3N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE GIVEN MATRIX IS EQUILIBRATED BY CALLING EQILBR (SEE SECTION 3.2.1.1.1) AND TRANSFORMED TO A SIMILAR UPPER-HESSENBERG MATRIX BY CALLING TFMREAHES (SEE SECTION 3.2.1.2.1.2). THE EIGENVALUES ARE THEN CALCULATED BY CALLING REAVALQRI, WHICH USES SINGLE QR ITERATION (SEE SECTION 3.3.1.2.1). THE PROCEDURE REAEIGVAL SHOULD BE USED ONLY IF ALL EIGENVALUES ARE REAL. FOR FURTHER DETAILS SEE REFERENCES [1], [2] AND [3]. SUBSECTION: REAEIG1. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" REAEIG1(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "CODE" 34184; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE MATRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED; EXIT: THE ARRAY ELEMENTS ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 4 EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF QR ITERATIONS; EM[6], THE TOLERANCE USED FOR THE EIGENVECTORS; FOR EACH EIGENVECTOR THE INVERSE ITERATION ENDS IF THE EUCLIDEAN NORM OF THE RESIDUE VECTOR IS SMALLER THAN EM[1] * EM[6]; EM[8], THE MAXIMUM ALLOWED NUMBER OF INVERSE ITERATIONS FOR THE CALCULATION OF EACH EIGENVECTOR; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N, EM[6] > EM[2] (E.G. EM[6] = "-10), EM[8] = 5; EXIT: EM[1], THE INFINITY NORM OF THE EQUILIBRATED MATRIX; EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF VAL AND COLUMNS OF VEC ARE APPROXIMATE EIGENVALUES AND EIGENVECTORS OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN REAEIG1; EM[7], THE MAXIMUM EUCLIDIAN NORM OF THE RESIDUES OF THE CALCULATED EIGENVECTORS (OF THE TRANS- FORMED MATRIX); EM[9], THE LARGEST NUMBER OF INVERSE ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGEN- VECTOR; IF, FOR SOME EIGENVECTOR THE EUCLIDEAN NORM OF THE RESIDUE REMAINS LARGER THAN EM[1] * EM[6], THE VALUE EM[8] + 1 IS DELIVERED; NEVERTHELESS THE EIGENVECTORS MAY THEN VERY WELL BE USEFUL, THIS SHOULD BE JUDGED FROM THE VALUE DELIVERED IN EM[7] OR FROM SOME OTHER TEST; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN MONOTONICALLY DECREASING ORDER; VEC: ; "ARRAY" VEC[1:N,1:N]; EXIT: THE CALCULATED EIGENVECTORS, CORRESPONDING TO THE EIGENVALUES IN ARRAY VAL[1:N], ARE DELIVERED IN THE COLUMNS OF ARRAY VEC; 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 5 MOREOVER: REAEIG1 DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE REAEIG1 DELIVERS K, THE NUMBER OF EIGENVALUES AND EIGENVECTORS NOT CALCULATED. PROCEDURES USED: EQILBR = CP34173, TFMREAHES = CP34170, BAKREAHES2 = CP34172, BAKLBR = CP34174, REAVALQRI = CP34180, REAVECHES = CP34181, REASCL = CP34183. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: N * N + 5N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. OPTIONS: F. METHOD AND PERFORMANCE: THE GIVEN MATRIX IS EQUILIBRATED BY CALLING EQILBR (SEE SECTION 3.2.1.1.1) AND TRANSFORMED TO A SIMILAR UPPER-HESSENBERG MATRIX BY CALLING TFMREAHES (SEE SECTION 3.2.1.2.1.2). THE EIGENVALUES ARE THEN CALCULATED BY CALLING REAVALQRI, WHICH USES SINGLE QR ITERATION (SEE SECTION 3.3.1.2.1). FURTHERMORE, TO FIND THE EIGENVECTORS WILKINSON'S DEVICE IS FIRST APPLIED [2, P.328 AND 628]. SUBSEQUENTLY THE EIGENVECTORS OF THE UPPER-HESSENBERG MATRIX ARE CALCULATED BY CALLING REAVECHES, WHICH USES INVERSE ITERATION (SEE SECTION 3.3.1.2.1). THE CALCULATED VECTORS ARE THEN BACK-TRANSFORMED TO THE CORRESPONDING EIGENVECTORS OF THE GIVEN MATRIX BY CALLING BAKREAHES2 AND BAKLBR (SEE SECTIONS 3.2.1.2.1.2 AND 3.2.1.1.1). FINALLY THE APPROXIMATE EIGENVECTORS ARE NORMALIZED BY CALLING REASCL (SEE SECTION 1.1.9) SUCH THAT, IN EACH EIGENVECTOR, AN ELEMENT OF MAXIMUM ABSOLUTE VALUE EQUALS 1. THE PROCEDURE REAEIG1 SHOULD BE USED ONLY IF ALL EIGENVALUES ARE REAL. FOR FURTHER DETAILS SEE THE GIVEN REFERENCES. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 6 SUBSECTION: REAEIG3. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" REAEIG3(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "CODE" 34187; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE MATRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED; EXIT: THE ARRAY ELEMENTS ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF QR ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[1], THE INFINITY NORM OF THE EQUILIBRATED MATRIX; EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED. IN THIS CASE ONLY THE LAST N - K ELEMENTS OF VAL ARE APPROXIMATE EIGENVALUES OF THE GIVEN MATRIX AND NO USEFUL EIGENVECTORS ARE DELIVERED. THE VALUE K IS DELIVERED IN REAEIG3; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED; VEC: ; "ARRAY" VEC[1:N,1:N]; EXIT: THE CALCULATED EIGENVECTORS, CORRESPONDING TO THE EIGENVALUES IN ARRAY VAL[1:N], ARE DELIVERED IN THE COLUMNS OF ARRAY VEC; MOREOVER: REAEIG3 DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE REAEIG3 DELIVERS K, THE NUMBER OF EIGENVALUES NOT CALCULATED. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 7 PROCEDURES USED: EQILBR = CP34173, TFMREAHES = CP34170, BAKREAHES2 = CP34172, BAKLBR = CP34174, REAQRI = CP34186, REASCL = CP34183. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: 4N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE GIVEN MATRIX IS EQUILIBRATED BY CALLING EQILBR (SEE SECTION 3.2.1.1.1) AND TRANSFORMED TO A SIMILAR UPPER-HESSENBERG MATRIX BY CALLING TFMREAHES (SEE SECTION 3.2.1.2.1.2). THE EIGENVALUES AND EIGENVECTORS OF THE UPPER-HESSENBERG MATRIX ARE THEN CALCULATED BY CALLING REAQRI, WHICH USES SINGLE QR ITERATION FOR THE EIGENVALUES AND A DIRECT METHOD FOR THE EIGENVECTORS (SEE SECTION 3.3.1.2.1). FINALLY THE EIGENVECTORS OF THE UPPER- HESSENBERG MATRIX ARE BACK-TRANSFORMED TO THE CORRESPONDING EIGEN- VECTORS OF THE GIVEN MATRIX BY CALLING BAKREAHES2 (SEE SECTION 3.1.2.1.2.1) AND NORMALIZED BY CALLING REASCL (SEE SECTION 1.1.9) SUCH THAT, IN EACH EIGENVECTOR, AN ELEMENT OF MAXIMUM ABSOLUTE VALUE EQUALS 1. THE PROCEDURE REAEIG3 SHOULD BE USED ONLY IF ALL EIGENVALUES ARE REAL. FOR FURTHER DETAILS SEE THE GIVEN REFERENCES. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 8 SUBSECTION: COMEIGVAL. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" COMEIGVAL(A, N, EM, RE, IM); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM; "CODE" 34192; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE MATRIX WHOSE EIGENVALUES ARE TO BE CALCULATED; EXIT: THE ARRAY ELEMENTS ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0], THE MACHINE PRECISION; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N; EXIT: EM[1], THE INFINITY NORM OF THE EQUILIBRATED MATRIX; EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF RE AND IM ARE APPROXIMATE EIGENVALUES OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN COMEIGVAL; RE,IM: ; "ARRAY" RE, IM[1:N]; EXIT: THE REAL AND IMAGINARY PARTS OF THE CALCULATED EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN ARRAY RE, IM[1:N], THE MEMBERS OF EACH NONREAL COMPLEX CONJUGATE PAIR BEING CONSECUTIVE; MOREOVER: COMEIGVAL DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE COMEIGVAL DELIVERS K, THE NUMBER OF EIGENVALUES NOT CALCULATED. PROCEDURES USED: EQILBR = CP34173, TFMREAHES = CP34170, COMVALQRI = CP34190. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 9 REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: 3N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE GIVEN MATRIX IS EQUILIBRATED BY CALLING EQILBR (SEE SECTION 3.2.1.1.1) AND TRANSFORMED TO A SIMILAR UPPER-HESSENBERG MATRIX BY CALLING TFMREAHES (SEE SECTION 3.2.1.2.1.2). THE EIGENVALUES ARE THEN CALCULATED BY CALLING COMVALQRI, WHICH USES DOUBLE QR ITERATION (SEE SECTION 3.3.1.2.1). FOR FURTHER DETAILS SEE REFERENCES [1], [2] AND [3]. SUBSECTION: COMEIG1. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "INTEGER" "PROCEDURE" COMEIG1(A, N, EM, RE, IM, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM, VEC; "CODE" 34194; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE MATRIX WHOSE EIGENVALUES AND EIGENVECTORS ARE TO BE CALCULATED; EXIT: THE ARRAY ELEMENTS ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0], THE MACHINE PRECISION; EM[2], THE RELATIVE TOLERANCE USED FOR THE QR ITERATION; EM[4], THE MAXIMUM ALLOWED NUMBER OF QR ITERATIONS; EM[6], THE TOLERANCE USED FOR THE EIGENVECTORS; FOR EACH EIGENVECTOR THE INVERSE ITERATION ENDS IF THE EUCLIDEAN NORM OF THE RESIDUE VECTOR IS SMALLER THAN EM[1] * EM[6]; EM[8], THE MAXIMUM ALLOWED NUMBER OF INVERSE ITERATIONS FOR THE CALCULATION OF EACH EIGENVECTOR; FOR THE CD CYBER 73-28 SUITABLE VALUES OF THE DATA TO BE GIVEN IN EM ARE: EM[0] = "-14, EM[2] > EM[0] (E.G. EM[2] = "-13), EM[4] = 10 * N, EM[6] > EM[2] (E.G. EM[6] = "-10), EM[8] = 5; 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 10 EXIT: EM[1], THE INFINITY NORM OF THE EQUILIBRATED MATRIX; EM[3], THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5], THE NUMBER OF QR ITERATIONS PERFORMED; IF THE ITERATION PROCESS IS NOT COMPLETED WITHIN EM[4] ITERATIONS, THE VALUE EM[4] + 1 IS DELIVERED AND IN THIS CASE ONLY THE LAST N - K ELEMENTS OF RE, IM AND COLUMNS OF VEC ARE APPROXIMATE EIGENVALUES AND EIGENVECTORS OF THE GIVEN MATRIX, WHERE K IS DELIVERED IN COMEIG1; EM[7], THE MAXIMUM EUCLIDIAN NORM OF THE RESIDUES OF THE CALCULATED EIGENVECTORS (OF THE TRANS- FORMED MATRIX); EM[9], THE LARGEST NUMBER OF INVERSE ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGEN- VECTOR; IF THE EUCLIDIAN NORM OF THE RESIDUE FOR ONE OR MORE EIGENVECTORS REMAINS LARGER THAN EM[1] * EM[6], THE VALUE EM[8]+1 IS DELIVERED; NEVERTHELESS THE EIGENVECTORS MAY THEN VERY WELL BE USEFUL, THIS SHOULD BE JUDGED FROM THE VALUE DELIVERED IN EM[7] OR FROM SOME OTHER TEST; RE,IM: ; "ARRAY" RE, IM[1:N]; EXIT: THE REAL AND IMAGINARY PARTS OF THE CALCULATED EIGENVALUES OF THE GIVEN MATRIX ARE DELIVERED IN ARRAY RE, IM[1:N], THE MEMBERS OF EACH NONREAL COMPLEX CONJUGATE PAIR BEING CONSECUTIVE; VEC: ; "ARRAY" VEC[1:N,1:N]; EXIT: THE CALCULATED EIGENVECTORS ARE DELIVERED IN THE COLUMNS OF ARRAY VEC; AN EIGENVECTOR, CORRESPONDING TO A REAL EIGENVALUE GIVEN IN ARRAY RE, IS DELIVERED IN THE CORRESPONDING COLUMN OF ARRAY VEC; THE REAL AND IMAGINARY PART OF AN EIGENVECTOR, CORRESPONDING TO THE FIRST MEMBER OF A NONREAL COMPLEX CONJUGATE PAIR OF EIGENVALUES GIVEN IN THE ARRAYS RE, IM, ARE DELIVERED IN THE TWO CONSECUTIVE COLUMNS OF ARRAY VEC CORRESPONDING TO THIS PAIR (THE EIGENVECTORS CORRESPONDING TO THE SECOND MEMBERS OF NONREAL COMPLEX CONJUGATE PAIRS ARE NOT DELIVERED, SINCE THEY ARE SIMPLY THE COMPLEX CONJUGATE OF THOSE CORRESPONDING TO THE FIRST MEMBER OF SUCH PAIRS); MOREOVER: COMEIG1 DELIVERS 0, PROVIDED THAT THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE COMEIG1 DELIVERS K, THE NUMBER OF EIGENVALUES AND EIGENVECTORS NOT CALCULATED. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 11 PROCEDURES USED: EQILBR = CP34173, TFMREAHES = CP34170, BAKREAHES2 = CP34172, BAKLBR = CP34174, REAVECHES = CP34181, COMVALQRI = CP34190, COMVECHES = CP34191, COMSCL = CP34193. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: N * N + 5N. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE GIVEN MATRIX IS EQUILIBRATED BY CALLING EQILBR (SEE SECTION 3.2.1.1.1) AND TRANSFORMED TO A SIMILAR UPPER-HESSENBERG MATRIX BY CALLING TFMREAHES (SEE SECTION 3.2.1.2.1.2). THE EIGENVALUES ARE THEN CALCULATED BY CALLING COMVALQRI, WHICH USES DOUBLE QR ITERATION (SEE SECTION 3.3.1.2.1). FURTHERMORE, TO FIND THE EIGENVECTORS WILKINSON'S DEVICE IS FIRST APPLIED [2, P.328 AND 628]. SUBSEQUENTLY THE EIGENVECTORS OF THE UPPER-HESSENBERG MATRIX ARE COMPUTED BY CALLING REAVECHES FOR THE REAL EIGENVALUES AND COMVECHES FOR THE OTHERS (SECTION 3.3.1.2.1.) THE COMPUTED VECTORS ARE THEN BACK-TRANSFORMED TO THE CORRESPONDING EIGENVECTORS OF THE GIVEN MATRIX BY CALLING BAKREAHES2 AND BAKLBR (SEE SECTIONS 3.2.1.2.1.2 AND 3.2.1.1.1). FINALLY THE APPROXIMATE EIGENVECTORS ARE NORMALIZED BY CALLING COMSCL (SEE SECTION 1.1.9) SUCH THAT, IN EACH EIGENVECTOR, AN ELEMENT OF MAXIMUM MODULUS EQUALS 1. FOR FURTHER DETAILS SEE THE GIVEN REFERENCES. REFERENCES: [1]. T.J. DEKKER AND W. HOFFMANN. ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2. MC TRACT 23, 1968, MATH. CENTR., AMSTERDAM. [2]. J.H. WILKINSON. THE ALGEBRAIC EIGENVALUE PROBLEM. CLARENDON PRESS, OXFORD, 1965. [3]. J.G. FRANCIS. THE QR TRANSFORMATION, PARTS 1 AND 2. COMP. J. 4 (1961), 265 - 271 AND 332 - 345. [4]. J.M. VARAH. EIGENVECTORS OF A REAL MATRIX BY INVERSE ITERATION. STANFORD UNIVERSITY, TECH. REP. NO. CS 34, 1966. 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 12 EXAMPLE OF USE: IN THIS SECTION WE ONLY GIVE AN EXAMPLE OF USE OF THE PROCEDURES REAEIG3 AND COMEIGVAL, BECAUSE A CALL OF THE OTHER PROCEDURES IS ALMOST SIMILAR. THE EIGENVALUES AND CORRESPONDING EIGENVECTORS OF A MATRIX, STORED IN ARRAY A, WITH A[I,J]:= "IF" I=1 "THEN" 1 "ELSE" 1 / (I + J - 1), MAY BE OBTAINED BY THE PROCEDURE REAEIG3 IN THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I, J, M; "ARRAY" A, VEC[1:4,1:4], EM[0:5], VAL[1:4]; "FOR" I:= 1, 2, 3, 4 "DO" "FOR" J:= 1, 2, 3, 4 "DO" A[I,J]:= "IF" I = 1 "THEN" 1 "ELSE" 1 / ( I + J - 1); EM[0]:= "-14; EM[2]:= "-13; EM[4]:= 40; M:= REAEIG3(A, 4, EM, VAL, VEC); OUTPUT(61, "("D, /")", M); "FOR" I:= M + 1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61, "("/, 2(+.13D"+2D, 2B), /, 3(21B, +.13D"+2D, /)")", VAL[I], VEC[1,I], VEC[2,I], VEC[3,I], VEC[4,I]); OUTPUT(61, "("/, 2(.2D"+2D, /), ZD")", EM[1], EM[3], EM[5]) "END" THE PROGRAM DELIVERS(THE RESULTS ARE CORRECT UP TO TWELVE DIGITS): THE NUMBER OF NOT CALCULATED EIGENVALUES: 0 THE EIGENVALUES AND CORRESPONDING EIGENVECTORS: +.1886632138548"+01 +.1000000000000"+01 +.3942239850770"+00 +.2773202862566"+00 +.2150878672143"+00 -.1980145931103"+00 +.1000000000000"+01 -.7388484093937"+00 -.3116238593839"+00 -.1475423243327"+00 -.1228293686543"-01 -.4634736456357"+00 +.1000000000000"+01 -.1542548002737"+00 -.3765787365625"+00 -.1441323817331"-03 +.1095712655340"+00 -.6208405341138"+00 +.1000000000000"+01 -.4887465241876"+00 EM[1] = .40"+01 EM[3] = .15"-14 EM[5] = 5 . 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 13 THE COMPLEX EIGENVALUES OF A MATRIX STORED IN ARRAY A WITH N = 3 AND THE ROWS (8, -1, -5), (-4, 4, -2) AND (18, -5, -7), MAY BE OBTAINED BY THE PROCEDURE COMEIGVAL IN THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I, M; "ARRAY" A[1:3,1:3], EM[0:5], RE, IM[1:3]; EM[0]:= "-14; EM[2]:= "-13; EM[4]:= 30; A[1,1]:= 8; A[1,2]:= -1; A[1,3]:= -5; A[2,1]:= -4; A[2,2]:= 4; A[2,3]:= -2; A[3,1]:= 18; A[3,2]:= -5; A[3,3]:= -7; M:= COMEIGVAL(A, 3, EM, RE, IM); OUTPUT(61, "("D, /")", M); "FOR" I:= M + 1 "STEP" 1 "UNTIL" 3 "DO" OUTPUT(61, "("2(+.13D"+2D, 2B), /")", RE[I], IM[I]); OUTPUT(61, "("/, 2(.2D"+2D, /), ZD")", EM[1], EM[3], EM[5]) "END" THE PROGRAM DELIVERS(THE RESULTS ARE CORRECT UP TO TWELVE DIGITS): THE NUMBER OF NOT CALCULATED EIGENVALUES: 0 THE EIGENVALUES: +.2000000000000"+01 +.4000000000000"+01 +.2000000000000"+01 -.4000000000000"+01 +.9999999999998"+00 +.0000000000000"+00 THE ARRAY EM: EM[1] = .30"+02 EM[3] = .78"-17 EM[5] = 6 . SOURCE TEXTS: 0"CODE" 34182; "COMMENT" MCA 2412; "INTEGER" "PROCEDURE" REAEIGVAL(A, N, EM, VAL); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL; "BEGIN" "INTEGER" I, J; "REAL" R; "ARRAY" D[1:N]; "INTEGER" "ARRAY" INT, INT0[1:N]; EQILBR(A, N, EM, D, INT0); TFMREAHES(A, N, EM, INT); J:= REAEIGVAL:= REAVALQRI(A, N, EM, VAL); "FOR" I:= J + 1 "STEP" 1 "UNTIL" N "DO" "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" VAL[J] > VAL[I] "THEN" "BEGIN" R:= VAL[I]; VAL[I]:= VAL[J]; VAL[J]:= R "END" "END" "END" REAEIGVAL 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 14 ; "EOP" 0"CODE" 34184; "COMMENT" MCA 2414; "INTEGER" "PROCEDURE" REAEIG1(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "BEGIN" "INTEGER" I, K, MAX, J, L; "REAL" RESIDU, R, MACHTOL; "ARRAY" D, V[1:N], B[1:N,1:N]; "INTEGER" "ARRAY" INT, INT0[1:N]; RESIDU:= 0; MAX:= 0; EQILBR(A, N, EM, D, INT0); TFMREAHES(A, N, EM, INT); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "FOR" J:= ("IF" I = 1 "THEN" 1 "ELSE" I - 1) "STEP" 1 "UNTIL" N "DO" B[I,J]:= A[I,J]; K:= REAEIG1:= REAVALQRI(B, N, EM, VAL); "FOR" I:= K + 1 "STEP" 1 "UNTIL" N "DO" "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" VAL[J] > VAL[I] "THEN" "BEGIN" R:= VAL[I]; VAL[I]:= VAL[J]; VAL[J]:= R "END" "END"; MACHTOL:= EM[0] * EM[1]; "FOR" L:= K + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" L > 1 "THEN" "BEGIN" "IF" VAL[L - 1] - VAL[L] < MACHTOL "THEN" VAL[L]:= VAL[L - 1] - MACHTOL "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "FOR" J:= ("IF" I = 1 "THEN" 1 "ELSE" I - 1) "STEP" 1 "UNTIL" N "DO" B[I,J]:= A[I,J]; REAVECHES(B, N, VAL[L], EM, V); "IF" EM[7] > RESIDU "THEN" RESIDU:= EM[7]; "IF" EM[9] > MAX "THEN" MAX:= EM[9]; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" VEC[J,L]:= V[J] "END"; EM[7]:= RESIDU; EM[9]:= MAX; BAKREAHES2(A, N, K + 1, N, INT, VEC); BAKLBR(N, K + 1, N, D, INT0, VEC); REASCL(VEC, N, K + 1, N) "END" REAEIG1 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 15 ; "EOP" 0"CODE" 34187; "COMMENT" MCA 2417; "INTEGER" "PROCEDURE" REAEIG3(A, N, EM, VAL, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, VAL, VEC; "BEGIN" "INTEGER" I; "REAL" S; "INTEGER" "ARRAY" INT, INT0[1:N]; "ARRAY" D[1:N]; EQILBR(A, N, EM, D, INT0); TFMREAHES(A, N, EM, INT); I:= REAEIG3:= REAQRI(A, N, EM, VAL, VEC); "IF" I = 0 "THEN" "BEGIN" BAKREAHES2(A, N, 1, N, INT, VEC); BAKLBR(N, 1, N, D, INT0, VEC); REASCL(VEC, N, 1, N) "END" "END" REAEIG3; "EOP" 0"CODE" 34192; "COMMENT" MCA 2422; "INTEGER" "PROCEDURE" COMEIGVAL(A, N, EM, RE, IM); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM; "BEGIN" "INTEGER" "ARRAY" INT, INT0[1:N]; "ARRAY" D[1:N]; EQILBR(A, N, EM, D, INT0); TFMREAHES(A, N, EM, INT); COMEIGVAL:= COMVALQRI(A, N, EM, RE, IM) "END" COMEIGVAL 1SECTION 3.3.1.2.2 (JULY 1974) PAGE 16 ; "EOP" 0"CODE" 34194; "COMMENT" MCA 2424; "INTEGER" "PROCEDURE" COMEIG1(A, N, EM, RE, IM, VEC); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM, VEC; "BEGIN" "INTEGER" I, J, K, PJ, ITT; "REAL" X, Y, MAX, NEPS; "ARRAY" AB[1:N,1:N], D, U, V[1:N]; "INTEGER" "ARRAY" INT, INT0[1:N]; "PROCEDURE" TRANSFER; "BEGIN" "INTEGER" I, J; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "FOR" J:= ("IF" I = 1 "THEN" 1 "ELSE" I - 1) "STEP" 1 "UNTIL" N "DO" AB[I,J]:= A[I,J] "END" TRANSFER; EQILBR(A, N, EM, D, INT0); TFMREAHES(A, N, EM, INT); TRANSFER; K:= COMEIG1:= COMVALQRI(AB, N, EM, RE, IM); NEPS:= EM[0] * EM[1]; MAX:= 0; ITT:= 0; "FOR" I:= K + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" X:= RE[I]; Y:= IM[I]; PJ:= 0; AGAIN: "FOR" J:= K + 1 "STEP" 1 "UNTIL" I - 1 "DO" "BEGIN" "IF" ((X - RE[J]) ** 2 + (Y - IM[J]) ** 2 <= NEPS ** 2) "THEN" "BEGIN" "IF" PJ = J "THEN" NEPS:= EM[2] * EM[1] "ELSE" PJ:= J; X:= X + 2 * NEPS; "GOTO" AGAIN "END" "END"; RE[I]:= X; TRANSFER; "IF" Y ^= 0 "THEN" "BEGIN" COMVECHES(AB, N, RE[I], IM[I], EM, U, V); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" VEC[J,I]:= U[J]; I:= I + 1; RE[I]:= X "END" "ELSE" REAVECHES(AB, N, X, EM, V); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" VEC[J,I]:= V[J]; "IF" EM[7] > MAX "THEN" MAX:= EM[7]; ITT:= "IF" ITT > EM[9] "THEN" ITT "ELSE" EM[9] "END"; EM[7]:= MAX; EM[9]:= ITT; BAKREAHES2(A, N, K + 1, N, INT, VEC); BAKLBR(N, K + 1, N, D, INT0, VEC); COMSCL(VEC, N, K + 1, N, IM) "END" COMEIG1; "EOP" 1SECTION 3.3.2.1 (JULY 1974) PAGE 1 AUTHOR : C.G. VAN DER LAAN. CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN. INSTITUTE : MATHEMATICAL CENTRE. RECEIVED: 730917. BRIEF DESCRIPTION: THIS SECTION CONTAINS FOUR PROCEDURES FOR CALCULATING THE EIGENVALUES OR THE EIGENVALUES AND EIGENVECTORS OF COMPLEX HERMITIAN MATRICES. EIGVALHRM CALCULATES THE EIGENVALUES OF A HERMITIAN MATRIX. EIGHRM CALCULATES THE EIGENVALUES AND EIGENVECTORS OF A HERMITIAN MATRIX. QRIVALHRM CALCULATES THE EIGENVALUES OF A HERMITIAN MATRIX. QRIHRM CALCULATES THE EIGENVALUES AND EIGENVECTORS OF A HERMITIAN MATRIX. WHEN A SMALL NUMBER OF EIGENVALUES OR EIGENVALUES AND EIGENVECTORS IS REQUIRED, THE USE OF EIGVALHRM OR EIGHRM IS RECOMMANDED; WHEN MORE THAN, SAY, 25 PERCENT OF THE EIGENSYSTEM IS REQUIRED. THE PROCEDURES QRIVALHRM OR QRIHRM ARE TO BE USED. 1SECTION 3.3.2.1 (JULY 1974) PAGE 2 SUBSECTION: EIGVALHRM. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "PROCEDURE" EIGVALHRM(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "CODE" 34368; THE MEANING OF THE FORMAL PARAMETERS IS : A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J); THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J); THE ELEMENTS AF A ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; EIGVALHRM CALCULATES THE LARGEST NUMVAL EIGENVALUES OF THE HERMITIAN MATRIX; VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: IN ARRAY VAL THE LARGEST NUMVAL EIGENVALUES ARE DELIVERED IN MONOTONICALLY NONINCREASING ORDER; EM: ; "ARRAY" EM[0:3]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE EIGENVALUES; MORE PRECISELY: THE TOLERANCE FOR EACH EIGENVALUE LAMBDA, IS ABS(LAMBDA)*EM[2]+EM[1]*EM[0]; EXIT: EM[1]: AN ESTIMATE OF A NORM OF THE ORIGINAL MATRIX; EM[3]: THE NUMBER OF ITERATIONS PERFORMED. PROCEDURES USED: HSHHRMTRIVAL = CP34364, VALSYMTRI = CP34151. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF ORDER N AND N - 1 RESPECTIVELY ARE DECLARED RUNNING TIME: PROPORTIONAL TO N CUBED. 1SECTION 3.3.2.1 (JULY 1974) PAGE 3 LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE QRIHRM (THIS SECTION). EXAMPLE OF USE: SEE EIGHRM (THIS SECTION). SUBSECTION: EIGHRM. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "PROCEDURE" EIGHRM(A, N, NUMVAL, VAL, VECR, VECI, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VECR, VECI, EM; "CODE" 34369; THE MEANING OF THE FORMAL PARAMETERS IS : A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J); THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J); THE ELEMENTS AF A ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; NUMVAL: ; EIGHRM CALCULATES THE LARGEST NUMVAL EIGENVALUES OF THE HERMITIAN MATRIX; VAL: ; "ARRAY" VAL[1:NUMVAL]; EXIT: IN ARRAY VAL THE LARGEST NUMVAL EIGENVALUES ARE DELIVERED IN MONOTONICALLY NONINCREASING ORDER; VECR,VECI: ; "ARRAY" VECR,VECI[1:N,1:NUMVAL]; EXIT: THE CALCULATED EIGENVECTORS; THE COMPLEX EIGENVECTOR WITH REAL PART VECR[1:N,I] AND IMAGINARY PART VECI[1:N,I] CORRESPONDS TO THE EIGENVALUE VAL[I], I=1,...,NUMVAL; 1SECTION 3.3.2.1 (JULY 1974) PAGE 4 EM: ; "ARRAY" EM[0:9]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE EIGENVALUES; MORE PRECISELY: THE TOLERANCE FOR EACH EIGENVALUE LAMBDA, IS ABS(LAMBDA)*EM[2]+EM[1]*EM[0]; EM[4]: THE ORTHOGONALIZATION PARAMETER (E.G. .01); EM[6]: THE TOLERANCE FOR THE EIGENVECTORS; EM[8]: THE MAXIMUM NUMBER OF INVERSE ITERATIONS ALLOWED FOR THE CALCULATION OF EACH EIGENVECTOR; EXIT: EM[1]: AN ESTIMATE OF A NORM OF THE ORIGINAL MATRIX; EM[3]: THE NUMBER OF ITERATIONS PERFORMED; EM[5]: THE NUMBER OF EIGENVECTORS INVOLVED IN THE LAST GRAM-SCHMIDT ORTHOGONALIZATION; EM[7]: THE MAXIMUM EUCLIDEAN NORM OF THE RESIDUES OF THE CALCULATED EIGENVECTORS; EM[9]: THE LARGEST NUMBER OF INVERSE ITERATIONS PERFORMED FOR THE CALCULATION OF SOME EIGENVECTOR; IF, HOWEVER, FOR SOME CALCULATED EIGENVECTOR, THE EUCLIDEAN NORM OF THE RESIDUES REMAINS GREATER THAN EM[1]*EM[6], THEN EM[9]:=EM[8]+1. PROCEDURES USED: HSHHRMTRI = CP34363, VALSYMTRI = CP34151, VECSYMTRI = CP34152, BAKHRMTRI = CP34365. REQUIRED CENTRAL MEMORY: THREE AUXILIARY ARRAYS OF ORDER N-1 AND TWO OF ORDER N ARE DECLARED RUNNING TIME: PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE QRIHRM (THIS SECTION). 1SECTION 3.3.2.1 (JULY 1974) PAGE 5 EXAMPLE OF USE: LET EIGHRM CALCULATE THE LARGEST EIGENVALUE AND THE CORRESPONDING EIGENVECTOR OF THE FOLLOWING MATRIX: (SEE GREGORY AND KARNEY, CHAPTER 6, EXAMPLE 6.6) 3 1 0 +2I 1 3 -2I 0 0 +2I 1 1 -2I 0 1 1 THE EIGENVECTORS ARE NORMALIZED BY THE PROCEDURE SCLCOM (SEE SECTION 1.2.11. ). "BEGIN" "COMMENT" GREGORY AND KARNEY,CHAPTER 6, EXAMPLE 6.6; "REAL" "ARRAY" A[1:4,1:4],VAL[1:1],VECR,VECI[1:4,1:1],EM[0:9]; "INTEGER" I; INIMAT(1,4,1,4,A,0); A[1,1]:=A[2,2]:=3; A[1,2]:=A[3,3]:=A[3,4]:=A[4,4]:=1; A[3,2]:=2;A[4,1]:=-2; EM[0]:=5"-14;EM[2]:="-12; EM[4]:=.01;EM[6]:="-10;EM[8]:=5; EIGHRM(A,4,1,VAL,VECR,VECI,EM); SCLCOM(VECR,VECI,4,1,1); OUTPUT(61,"(""("LARGEST EIGENVALUE: ")",N/")",VAL[1]); OUTPUT(61,"(""("CORRESPONDING EIGENVECTOR:")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("+D.D,+D.DDD,"("*I")",/")",VECR[I,1],VECI[I,1]); "FOR" I:=1,3,5,7,9 "DO" OUTPUT(61,"("/,"("EM[")",D,"("]: ")",+D.DDD"+DD")",I,EM[I]); "END" DELIVERS: LARGEST EIGENVALUE: +4.8284271247462"000 CORRESPONDING EIGENVECTOR: +1.0+0.000*I +1.0+0.000*I -0.0+0.414*I +0.0-0.414*I EM[1]: +6.000"+00 EM[3]: +1.800"+01 EM[5]: +1.000"+00 EM[7]: +5.303"-14 EM[9]: +1.000"+00 1SECTION 3.3.2.1 (JULY 1974) PAGE 6 SUBSECTION: QRIVALHRM. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "INTEGER" "PROCEDURE" QRIVALHRM(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "CODE" 34370; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J); THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J); THE ELEMENTS AF A ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE CALCULATED EIGENVALUES; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE QR ITERATION; EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; EXIT: EM[1]: AN ESTIMATE OF A NORM OF THE ORIGINAL MATRIX; EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[5]:= EM[4]+1 IN THE CASE QRIVALHRM^=0; QRIVALHRM:=0, PROVIDED THE QR ITERATION IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, QRIVALHRM:=THE NUMBER OF EIGENVALUES, K, NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF VAL ARE APPROXIMATE EIGENVALUES OF THE ORIGINAL HERMITEAN MATRIX. PROCEDURES USED: HSHHRMTRIVAL = CP34364, QRIVALSYMTRI = CP34160. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF ORDER N ARE DECLARED. RUNNING TIME: PROPORTIONAL TO N CUBED. 1SECTION 3.3.2.1 (JULY 1974) PAGE 7 LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE QRIHRM (THIS SECTION). EXAMPLE OF USE: SEE QRIHRM (THIS SECTION). SUBSECTION: QRIHRM. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "INTEGER" "PROCEDURE" QRIHRM(A, N, VAL, VR, VI, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, VR, VI, EM; "CODE" 34371; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE REAL PART OF THE UPPER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE UPPER TRIANGULAR PART OF A (THE ELEMENTS A[I,J], I<=J); THE IMAGINARY PART OF THE STRICT LOWER TRIANGLE OF THE HERMITIAN MATRIX MUST BE GIVEN IN THE STRICT LOWER PART OF A (THE ELEMENTS A[I,J],I>J); THE ELEMENTS AF A ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE CALCULATED EIGENVALUES; VR,VI: ; "ARRAY" VR,VI[1:N,1:N]; EXIT: THE CALCULATED EIGENVECTORS; THE COMPLEX EIGENVECTOR WITH REAL PART VR[1:N,I] AND IMAGINARY PART VI[1:N,I] CORRESPONDS TO THE EIGENVALUE VAL[I], I=1,...,N; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE QR ITERATION; (E.G. THE MACHINE PRECISION); EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; (E.G. 10 * N); EXIT: EM[1]: AN ESTIMATE OF A NORM OF THE ORIGINAL MATRIX; EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[5]:=EM[4]+1 IN THE CASE QRIHRM^=0; 1SECTION 3.3.2.1 (JULY 1974) PAGE 8 QRIHRM:=0, PROVIDED THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, QRIHRM:= THE NUMBER OF EIGENVALUES, K, NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF VAL ARE APPROXIMATE EIGENVALUES AND THE COLUMNS OF THE ARRAYS VR,VI[1:N,N-K:N] ARE APPROXIMATE EIGENVECTORS OF THE ORIGINAL HERMITEAN MATRIX . PROCEDURES USED: HSHHRMTRI = CP34363, QRISYMTRI = CP34161, BAKHRMTRI = CP34365. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF ORDER N - 1 AND TWO OF ORDER N ARE DECLARED RUNNING TIME: PROPORTIONAL TO N CUBED. LANGUAGE: ALGOL 60. THE FOLLOWING HOLDS FOR THE FOUR PROCEDURES OF THIS SECTION: METHOD AND PERFORMANCE: FOR THE TRANSFORMATION OF THE GIVEN HERMITIAN MATRIX INTO A REAL SYMMETRIC TRIDIAGONAL MATRIX, AND FOR THE CORRESPONDING BACK TRANSFORMATION, PROCEDURES OF SECTION 3.2.1.2.2.1. ARE USED. FOR THE CALCULATION OF THE EIGENVALUES AND EIGENVECTORS OF THE RESULTING SYMMETRIC TRIDIAGONAL MATRIX, PROCEDURES OF SECTION 3.3.1.1.1. ARE USED. EXAMPLE OF USE: QRIHRM CALCULATES THE EIGENVALUES AND EIGENVECTORS OF THE FOLLOWING MATRIX: (SEE GREGORY AND KARNEY, CHAPTER 6, EXAMPLE 6.6) 3 1 0 +2I 1 3 -2I 0 0 +2I 1 1 -2I 0 1 1 THE EIGENVECTORS ARE NORMALIZED BY THE PROCEDURE SCLCOM (SEE SECTION 1.2.11. ). ONLY THE EIGENVECTORS CORRESPONDING TO VAL[2] AND VAL[3] ARE PRINTED BY THE FOLLOWING PROGRAM: 1SECTION 3.3.2.1 (JULY 1974) PAGE 9 "BEGIN" "COMMENT" GREGORY AND KARNEY,CHAPTER 6, EXAMPLE 6.6; "REAL" "ARRAY" A,VR,VI[1:4,1:4],VAL[1:4],EM[0:5];"INTEGER" I; INIMAT(1,4,1,4,A,0); A[1,1]:=A[2,2]:=3; A[3,2]:=2;A[4,1]:=-2; A[1,2]:=A[3,3]:=A[3,4]:=A[4,4]:=1; EM[0]:=EM[2]:=5"-14;EM[4]:=20; OUTPUT(61,"(""("QRIHRM: ")",D/")",QRIHRM(A,4,VAL,VR,VI,EM)); SCLCOM(VR,VI,4,2,3); OUTPUT(61,"(""("EIGENVALUES: ")"")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("/,"("VAL[")",D,"("]: ")", +D.3DBB")",I,VAL[I]); OUTPUT(61,"("/,"("EIGENVECTORS CORRESPONDING TO")",/, "(" VAL[2] , VAL[3] ")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("+D,+D,"("*I , ")",+D,+D,"("*I")",/")", VR[I,2],VI[I,2],VR[I,3],VI[I,3]); "FOR" I:=1,3,5 "DO" OUTPUT(61,"("/,"("EM[")",D,"("]: ")",+D.DDD"+DD")",I,EM[I]) "END" OUTPUT: QRIHRM: 0 EIGENVALUES: VAL[1]: +4.828 VAL[2]: +4.000 VAL[3]: -0.000 VAL[4]: -0.828 EIGENVECTORS CORRESPONDING TO VAL[2] , VAL[3] +1+0*I , +0-1*I -1+0*I , +0+1*I +0-1*I , +1+0*I +0-1*I , +1+0*I EM[1]: +6.000"+00 EM[3]: +3.804"-22 EM[5]: +6.000"+00 SOURCE TEXT(S) : 0"CODE" 34368; "PROCEDURE" EIGVALHRM(A, N, NUMVAL, VAL, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" D[1:N], BB[1:N - 1]; HSHHRMTRIVAL(A, N, D, BB, EM); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM) "END" EIGVALHRM 1SECTION 3.3.2.1 (JULY 1974) PAGE 10 ; "EOP" 0"CODE" 34369; "PROCEDURE" EIGHRM(A, N, NUMVAL, VAL, VECR, VECI, EM); "VALUE" N, NUMVAL; "INTEGER" N, NUMVAL; "ARRAY" A, VAL, VECR, VECI, EM; "BEGIN" "ARRAY" BB, TR, TI[1:N - 1], D, B[1:N]; HSHHRMTRI(A, N, D, B, BB, EM, TR, TI); VALSYMTRI(D, BB, N, 1, NUMVAL, VAL, EM); B[N]:= 0; VECSYMTRI(D, B, N, 1, NUMVAL, VAL, VECR, EM); BAKHRMTRI(A, N, 1, NUMVAL, VECR, VECI, TR, TI) "END" EIGHRM; "EOP" 0"CODE" 34370; "INTEGER" "PROCEDURE" QRIVALHRM(A, N, VAL, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B,BB[1:N]; "INTEGER" I; HSHHRMTRIVAL(A, N, VAL, BB, EM); B[N]:=BB[N]:= 0; "FOR" I:=1 "STEP" 1 "UNTIL" N-1 "DO" B[I]:=SQRT(BB[I]); QRIVALHRM:=QRIVALSYMTRI(VAL, BB, N, EM) "END" QRIVALHRM; "EOP" 0"CODE" 34371; "INTEGER" "PROCEDURE" QRIHRM(A, N, VAL, VR, VI, EM); "VALUE" N; "INTEGER" N; "ARRAY" A, VAL, VR, VI, EM; "BEGIN" "INTEGER" I, J; "ARRAY" B, BB[1:N], TR, TI[1:N - 1]; HSHHRMTRI(A, N, VAL, B, BB, EM, TR, TI); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" VR[I,I]:= 1; "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" VR[I,J]:= VR[J,I]:= 0 "END"; B[N]:= BB[N]:= 0; I:= QRIHRM:= QRISYMTRI(VR, N, VAL, B, BB, EM); BAKHRMTRI(A, N, I+1, N, VR, VI, TR, TI); "END" QRIHRM; "EOP" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 1 AUTHOR : C.G. VAN DER LAAN. CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN. INSTITUTE : MATHEMATICAL CENTRE. RECEIVED: 731016. BRIEF DESCRIPTION : THIS SECTION CONTAINS THE PROCEDURES VALQRICOM AND QRICOM. VALQRICOM CALCULATES THE EIGENVALUES OF A COMPLEX UPPER-HESSENBERG MATRIX WITH A REAL SUBDIAGONAL. QRICOM CALCULATES THE EIGENVECTORS AS WELL. KEYWORDS : EIGENVALUES, EIGENVECTORS, COMPLEX UPPER-HESSENBERG MATRIX, QR-ITERATION. 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 2 SUBSECTION: VALQRICOM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "INTEGER" "PROCEDURE" VALQRICOM(A1, A2, B, N, EM, VAL1, VAL2); "VALUE" N; "INTEGER" N; "ARRAY" A1, A2, B, EM, VAL1, VAL2; "CODE" 34372; THE MEANING OF THE FORMAL PARAMETERS IS: A1,A2: ; "ARRAY" A1,A2[1:N,1:N]; ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE UPPER TRIANGLE OF THE UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE CORRESPONDING PARTS OF THE ARRAYS A1 AND A2; THE ELEMENTS IN THE UPPER TRIANGLE OF THE ARRAYS A1 AND A2 ARE ALTERED; B: ; "ARRAY" B[1:N-1]; ENTRY: THE REAL SUBDIAGONAL OF THE UPPER-HESSENBERG MATRIX; THE ELEMENTS OF THE ARRAY B ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[1]: AN ESTIMATE OF THE NORM OF THE UPPER-HESSENBERG MATRIX (E.G. THE SUM OF THE INFINITY NORMS OF THE REAL AND IMAGINARY PARTS OF THE MATRIX); EM[2]: THE RELATIVE TOLERANCE FOR THE QR-ITERATION; (E.G. THE MACHINE PRECISION); EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; (E.G. 10 * N); EXIT: EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[5]:=EM[4]+1 IN THE CASE VALQRICOM^=0; VAL1,VAL2: ; "ARRAY" VAL1,VAL2[1:N]; EXIT: THE REAL PART AND THE IMAGINARY PART OF THE CALCULATED EIGENVALUES ARE DELIVERED IN THE ARRAYS VAL1 AND VAL2, RESPECTIVELY; VALQRICOM:=0, PROVIDED THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, VALQRICOM:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF THE ARRAYS VAL1 AND VAL2 ARE APPROXIMATE EIGENVALUES OF THE UPPER-HESSENBERG MATRIX. 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 3 PROCEDURES USED: COMKWD = CP34345, ROTCOMROW = CP34358, ROTCOMCOL = CP34357, COMCOLCST = CP34352. RUNNING TIME: PROPORTIONAL TO N**2 * NUMBER OF ITERATIONS. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE QRICOM (THIS SECTION). EXAMPLE OF USE: AS A FORMAL TEST OF THE PROCEDURE VALQRICOM THE ZEROS OF THE POLYNOMIAL X**4 + (4+2*I)* X**3 + (5+6*I)* X**2 + (2+6*I)* X + 2*I ARE OBTAINED BY MEANS OF THE CALCULATION OF THE EIGENVALUES OF THE FOLLOWING COMPANION MATRIX: (SEE WILKINSON AND REINSCH, 1971, CONTRIBUTION II/15) -4-2*I -5-6*I -2-6*I -2*I 1 0 0 0 0 1 0 0 0 0 1 0 "BEGIN" "REAL" "ARRAY" A1,A2[1:4,1:4],B[1:3],EM[0:5],VAL1,VAL2[1:4]; "INTEGER" I; INIMAT(1,4,1,4,A1,0);INIMAT(1,4,1,4,A2,0); A1[1,1]:=-4;A1[1,2]:=-5;A1[1,3]:=A2[1,1]:=A2[1,4]:=-2; A2[1,2]:=A2[1,3]:=-6; B[1]:=B[2]:=B[3]:=1; EM[0]:=5"-14;EM[1]:=27;EM[2]:="-12;EM[4]:=15; OUTPUT(61,"(""("VALQRICOM: ")",D/")", VALQRICOM(A1,A2,B,4,EM,VAL1,VAL2)); OUTPUT(61,"(""("EIGENVALUES:")",/,"("REAL PART")",14B, "("IMAGINARY PART")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("N,N/")",VAL1[I],VAL2[I]); OUTPUT(61,"(""("EM[3]: ")",D.D"+DD/,"("EM[5]: ")",3D")", EM[3],EM[5]) "END" OUTPUT: VALQRICOM: 0 EIGENVALUES: REAL PART IMAGINARY PART -1.0000001920467"+000 -1.0000000831019"+000 -9.9999980795324"-001 -9.9999991689805"-001 -1.0000000047492"+000 +1.6824958523393"-007 -9.9999999525076"-001 -1.6824956397352"-007 EM[3]: 3.2"-14 EM[5]: 010 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 4 SUBSECTION: QRICOM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "INTEGER" "PROCEDURE" QRICOM(A1,A2,B,N,EM,VAL1,VAL2,VEC1,VEC2); "VALUE" N; "INTEGER" N; "ARRAY" A1, A2, B, EM, VAL1, VAL2, VEC1, VEC2; "CODE" 34373; THE MEANING OF THE FORMAL PARAMETERS IS: A1,A2: ; "ARRAY" A1,A2[1:N,1:N]; ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE UPPER TRIANGLE OF THE UPPER-HESSENBERG MATRIX MUST BE GIVEN IN THE CORRESPONDING PARTS OF THE ARRAYS A1 AND A2; THE ELEMENTS IN THE UPPER TRIANGLE OF THE ARRAYS A1 AND A2 ARE ALTERED; B: ; "ARRAY" B[1:N-1]; ENTRY: THE REAL SUBDIAGONAL OF THE UPPER-HESSENBERG MATRIX; THE ELEMENTS OF THE ARRAY B ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[1]: AN ESTIMATE OF THE NORM OF THE UPPER-HESSENBERG MATRIX (E.G. THE SUM OF THE INFINITY NORMS OF THE REAL AND IMAGINARY PARTS OF THE MATRIX); EM[2]: THE RELATIVE TOLERANCE FOR THE QR-ITERATION; (E.G. THE MACHINE PRECISION); EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS; (E.G. 10 * N); EXIT: EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED; EM[5] THE NUMBER OF ITERATIONS PERFORMED; EM[5]:=EM[4]+1 IN THE CASE QRICOM^=0; VAL1,VAL2: ; "ARRAY" VAL1,VAL2[1:N]; EXIT: THE REAL PART AND THE IMAGINARY PART OF THE CALCULATED EIGENVALUES ARE DELIVERED IN THE ARRAYS VAL1 AND VAL2, RESPECTIVELY; VEC1,VEC2: ; "ARRAY" VEC1,VEC2[1:N,1:N]; EXIT: THE EIGENVECTORS OF THE UPPER-HESSENBERG MATRIX; THE EIGENVECTOR WITH REAL PART VEC1[1:N,J] AND IMAGINARY PART VEC2[1:N,J] CORRESPONDS TO THE EIGENVALUE VAL1[J] + VAL2[J] * I, J=1,...,N; 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 5 QRICOM:=0, PROVIDED THE PROCESS IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, QRICOM:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF THE ARRAYS VAL1 AND VAL2 ARE APPROXIMATE EIGENVALUES OF THE UPPER-HESSENBERG MATRIX, AND NO USEFUL EIGENVECTORS ARE DELIVERED. PROCEDURES USED: COMKWD = CP34345, ROTCOMROW = CP34358, ROTCOMCOL = CP34357, COMCOLCST = CP34352, COMROWCST = CP34353, MATVEC = CP34011, COMMATVEC = CP34354, COMDIV = CP34342. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF ORDER N ARE DECLARED. RUNNING TIME: PROPORTIONAL TO N**2 * NUMBER OF ITERATIONS. LANGUAGE: ALGOL 60. THE FOLLOWING HOLDS FOR BOTH PROCEDURES: METHOD AND PERFORMANCE: THE UPPER-HESSENBERG MATRIX IS TRANSFORMED BY MEANS OF FRANCIS' QR ITERATION ( FRANCIS, 1961, AND WILKINSON, 1965 ) INTO A COMPLEX UPPER TRIANGULAR MATRIX. THE EIGENVALUES ARE THE DIAGONAL ELEMENTS OF THE LATTER MATRIX. TO CALCULATE THE EIGENVECTORS WE FIRST SOLVE THE RESULTING TRIANGULAR SYSTEM OF LINEAR EQUATIONS AND SUBSEQUENTLY PERFORM THE CORRESPONDING BACKTRANSFORMATION. 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 6 EXAMPLE OF USE: QRICOM CALCULATES THE EIGENVALUES AND EIGENVECTORS OF THE FOLLOWING MATRIX: (SEE WILKINSON AND REINSCH, 1971, CONTRIBUTION II/15) -4-2*I -5-6*I -2-6*I -2*I 1 0 0 0 0 1 0 0 0 0 1 0 THE EIGENVECTORS ARE NORMALIZED BY THE PROCEDURE SCLCOM. (SEE SECTION 1.2.11.). ONLY THE EIGENVECTOR CORRESPONDING TO VAL1[1] + VAL2[1] * I IS PRINTED BY THE FOLLOWING PROGRAM. "BEGIN" "REAL" "ARRAY" A1,A2,VEC1,VEC2[1:4,1:4],B[1:3], EM[0:5],VAL1,VAL2[1:4]; "INTEGER" I; INIMAT(1,4,1,4,A1,0);INIMAT(1,4,1,4,A2,0); A1[1,1]:=-4;A1[1,2]:=-5;A1[1,3]:=A2[1,1]:=A2[1,4]:=-2; A2[1,2]:=A2[1,3]:=-6; B[1]:=B[2]:=B[3]:=1; EM[0]:=5"-14;EM[1]:=27;EM[2]:="-12;EM[4]:=15; OUTPUT(61,"(""("QRICOM: ")",D/")", QRICOM(A1,A2,B,4,EM,VAL1,VAL2,VEC1,VEC2)); OUTPUT(61,"(""("EIGENVALUES:")",/,"("REAL PART")",14B, "("IMAGINARY PART")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("N,N/")",VAL1[I],VAL2[I]); SCLCOM(VEC1,VEC2,4,1,4); OUTPUT(61,"(""("FIRST EIGENVECTOR:")",/,"("REAL PART")",14B, "("IMAGINARY PART")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("N,N/")",VEC1[I,1],VEC2[I,1]); OUTPUT(61,"(""("EM[3]: ")",D.D"+DD/,"("EM[5]: ")",3D")", EM[3],EM[5]) "END" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 7 OUTPUT: QRICOM: 0 EIGENVALUES: REAL PART IMAGINARY PART -9.9999980795324"-001 -9.9999991689805"-001 -1.0000001920467"+000 -1.0000000831019"+000 -1.0000000047492"+000 +1.6824958523393"-007 -9.9999999525076"-001 -1.6824956397352"-007 FIRST EIGENVECTOR: REAL PART IMAGINARY PART +1.0000000000000"+000 -1.7763568394003"-015 -5.0000004155098"-001 +5.0000009602339"-001 -5.4472417687634"-008 -5.0000013757436"-001 +2.5000014403510"-001 +2.5000006232645"-001 EM[3]: 3.2"-14 EM[5]: 010 REFERENCES: DEKKER, T.J. (1968), ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1, MATH. CENTRE TRACTS 22, MATHEMATISCH CENTRUM; DEKKER, T.J. AND W.HOFFMANN (1968), ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2, MATH. CENTRE TRACTS 23, MATHEMATISCH CENTRUM; FRANCIS, J.G.F. (1961), THE QR-TRANSFORMATION, PART 1 AND 2, COMP.J. 4, P.265-271 AND P.332-345; RUHE, A. (1966), EIGENVALUES OF A COMPLEX MATRIX BY THE QR METHOD. BIT, 6, P.350-358; WILKINSON, J.H. (1965), THE ALGEBRAIC EIGENVALUE PROBLEM, CLARENDOM PRESS, OXFORD; 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 8 SOURCE TEXT(S) : 0"CODE" 34372; "INTEGER" "PROCEDURE" VALQRICOM(A1, A2, B, N, EM, VAL1, VAL2); "VALUE" N; "INTEGER" N; "ARRAY" A1, A2, B, EM, VAL1, VAL2; "BEGIN" "INTEGER" M, NM1, I, I1, Q, Q1, MAX, COUNT; "REAL" R, Z1, Z2, DD1, DD2, CC, G1, G2, K1, K2, HC, A1NN, A2NN, AIJ1, AIJ2, AI1I, KAPPA, NUI, MUI1, MUI2, MUIM11, MUIM12, NUIM1, TOL; TOL:= EM[1] * EM[2]; MAX:= EM[4]; COUNT:= 0; R:= 0; M:= N; "IF" N > 1 "THEN" HC:= B[N - 1]; IN: NM1:= N - 1; "FOR" I:= N, I - 1 "WHILE" ("IF" I >= 1 "THEN" ABS(B[I]) > TOL "ELSE" "FALSE") "DO" Q:= I; "IF" Q > 1 "THEN" "BEGIN" "IF" ABS(B[Q - 1]) > R "THEN" R:= ABS(B[Q - 1]) "END"; "IF" Q = N "THEN" "BEGIN" VAL1[N]:= A1[N,N]; VAL2[N]:= A2[N,N]; N:= NM1; "IF" N > 1 "THEN" HC:= B[N - 1]; "END" "ELSE" "BEGIN" DD1:= A1[N,N]; DD2:= A2[N,N]; CC:= B[NM1]; COMKWD((A1[NM1,NM1] - DD1) / 2, (A2[NM1,NM1] - DD2) / 2, CC * A1[NM1,N], CC * A2[NM1,N], G1, G2, K1, K2); "IF" Q = NM1 "THEN" "BEGIN" VAL1[NM1]:= G1 + DD1; VAL2[NM1]:= G2 + DD2; VAL1[N]:= K1 + DD1; VAL2[N]:= K2 + DD2; N:= N - 2; "IF" N > 1 "THEN" HC:= B[N - 1]; "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" OUT; Z1:= K1 + DD1; Z2:= K2 + DD2; "IF" ABS(CC) > ABS(HC) "THEN" Z1:= Z1 + ABS(CC); HC:= CC / 2; I:= Q1:= Q + 1; AIJ1:= A1[Q,Q] - Z1; AIJ2:= A2[Q,Q] - Z2; AI1I:= B[Q]; KAPPA:= SQRT(AIJ1 ** 2 + AIJ2 ** 2 + AI1I ** 2); MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; NUI:= AI1I / KAPPA; A1[Q,Q]:= KAPPA; A2[Q,Q]:= 0; A1[Q1,Q1]:= A1[Q1,Q1] - Z1; A2[Q1,Q1]:= A2[Q1,Q1] - Z2; ROTCOMROW(Q1, N, Q, Q1, A1, A2, MUI1, MUI2, NUI); ROTCOMCOL(Q, Q, Q, Q1, A1, A2, MUI1, - MUI2, - NUI); A1[Q,Q]:= A1[Q,Q] + Z1; A2[Q,Q]:= A2[Q,Q] + Z2;"COMMENT" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 9 ; "FOR" I1:= Q1 + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" AIJ1:= A1[I,I]; AIJ2:= A2[I,I]; AI1I:= B[I]; KAPPA:= SQRT(AIJ1 ** 2 + AIJ2 ** 2 + AI1I ** 2); MUIM11:= MUI1; MUIM12:= MUI2; NUIM1:= NUI; MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; NUI:= AI1I / KAPPA; A1[I1,I1]:= A1[I1,I1] - Z1; A2[I1,I1]:= A2[I1,I1] - Z2; ROTCOMROW(I1, N, I, I1, A1, A2, MUI1, MUI2, NUI); A1[I,I]:= MUIM11 * KAPPA; A2[I,I]:= - MUIM12 * KAPPA; B[I - 1]:= NUIM1 * KAPPA; ROTCOMCOL(Q, I, I, I1, A1, A2, MUI1, - MUI2, - NUI); A1[I,I]:= A1[I,I] + Z1; A2[I,I]:= A2[I,I] + Z2; I:= I1; "END"; AIJ1:= A1[N,N]; AIJ2:= A2[N,N]; KAPPA:= SQRT(AIJ1 ** 2 + AIJ2 ** 2); "IF" ("IF" KAPPA < TOL "THEN" "TRUE" "ELSE" AIJ2 ** 2 <= EM[0] * AIJ1 ** 2) "THEN" "BEGIN" B[NM1]:= NUI * AIJ1; A1[N,N]:= AIJ1 * MUI1 + Z1; A2[N,N]:= - AIJ1 * MUI2 + Z2 "END" "ELSE" "BEGIN" B[NM1]:= NUI * KAPPA; A1NN:= MUI1 * KAPPA; A2NN:= - MUI2 * KAPPA; MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; COMCOLCST(Q, NM1, N, A1, A2, MUI1, MUI2); A1[N,N]:= MUI1 * A1NN - MUI2 * A2NN + Z1; A2[N,N]:= MUI1 * A2NN + MUI2 * A1NN + Z2; "END"; "END" "END"; "IF" N > 0 "THEN" "GOTO" IN; OUT: EM[3]:= R; EM[5]:= COUNT; VALQRICOM:= N; "END" VALQRICOM; "EOP" 0"CODE" 34373; "INTEGER" "PROCEDURE" QRICOM(A1, A2, B, N, EM, VAL1, VAL2, VEC1, VEC2); "VALUE" N; "INTEGER" N; "ARRAY" A1, A2, B, EM, VAL1, VAL2, VEC1, VEC2; "BEGIN" "INTEGER" M, NM1, I, I1, J, Q, Q1, MAX, COUNT; "REAL" R, Z1, Z2, DD1, DD2, CC, P1, P2, T1, T2, DELTA1, DELTA2, MV1, MV2, H, H1, H2, G1, G2, K1, K2, HC, AIJ12, AIJ22, A1NN, A2NN, AIJ1, AIJ2, AI1I, KAPPA, NUI, MUI1, MUI2, MUIM11, MUIM12, NUIM1, TOL, MACHTOL; "ARRAY" TF1, TF2[1:N];"COMMENT" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 10 ; TOL:= EM[1] * EM[2]; MACHTOL:= EM[0] * EM[1]; MAX:= EM[4]; COUNT:= 0; R:= 0; M:= N; "IF" N > 1 "THEN" HC:= B[N - 1]; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" VEC1[I,I]:= 1; VEC2[I,I]:= 0; "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" VEC1[I,J]:= VEC1[J,I]:= VEC2[I,J]:= VEC2[J,I]:= 0 "END"; IN: NM1:= N - 1; "FOR" I:= N, I - 1 "WHILE" ("IF" I >= 1 "THEN" ABS(B[I]) > TOL "ELSE" "FALSE") "DO" Q:= I; "IF" Q > 1 "THEN" "BEGIN" "IF" ABS(B[Q - 1]) > R "THEN" R:= ABS(B[Q - 1]) "END"; "IF" Q = N "THEN" "BEGIN" VAL1[N]:= A1[N,N]; VAL2[N]:= A2[N,N]; N:= NM1; "IF" N > 1 "THEN" HC:= B[N - 1]; "END" "ELSE" "BEGIN" DD1:= A1[N,N]; DD2:= A2[N,N]; CC:= B[NM1]; P1:= (A1[NM1,NM1] - DD1) * .5; P2:= (A2[NM1,NM1] - DD2) * .5; COMKWD(P1, P2, CC * A1[NM1,N], CC * A2[NM1,N], G1, G2, K1, K2); "IF" Q = NM1 "THEN" "BEGIN" A1[N,N]:= VAL1[N]:= G1 + DD1; A2[N,N]:= VAL2[N]:= G2 + DD2; A1[Q,Q]:= VAL1[Q]:= K1 + DD1; A2[Q,Q]:= VAL2[Q]:= K2 + DD2; KAPPA:= SQRT(K1 ** 2 + K2 ** 2 + CC ** 2); NUI:= CC / KAPPA; MUI1:= K1 / KAPPA; MUI2:= K2 / KAPPA; AIJ1:= A1[Q,N]; AIJ2:= A2[Q,N]; H1:= MUI1 ** 2 - MUI2 ** 2; H2:= 2 * MUI1 * MUI2; H:= - NUI * 2; A1[Q,N]:= H * (P1 * MUI1 + P2 * MUI2) - NUI * NUI * CC + AIJ1 * H1 + AIJ2 * H2; A2[Q,N]:= H * (P2 * MUI1 - P1 * MUI2) + AIJ2 * H1 - AIJ1 * H2; ROTCOMROW(Q + 2, M, Q, N, A1, A2, MUI1, MUI2, NUI); ROTCOMCOL(1, Q - 1, Q, N, A1, A2, MUI1, - MUI2, - NUI); ROTCOMCOL(1, M, Q, N, VEC1, VEC2, MUI1, - MUI2, - NUI); N:= N - 2; "IF" N > 1 "THEN" HC:= B[N - 1]; B[Q]:= 0 "END" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 11 "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" OUT; Z1:= K1 + DD1; Z2:= K2 + DD2; "IF" ABS(CC) > ABS(HC) "THEN" Z1:= Z1 + ABS(CC); HC:= CC / 2; Q1:= Q + 1; AIJ1:= A1[Q,Q] - Z1; AIJ2:= A2[Q,Q] - Z2; AI1I:= B[Q]; KAPPA:= SQRT(AIJ1 ** 2 + AIJ2 ** 2 + AI1I ** 2); MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; NUI:= AI1I / KAPPA; A1[Q,Q]:= KAPPA; A2[Q,Q]:= 0; A1[Q1,Q1]:= A1[Q1,Q1] - Z1; A2[Q1,Q1]:= A2[Q1,Q1] - Z2; ROTCOMROW(Q1, M, Q, Q1, A1, A2, MUI1, MUI2, NUI); ROTCOMCOL(1, Q, Q, Q1, A1, A2, MUI1, - MUI2, - NUI); A1[Q,Q]:= A1[Q,Q] + Z1; A2[Q,Q]:= A2[Q,Q] + Z2; ROTCOMCOL(1, M, Q, Q1, VEC1, VEC2, MUI1, - MUI2, - NUI); "FOR" I:= Q1 "STEP" 1 "UNTIL" NM1 "DO" "BEGIN" I1:= I + 1; AIJ1:= A1[I,I]; AIJ2:= A2[I,I]; AI1I:= B[I]; KAPPA:= SQRT(AIJ1 ** 2 + AIJ2 ** 2 + AI1I ** 2); MUIM11:= MUI1; MUIM12:= MUI2; NUIM1:= NUI; MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; NUI:= AI1I / KAPPA; A1[I1,I1]:= A1[I1,I1] - Z1; A2[I1,I1]:= A2[I1,I1] - Z2; ROTCOMROW(I1, M, I, I1, A1, A2, MUI1, MUI2, NUI); A1[I,I]:= MUIM11 * KAPPA; A2[I,I]:= - MUIM12 * KAPPA; B[I - 1]:= NUIM1 * KAPPA; ROTCOMCOL(1, I, I, I1, A1, A2, MUI1, - MUI2, - NUI); A1[I,I]:= A1[I,I] + Z1; A2[I,I]:= A2[I,I] + Z2; ROTCOMCOL(1, M, I, I1, VEC1, VEC2, MUI1, - MUI2, - NUI); "END"; "COMMENT" 1SECTION 3.3.2.2.1 (JULY 1974) PAGE 12 ; AIJ1:= A1[N,N]; AIJ2:= A2[N,N]; AIJ12:= AIJ1 ** 2; AIJ22:= AIJ2 ** 2; KAPPA:= SQRT(AIJ12 + AIJ22); "IF" ("IF" KAPPA < TOL "THEN" "TRUE" "ELSE" AIJ22 <= EM[0] * AIJ12) "THEN" "BEGIN" B[NM1]:= NUI * AIJ1; A1[N,N]:= AIJ1 * MUI1 + Z1; A2[N,N]:= - AIJ1 * MUI2 + Z2 "END" "ELSE" "BEGIN" B[NM1]:= NUI * KAPPA; A1NN:= MUI1 * KAPPA; A2NN:= - MUI2 * KAPPA; MUI1:= AIJ1 / KAPPA; MUI2:= AIJ2 / KAPPA; COMCOLCST(1, NM1, N, A1, A2, MUI1, MUI2); COMCOLCST(1, NM1, N, VEC1, VEC2, MUI1, MUI2); COMROWCST(N + 1, M, N, A1, A2, MUI1, - MUI2); COMCOLCST(N, M, N, VEC1, VEC2, MUI1, MUI2); A1[N,N]:= MUI1 * A1NN - MUI2 * A2NN + Z1; A2[N,N]:= MUI1 * A2NN + MUI2 * A1NN + Z2; "END"; "END"; "END"; "IF" N > 0 "THEN" "GOTO" IN; "FOR" J:= M "STEP" - 1 "UNTIL" 2 "DO" "BEGIN" TF1[J]:= 1; TF2[J]:= 0; T1:= A1[J,J]; T2:= A2[J,J]; "FOR" I:= J - 1 "STEP" - 1 "UNTIL" 1 "DO" "BEGIN" DELTA1:= T1 - A1[I,I]; DELTA2:= T2 - A2[I,I]; COMMATVEC(I + 1, J, I, A1, A2, TF1, TF2, MV1, MV2); "IF" ABS(DELTA1) < MACHTOL "AND" ABS(DELTA2) < MACHTOL "THEN" "BEGIN" TF1[I]:= MV1 / MACHTOL; TF2[I]:= MV2 / MACHTOL "END" "ELSE" COMDIV(MV1, MV2, DELTA1, DELTA2, TF1[I], TF2[I]); "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" COMMATVEC(1, J, I, VEC1, VEC2, TF1, TF2, VEC1[I,J], VEC2[I,J]); "END"; OUT: EM[3]:= R; EM[5]:= COUNT; QRICOM:= N; "END" QRICOM; "EOP" 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 1 AUTHOR : C.G. VAN DER LAAN. CONTRIBUTORS : H.FIOLET, C.G. VAN DER LAAN. INSTITUTE : MATHEMATICAL CENTRE. RECEIVED: 731016. BRIEF DESCRIPTION : THIS SECTION CONTAINS THE PROCEDURES EIGVALCOM AND EIGCOM. EIGVALCOM CALCULATES THE EIGENVALUES OF A COMPLEX MATRIX AND EIGCOM CALCULATES THE EIGENVECTORS AS WELL. KEYWORDS : EIGENVALUES, EIGENVECTORS, COMPLEX MATRICES, EQUILIBRATION, REDUCTION HESSENBERG FORM, HOUSEHOLDER TRANSFORMATION, QR-ITERATION. 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 2 SUBSECTION: EIGVALCOM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "INTEGER" "PROCEDURE" EIGVALCOM(AR, AI, N, EM, VALR, VALI); "VALUE" N; "INTEGER" N; "ARRAY" AR, AI, EM, VALR, VALI; "CODE" 34374; THE MEANING OF THE FORMAL PARAMETERS IS: AR,AI: ; "ARRAY" AR,AI[1:N,1:N]; ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE MATRIX MUST BE GIVEN IN THE ARRAYS AR AND AI, RESPECTIVELY; THE ELEMENTS OF THE ARRAYS AR AND AI ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:7]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE QR-ITERATION; (E.G. THE MACHINE PRECISION); EM[4]: THE MAXIMUM ALLOWED NUMBER OF QR-ITERATIONS; (E.G. 10 * N); EM[6]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS FOR EQUILIBRATING THE ORIGINAL MATRIX (E.G. N**2/2); EXIT: EM[1]: THE EUCLIDEAN NORM OF THE EQUILIBRATED MATRIX; EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED IN THE QR-ITERATION; EM[5]: THE NUMBER OF QR-ITERATIONS PERFORMED; EM[5]:=EM[4]+1 IN THE CASE EIGVALCOM^=0; EM[7]: THE NUMBER OF ITERATIONS PERFORMED FOR EQUILIBRATING THE ORIGINAL MATRIX; VALR,VALI: ; "ARRAY" VALR,VALI[1:N]; EXIT: THE REAL PART AND THE IMAGINARY PART OF THE CALCULATED EIGENVALUES ARE DELIVERED IN THE ARRAYS VALR AND VALI, RESPECTIVELY; EIGVALCOM:=0, PROVIDED THE QR-ITERATION IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, EIGVALCOM:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF THE ARRAYS VALR AND VALI ARE APPROXIMATE EIGENVALUES OF THE ORIGINAL MATRIX. PROCEDURES USED: EQILBRCOM = CP34361, COMEUCNRM = CP34359, HSHCOMHES = CP34366, VALQRICOM = CP34372. 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 3 REQUIRED CENTRAL MEMORY: FIVE REAL ARRAYS OF ORDER N AND ONE INTEGER ARRAY OF ORDER N ARE DECLARED. RUNNING TIME: PROPORTIONAL TO N ** 2 * MAX(N,NUMBER OF ITERATIONS). LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE EIGCOM (THIS SECTION). EXAMPLE OF USE: SEE EIGCOM (THIS SECTION). SUBSECTION: EIGCOM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "INTEGER" "PROCEDURE" EIGCOM(AR, AI, N, EM, VALR, VALI, VR, VI); "VALUE" N; "INTEGER" N; "ARRAY" AR, AI, EM, VALR, VALI, VR, VI; "CODE" 34375; THE MEANING OF THE FORMAL PARAMETERS IS: AR,AI: ; "ARRAY" AR,AI[1:N,1:N]; ENTRY: THE REAL PART AND THE IMAGINARY PART OF THE MATRIX MUST BE GIVEN IN THE ARRAYS AR AND AI, RESPECTIVELY; THE ELEMENTS OF THE ARRAYS AR AND AI ARE ALTERED; N: ; THE ORDER OF THE GIVEN MATRIX; EM: ; "ARRAY" EM[0:7]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE FOR THE QR-ITERATION; (E.G. THE MACHINE PRECISION); EM[4]: THE MAXIMUM ALLOWED NUMBER OF QR-ITERATIONS; (E.G. 10 * N); EM[6]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS FOR EQUILIBRATING THE ORIGINAL MATRIX (E.G. N**2/2); EXIT: EM[1]: THE EUCLIDEAN NORM OF THE EQUILIBRATED MATRIX; EM[3]: THE MAXIMUM ABSOLUTE VALUE OF THE SUBDIAGONAL ELEMENTS NEGLECTED IN THE QR-ITERATION; EM[5]: THE NUMBER OF QR-ITERATIONS PERFORMED; EM[5]:=EM[4]+1 IN THE CASE EIGCOM^=0; EM[7]: THE NUMBER OF ITERATIONS PERFORMED FOR EQUILIBRATING THE ORIGINAL MATRIX; 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 4 VALR,VALI: ; "ARRAY" VALR,VALI[1:N]; EXIT: THE REAL PART AND THE IMAGINARY PART OF THE CALCULATED EIGENVALUES ARE DELIVERED IN THE ARRAYS VALR AND VALI, RESPECTIVELY; VR,VI: ; "ARRAY" VR,VI[1:N,1:N]; EXIT: THE EIGENVECTORS OF THE MATRIX; THE NORMALIZED EIGENVECTOR WITH REAL PART VR[1:N,J] AND IMAGINARY PART VI[1:N,J] CORRESPONDS TO THE EIGENVALUE VALR[J] + VALI[J] * I, J=1,...,N; EIGCOM:=0, PROVIDED THE QR-ITERATION IS COMPLETED WITHIN EM[4] ITERATIONS; OTHERWISE, EIGCOM:= THE NUMBER, K, OF EIGENVALUES NOT CALCULATED AND ONLY THE LAST N-K ELEMENTS OF THE ARRAYS VALR AND VALI ARE APPROXIMATE EIGENVALUES OF THE ORIGINAL MATRIX AND NO USEFUL EIGENVECTORS ARE DELIVERED. PROCEDURES USED: EQILBRCOM = CP34361, COMEUCNRM = CP34359, HSHCOMHES = CP34366, QRICOM = CP34373, BAKCOMHES = CP34367, BAKLBRCOM = CP34362, SCLCOM = CP34360. REQUIRED CENTRAL MEMORY: FIVE REAL ARRAYS OF ORDER N AND ONE INTEGER ARRAY OF ORDER N ARE DECLARED. RUNNING TIME: PROPORTIONAL TO N**2 * MAX(N, NUMBER OF ITERATIONS). LANGUAGE : ALGOL 60. 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 5 THE FOLLOWING HOLDS FOR BOTH PROCEDURES: METHOD AND PERFORMANCE: FOR CALCULATING THE EIGENVALUES AND EIGENVECTORS OF A COMPLEX MATRIX WE DISTINGUISH THE FOLLOWING STEPS: 1) THE MATRIX IS EQUILIBRATED (SEE ALSO SECTION 3.2.1.1.2.). 2) THE EQUILIBRATED MATRIX IS TRANSFORMED INTO HESSENBERG FORM BY MEANS OF HOUSEHOLDER MATRICES (SEE ALSO SECTION 3.2.1.2.2.2.). 3) THE HESSENBERG MATRIX IS TRANSFORMED INTO AN UPPER TRIANGULAR MATRIX BY MEANS OF QR-ITERATION WITH SHIFT OF ORIGIN AND DEFLATION (SEE ALSO SECTION 3.3.2.2.1.). THE DIAGONAL ELEMENTS OF THE UPPER TRIANGULAR MATRIX ARE THE EIGENVALUES OF THE ORIGINAL MATRIX. THE EIGENVECTORS OF THE ORIGINAL MATRIX ARE OBTAINED BY CALCULATING THE EIGENVECTORS OF THE UPPER TRIANGULAR MATRIX (3) FOLLOWED BY BACKTRANSFORMATIONS CORRESPONDING TO (2) AND (1). REFERENCES: DEKKER, T.J. (1968), ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 1, MATH. CENTRE TRACTS 22, MATHEMATISCH CENTRUM; DEKKER, T.J. AND W.HOFFMANN (1968), ALGOL 60 PROCEDURES IN NUMERICAL ALGEBRA, PART 2, MATH. CENTRE TRACTS 23, MATHEMATISCH CENTRUM; FRANCIS, J.G.F. (1961), THE QR-TRANSFORMATION, PART 1 AND 2, COMP.J. 4, P.265-271 AND P.332-345; MUELLER, D.J. (1966), HOUSEHOLDER,S METHOD FOR COMPLEX MATRICES AND EIGENSYSTEMS OF HERMITIAN MATRICES, NUMER.MATH., 8, P.72-92; OSBORNE, E.E. (1960), ON PRECONDITIONING OF MATRICES, JACM., 7, P.338-354; PARLETT, B.N. AND C.REINSCH (1969), BALANCING A MATRIX FOR CALCULATION OF EIGENVALUES AND EIGENVECTORS, NUM. MATH., 13, P.293-304; WILKINSON, J.H. (1965), THE ALGEBRAIC EIGENVALUE PROBLEM, CLARENDOM PRESS, OXFORD. 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 6 EXAMPLE OF USE: EIGCOM CALCULATES THE EIGENVALUES AND THE EIGENVECTORS OF THE FOLLOWING MATRIX: (SEE WILKINSON AND REINSCH, 1971, CONTRIBUTION II/15) 1+3*I 2+1*I 3+2*I 1+1*I 3+4*I 1+2*I 2+1*I 4+3*I 2+3*I 1+5*I 3+1*I 5+2*I 1+2*I 3+1*I 1+4*I 5+3*I ONLY THE EIGENVECTOR CORRESPONDING TO VALR[1] + VALI[1] * I IS PRINTED BY THE FOLLOWING PROGRAM. "BEGIN" "REAL" "ARRAY" AR,AI,VR,VI[1:4,1:4],EM[0:7],VALR,VALI[1:4]; "INTEGER" I; AR[1,1]:=AR[1,4]:=AR[2,2]:=AR[3,2]:=AR[4,1]:=AR[4,3]:= AI[1,2]:=AI[1,4]:=AI[2,3]:=AI[3,3]:=AI[4,2]:=1; AR[1,2]:=AR[2,3]:=AR[3,1]:=AI[1,3]:=AI[2,2]:=AI[3,4]:=AI[4,1]:=2; AR[1,3]:=AR[2,1]:=AR[3,3]:=AR[4,2]:= AI[1,1]:=AI[2,4]:=AI[3,1]:=AI[4,4]:=3; AR[2,4]:=AI[2,1]:=AI[4,3]:=4; AR[3,4]:=AR[4,4]:=AI[3,2]:=5; EM[0]:=5"-14;EM[2]:="-12;EM[4]:=10;EM[6]:=10; OUTPUT(61,"(""("EIGCOM: ")",D")", EIGCOM(AR,AI,4,EM,VALR,VALI,VR,VI)); OUTPUT(61,"("/,"("EIGENVALUES:")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("2(+D.4D),"(" * I")",/")", VALR[I],VALI[I]); OUTPUT(61,"(""("FIRST EIGENVECTOR:")",/")"); "FOR" I:=1,2,3,4 "DO" OUTPUT(61,"("2(+D.4D),"(" * I")",/")", VR[I,1],VI[I,1]); OUTPUT(61,"(""("EM[1]: ")",+DD.DD/,"("EM[3]: ")",+D.D"+DD/, "("EM[5]: ")",+ZD/,"("EM[7]: ")",+ZD")",EM[1],EM[3],EM[5],EM[7]); "END" OUTPUT: EIGCOM: 0 EIGENVALUES: -3.3710-0.7705 * I +9.7837+9.3225 * I +1.3657-1.4011 * I +2.2217+1.8490 * I FIRST EIGENVECTOR: -0.5061+0.5835 * I +1.0000+0.0000 * I +0.5183-0.7147 * I -0.5535+0.0188 * I EM[1]: +15.30 EM[3]: +6.0"-12 EM[5]: +7 EM[7]: +4 1SECTION 3.3.2.2.2 (JULY 1974) PAGE 7 SOURCE TEXT(S) : 0"CODE" 34374; "INTEGER" "PROCEDURE" EIGVALCOM(AR, AI, N, EM, VALR, VALI); "VALUE" N; "INTEGER" N; "ARRAY" AR, AI, EM, VALR, VALI; "BEGIN" "INTEGER" "ARRAY" INT[1:N]; "ARRAY" D, B, DEL, TR, TI[1:N]; EQILBRCOM(AR, AI, N, EM, D, INT); EM[1]:= COMEUCNRM(AR, AI, N - 1, N); HSHCOMHES(AR, AI, N, EM, B, TR, TI, DEL); EIGVALCOM:= VALQRICOM(AR, AI, B, N, EM, VALR, VALI) "END" EIGVALCOM; "EOP" 0"CODE" 34375; "INTEGER" "PROCEDURE" EIGCOM(AR, AI, N, EM, VALR, VALI, VR, VI); "VALUE" N; "INTEGER" N; "ARRAY" AR, AI, EM, VALR, VALI, VR, VI; "BEGIN" "INTEGER" I; "INTEGER" "ARRAY" INT[1:N]; "ARRAY" D, B, DEL, TR, TI[1:N]; EQILBRCOM(AR, AI, N, EM, D, INT); EM[1]:= COMEUCNRM(AR, AI, N - 1, N); HSHCOMHES(AR, AI, N, EM, B, TR, TI, DEL); I:= EIGCOM:= QRICOM(AR, AI, B, N, EM, VALR, VALI, VR, VI); "IF" I = 0 "THEN" "BEGIN" BAKCOMHES(AR, AI, TR, TI, DEL, VR, VI, N, 1, N); BAKLBRCOM(N, 1, N, D, INT, VR, VI); SCLCOM(VR, VI, N, 1, N) "END" "END" EIGCOM; "EOP" 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 1 AUTHOR: J.J.G. ADMIRAAL. INSTITUTE: UNIVERSITY OF AMSTERDAM. RECEIVED: 751101. BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO MAIN AND NINE AUXILIARY PROCEDURES: THE TWO MAIN PROCEDURES ARE: A. QZIVAL FINDS N PAIRS OF SCALARS (ALFA[M],BETA[M]),WHERE BETA[M] IS REAL,SUCH THAT THE MATRIX BETA[M] * A - ALFA[M] * B IS SINGULAR. B. QZI FINDS N PAIRS OF SCALARS (ALFA[M], BETA[M]),WHERE BETA[M] IS REAL,SUCH THAT THE MATRIX BETA[M] * A - ALFA[M] * B IS SINGULAR; MOREOVER THE GENERALIZED EIGENVECTORS (THE HOMOGENOUS SOLUTION OF ( BETA[M] * A - ALFA[M] * B ) * X = 0 ) ARE CALCULATED. THE AUXILIARY PROCEDURES ARE: A. HSHDECMUL: THIS PROCEDURE CALCULATES REAL MATRICES Q AND R SUCH THAT Q.A=R WHERE A IS A GIVEN REAL SQUARE MATRIX, Q IS A PRODUCT OF HOUSEHOLDER MATRICES AND R AN UPPERTRIANGULAR MATRIX. MOREOVER Q.B IS FORMED WITH B, A GIVEN MATRIX. B. HESTGL3: GIVEN THE REAL SQUARE MATRICES A,B AND X, WITH B AN UPPER TRIANGULAR MATRIX,HESTGL3 CALCULATES THE MATRICES Q,Z,H,R, WHERE Q,Z ARE ORTHOGONAL, H UPPER HESSENBERG AND R AN UPPER TRIANGULAR MATRIX SUCH THAT Q.A.Z = H AND Q.B.Z = R. FURTHER: A:= Q.A.Z ; B:= Q.B.Z AND X:= Q.X.Z. C. HESTGL2: SEE HESTGL3,BUT HERE THE MATRIX X HAS BEEN LEFT OUT. D. HSH2COL: THIS PROCEDURE CALCULATES A HOUSEHOLDER MATRIX Q SUCH THAT BY PREMULTIPLYING A GIVEN COLUMN VECTOR V BY Q A ZERO ELEMENT IS FORMED IN V. HERE THE VECTOR V IS A COLUMN OF A MATRIX. FURTHER: A:= Q.A AND B:= Q.B WHERE A,B ARE TWO GIVEN REAL MATRICES. 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 2 E. HSH3COL: THIS PROCEDURE CALCULATES A HOUSEHOLDER MATRIX Q SUCH THAT BY PREMULTIPLYING A GIVEN COLUMN VECTOR V BY Q TWO SUCCESSIVE ZERO ELEMENTS ARE FORMED IN V. HERE THE VECTOR V IS A COLUMN OF A MATRIX. FURTHER: A:= Q.A AND B:= Q.B WHERE A AND B ARE TWO GIVEN REAL MATRICES. F. HSH2ROW3: THIS PROCEDURE CALCULATES A HOUSEHOLDER MATRIX Z SUCH THAT BY POSTMULTIPLYING A GIVEN ROWVECTOR V BY Z A ZERO ELEMENT IS FORMED IN V. HERE THE VECTOR V IS A ROW OF A MATRIX. FURTHER: A:= A.Z; B:= B.Z AND X:= X.Z WHERE A,B,X ARE THREE GIVEN REAL MATRICES. G. HSH2ROW2: SEE HSH2ROW3, BUT HERE THE MATRIX X HAS BEEN LEFT OUT. H. HSH3ROW3: THIS PROCEDURE CALCULATES A HOUSEHOLDER MATRIX Z SUCH THAT BY POSTMULTIPLYING A GIVEN ROWVECTOR V BY Z, TWO SUCCESSIVE ZERO ELEMENTS ARE FORMED IN V. HERE THE VECTOR V IS A ROW OF A MATRIX. FURTHER: A:= A.Z; B:= B.Z AND X:= X.Z WHERE A,B AND X ARE THREE GIVEN REAL MATRICES. I: HSH3ROW2: SEE HSH3ROW3, BUT HERE THE MATRIX X HAS BEEN LEFT OUT. KEYWORDS: HOUSEHOLDER'S TRANSFORMATION, GENERALIZED EIGENVALUES, GENERALIZED EIGENVECTORS, UPPER HESSENBERG MATRIX, UPPER TRIANGULAR MATRIX. REFERENCES: [1]. C.B. MOLER AND G.W. STEWART. AN ALGORITHM FOR THE GENERALIZED MATRIX EIGENVALUE PROBLEM A * X = LAMBDA * B * X. REPORT STANFORD UNIVERSITY STAN-CS-232-71; 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 3 SUBSECTION: QZIVAL CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" QZIVAL(N,A,B,ALFR,ALFI,BETA,ITER,EM); "VALUE" N; "INTEGER" N; "ARRAY" A,B,ALFR,ALFI,BETA,EM; "INTEGER" "ARRAY" ITER; "CODE" 34600; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF ROWS AND COLUMNS OF THE MATRICES A,B A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE GIVEN MATRIX; EXIT: A QUASI UPPER-TRIANGULAR MATRIX (SEE METHOD AND PERFORMANCE); B: ; "ARRAY" B[1:N,1:N]; ENTRY: THE GIVEN MATRIX; EXIT: AN UPPER-TRIANGULAR MATRIX; ALFR: ; "ARRAY" ALFR[1:N]; EXIT : THE REAL PARTS OF ALFA[1:N] (SEE METHOD AND PERFORMANCE); ALFI: ; "ARRAY" ALFI[1:N]; EXIT : THE IMAGINARY PARTS OF ALFA[1:N]; BETA: ; "ARRAY" BETA[1:N]; EXIT : THE REAL SCALARS BETA[N] ITER: ; "INTEGER" "ARRAY" ITER[1:N]; TROUBLE INDICATOR AND ITERATION COUNTER; IF ITER[1]=0 THEN NO TROUBLE IS SIGNALIZED, FURTHER SEE METHOD AND PERFORMANCE; EM: ; "ARRAY" EM[0:1]; ENTRY: EM[0]: THE SMALLEST POSITIVE MACHINE NUMBER; EM[1]: THE RELATIVE PRECISION OF ELEMENTS OF A AND B; 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 4 PROCEDURES USED: TAMMAT = CP 34014 ELMCOL = CP 34023 HSHDECMUL = CP 34602 HESTGL2 = CP 34604 HSH2COL = CP 34605 HSH3COL = CP 34606 HSH2ROW2 = CP 34608 HSH3ROW2 = CP 34610 CHSH2 = CP 34611 HSHVECMAT = CP 31070 HSHVECTAM = CP 31073 REQUIRED CENTRAL MEMORY : HSHDECMUL DECLARES AN ARRAY OF N REALS. RUNNING TIME: PROPORTIONAL TO N ** 3 METHOD AND PERFORMANCE; THE PROCEDURE QZIVAL SOLVES THE GENERALIZED MATRIX EIGENVALUE PROBLEM A * X = LAMBDA * B * X BY MEANS OF QZ ITERATION (SEE REF[1]); QZIVAL FINDS N PAIRS OF SCALARS (ALFA[M],BETA[M]) SUCH THAT BETA[M] * A - ALFA[M] * B IS SINGULAR. THE EIGENVALUES OF A * X - LAMBDA * B * X CAN BE OBTAINED BY DIVIDING ALFA[M] BY BETA[M],EXCEPT BETA[M] MIGHT BE ZERO. IN THIS ALGORITHM ONLY UNITARY TRANSFORMATIONS ARE APPLIED; A FORTIORI NO INVERSES ARE CALCULATED, SO EITHER A OR B (OR BOTH) MAY BE SINGULAR. BETA[M] IS REAL, ALFA[M] IS COMPLEX. REAL AND IMAGINARY PARTS ARE GIVEN IN ALFR[M] AND ALFI[M]. THE OCCURRENCE OF COMPLEX PAIRS IS ALWAYS IN SUCCESSIVE ELEMENTS, SUCH THAT ALFA[M]/BETA[M] AND ALFA[M+1]/BETA[M+1] ARE COMPLEX CONJUGATE, BUT ALFA[M] AND ALFA[M+1] ARE NOT NECESSARILY CONJUGATE. ONLY REAL ARITHMETIC IS USED IN THE PROCEDURE. IF A AND B WERE REDUCED TO TRIANGULAR FORM BY UNITARY TRANSFORMATIONS,ALFA AND BETA WOULD BE THE DIAGONALS. A AND B ARE ACTUALLY REDUCED TO QUASI-TRIANGULAR FORM HAVING ONLY 1-BY-1 AND 2-BY-2 BLOCKS ON THE DIAGONAL OF A. IF ALFA[M] IS NOT REAL, THEN BETA[M] IS NOT ZERO. ITER IS THE TROUBLE INDICATOR AND ITERATION COUNTER. IF ITER[1]=0 THEN EVERYTHING IS O.K. ITER[M] IS THE NUMBER OF ITERATIONS NEEDED FOR THE M-TH EIGENVALUE. IF ITER[1] THROUGH ITER[M]= -1 THEN THE ITERATION FOR THE M-TH EIGENVALUE DID NOT CONVERGE AND ALFA[1] THROUGH ALFA[M] AND BETA[1] THROUGH BETA[M] ARE PROBABLY INACCURATE. EXAMPLE OF USE: "BEGIN" "ARRAY" A,B[1:4,1:4],ALFR,ALFI,BETA[1:4],EM[0:1]; "INTEGER" "ARRAY" ITER[1:4];"INTEGER" K,L; A[1,1]:=2; A[1,2]:=3; A[1,3]:=-3; A[1,4]:=4; 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 5 A[2,1]:=1; A[2,2]:=-1; A[2,3]:=5; A[2,4]:=1; A[3,1]:=0; A[3,2]:=2; A[3,3]:=6; A[3,4]:=8; A[4,1]:=1; A[4,2]:=1; A[4,3]:=0; A[4,4]:=4; B[1,1]:=1; B[1,2]:=5; B[1,3]:=9; B[1,4]:=0; B[2,1]:=2; B[2,2]:=6; B[2,3]:=10; B[2,4]:=2; B[3,1]:=3; B[3,2]:=7; B[3,3]:=11; B[3,4]:=-1; B[4,1]:=4; B[4,2]:=8; B[4,3]:=12; B[4,4]:=3; OUTPUT(61,"(""("A")",/,4(4(+ZDBB),/),/")",A); OUTPUT(61,"(""("B")",/,4(4(+ZDBB),/),/")",B); EM[0]:=DWARF;EM[1]:="-15; QZIVAL(4,A,B,ALFR,ALFI,BETA,ITER,EM); "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61,"(""("ITER[")",D,"("]=")",ZD,/")",K,ITER[K]); OUTPUT(61,"(""("ALFA(REAL PART)")"8B,"("ALFA(IMAGINARY PART)")" 3B,"("BETA")",/")"); "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61,"("3(N),/")",ALFR[K],ALFI[K],BETA[K]); OUTPUT(61,"("/"("LAMBDA(REAL PART)")"6B, "("LAMBDA(IMAGINARY PART)")"/")"); "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" "BEGIN" "IF" BETA[K]=0 "THEN" OUTPUT(61,"(""("INFINITE")"15B,"("INDEFINITE")"/,")") "ELSE" OUTPUT(61,"("2(N),/")",ALFR[K]/BETA[K],ALFI[K]/BETA[K]) "END" "END" A +2 +3 -3 +4 +1 -1 +5 +1 +0 +2 +6 +8 +1 +1 +0 +4 B +1 +5 +9 +0 +2 +6 +10 +2 +3 +7 +11 -1 +4 +8 +12 +3 ITER[1]= 0 ITER[2]= 0 ITER[3]= 0 ITER[4]= 5 ALFA(REAL PART) ALFA(IMAGINARY PART) BETA -4.4347115652167"+000 +0.0000000000000"+000 +0.0000000000000"+000 -5.7288406521003"+000 +0.0000000000000"+000 +2.8441121744896"+000 -8.6671777386054"-001 +2.7607904944916"+000 +8.7617886336960"+000 -4.7262205157527"-001 -1.5054617625576"+000 +4.7778119295757"+000 LAMBDA(REAL PART) LAMBDA(IMAGINARY PART) INFINITE INDEFINITE -2.0142808372628"+000 +0.0000000000000"+000 -9.8920187429234"-002 +3.1509439566644"-001 -9.8920187429236"-002 -3.1509439566645"-001 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 6 SUBSECTION: QZI CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" QZI(N,A,B,X,ALFR,ALFI,BETA,ITER,EM); "VALUE" N; "INTEGER" N; "ARRAY" A,B,X,ALFR,ALFI,BETA,EM; "INTEGER" "ARRAY" ITER; "CODE" 34601; THE MEANING OF THE FORMAL PARAMETERS IS; N: ; THE NUMBER OF ROWS AND COLUMNS OF THE MATRICES A,B AND X; A: "ARRAY" A[1:N,1:N]; ENTRY: THE GIVEN MATRIX A; EXIT: A QUASI UPPER TRIANGULAR MATRIX; (SEE METHOD AND PERFORMANCE); B: ; "ARRAY" B[1:N,1:N]; ENTRY: THE GIVEN MATRIX B; EXIT: AN UPPER-TRIANGULAR MATRIX; X: ; "ARRAY" X[1:N,1:N]; ENTR: THE N*N UNIT MATRIX; EXIT: THE MATRIX OF EIGENVECTORS, THE EIGENVECTORS ARE STORED IN THE ARRAY X AS FOLLOWS: IF ALFI[M]=0 THEN X[.,M] IS THE M-TH REAL EIGENVECTOR; OTHERWISE, FOR EACH PAIR OF CONSECUTIVE COLUMNS X[.,M] AND X[.,M+1] ARE THE REAL AND IMAGINARY PARTS OF THE M-TH COMPLEX EIGENVECTOR. X[.,M] AND -X[.,M+1] ARE THE REAL AND IMAGINARY PARTS OF THE M+1 -ST COMPLEX EIGENVECTOR. THE EIGENVECTORS ARE NORMALIZED SUCH THAT THE LARGEST COMPONENT IS 1 OR 1 + 0 * I. ALFR: ; "ARRAY" ALFR[1:N]; EXIT: THE REAL PARTS OF ALFA[1:N]; ALFI: ; "ARRAY" ALFI[1:N]; EXIT: THE IMAGINARY PARTS OF ALFA[1:N]; BETA: ; "ARRAY" BETA[1:N]; ITER: ; "INTEGER" "ARRAY" ITER[1:N]; TROUBLE INDICATOR AND ITERATION COUNTER; IF ITER[1]=0 THEN NO TROUBLE IS SIGNALIZED, FOR FURTHER INFORMATION SEE METHOD AND PEFORMANCE OF PROCEDURE QZIVAL (THIS SECTION). EM: ; "ARRAY" EM[0:1]; ENTRY: EM[0]: THE SMALLEST POSITIVE MACHINE NUMBER; EM[1]: THE RELATIVE PRECISION OF ELEMENTS OF A AND B; 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 7 PROCEDURES USED: MATMAT = CP 34013 TAMMAT = CP 34014 ELMCOL = CP 34023 HSHDECMUL = CP 34602 HESTGL3 = CP 34603 HSH2COL = CP 34605 HSH2ROW3 = CP 34607 HSH3ROW3 = CP 34609 HSH3COL = CP 34606 CHSH2 = CP 34611 COMDIV = CP 34342 HSHVECMAT = CP 31070 HSHVECTAM = CP 31073 RUNNING TIME: PROPORTIONAL TO N ** 3; REQUIRED CENTRAL MEMORY : HSHDECMUL DECLARES AN ARRAY OF N REALS. METHOD AND PERFORMANCE; THE PROCEDURE QZI APPLIES THE SAME METHOD AS QZIVAL. EXAMPLE OF USE: "BEGIN" "ARRAY" A,B,X[1:4,1:4],ALFR,ALFI,BETA[1:4],EM[0:1]; "INTEGER" "ARRAY" ITER[1:4];"INTEGER" K,L; A[1,1]:=2; A[1,2]:=3; A[1,3]:=-3; A[1,4]:=4; A[2,1]:=1; A[2,2]:=-1; A[2,3]:=5; A[2,4]:=1; A[3,1]:=0; A[3,2]:=2; A[3,3]:=6; A[3,4]:=8; A[4,1]:=1; A[4,2]:=1; A[4,3]:=0; A[4,4]:=4; B[1,1]:=1; B[1,2]:=5; B[1,3]:=9; B[1,4]:=0; B[2,1]:=2; B[2,2]:=6; B[2,3]:=10; B[2,4]:=2; B[3,1]:=3; B[3,2]:=7; B[3,3]:=11; B[3,4]:=-1; B[4,1]:=4; B[4,2]:=8; B[4,3]:=12; B[4,4]:=3; "FOR" K:=1,2,3,4 "DO" "FOR" L:=1,2,3,4 "DO" X[K,L]:="IF" K=L "THEN" 1 "ELSE" 0; OUTPUT(61,"(""("A")",/,4(4(+ZDBB),/),/")",A); OUTPUT(61,"(""("B")",/,4(4(+ZDBB),/),/")",B); EM[0]:=1.0"-280;EM[1]:="-15; QZI(4,A,B,X,ALFR,ALFI,BETA,ITER,EM); 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 8 "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61,"(""("ITER[")",D,"("]=")",ZD,/")",K,ITER[K]); OUTPUT(61,"("/"("EIGENVECTORS")",/,4(4(+D.8D"+2D2B),/),/")",X); OUTPUT(61,"(""("ALFA(REAL PART)")"8B,"("ALFA(IMAGINARY PART)")" 9B,"("BETA")",/")"); "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" OUTPUT(61,"("3(N),/")",ALFR[K],ALFI[K],BETA[K]); OUTPUT(61,"("/"("LAMBDA(REAL PART)")"6B, "("LAMBDA(IMAGINARY PART)")"/")"); "FOR" K:=1 "STEP" 1 "UNTIL" 4 "DO" "BEGIN" "IF" BETA[K]=0 "THEN" OUTPUT(61,"(""("INFINITE")"15B,"("INDEFINITE")"/")") "ELSE" OUTPUT(61,"("2(N),/")",ALFR[K]/BETA[K],ALFI[K]/BETA[K]) "END" "END" A +2 +3 -3 +4 +1 -1 +5 +1 +0 +2 +6 +8 +1 +1 +0 +4 B +1 +5 +9 +0 +2 +6 +10 +2 +3 +7 +11 -1 +4 +8 +12 +3 ITER[1]= 0 ITER[2]= 0 ITER[3]= 0 ITER[4]= 5 EIGENVECTORS -5.00000000"-01 +1.00000000"+00 -6.29204867"-01 +6.52026261"-01 +1.00000000"+00 -3.82541766"-02 +1.00000000"+00 +0.00000000"+00 -5.00000000"-01 -3.04677732"-02 +1.65896051"-01 +1.09306265"-01 -4.35116786"-15 -7.63328122"-01 -5.84845537"-01 +1.77430910"-01 ALFA(REAL PART) ALFA(IMAGINARY PART) BETA -4.4347115652167"+000 +0.0000000000000"+000 +0.0000000000000"+000 -5.7288406521003"+000 +0.0000000000000"+000 +2.8441121744896"+000 -8.6671777386054"-001 +2.7607904944916"+000 +8.7617886336960"+000 -4.7262205157527"-001 -1.5054617625576"+000 +4.7778119295757"+000 LAMBDA(REAL PART) LAMBDA(IMAGINARY PART) INFINITE INDEFINITE -2.0142808372628"+000 +0.0000000000000"+000 -9.8920187429234"-002 +3.1509439566644"-001 -9.8920187429236"-002 -3.1509439566645"-001 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 9 SUBSECTION: HSHDECMUL. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "PROCEDURE" HSHDECMUL(N,A,B,DWARF); "VALUE" N,DWARF; "INTEGER"N; "REAL"DWARF; "ARRAY" A,B; "CODE" 34602; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRICES; A: ; "REAL" "ARRAY" A[1:N,1:N]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX Q.A (SEE BRIEF DESCRIPTION); B: ; "REAL" "ARRAY" B[1:N,1:N]; ENTRY: THE GIVEN MATRIX B; EXIT: THE UPPER TRIANGULAR MATRIX Q.B (SEE BRIEF DESCRIPTION); DWARF: < ARITHMETIC EXPRESSION>; THE SMALLEST POSITIVE MACHINE NUMBER. PROCEDURES USED: TAMMAT = CP 34014; HSHVECMAT = CP 31070. REQUIRED CENTRAL MEMORY : AN ARRAY OF N REALS IS DECLARED. METHOD AND PERFORMANCE: FOR EACH MATRIX COLUMN A[I] A HOUSEHOLDERMATRIX Q IS FORMED, SUCH THAT Q.A[I] HAS ONLY ZERO ELEMENTS BELOW THE DIAGONAL ELEMENT. WHEN SUCCESSIVELY FOR I = 1,2,....,N-1 THESE TRANSFORMATIONS HAVE BEEN PERFORMED,THE MATRIX A HAS BEEN CHANGED INTO AN UPPER TRIANGULAR MATRIX. THE SAME TRANSFORMATIONS ARE PERFORMED ON THE MATRIX B EXAMPLE OF USE: THE PROCEDURE HSHDECMUL IS USED IN QZI AND QZIVAL (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 10 SUBSECTION: HESTGL3: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HESTGL3(N,A,B,X); "VALUE" N; "INTEGER" N; "ARRAY" A,B,X; "CODE" 34603; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRICES; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE GIVEN MATRIX A; EXIT: THE UPPER HESSENBERG MATRIX Q.A.Z (SEE BRIEF DESCRIPTION); B: ; "ARRAY" B[1:N,1:N]; ENTRY: THE GIVEN UPPER TRIANGULAR MATRIX B; EXIT: THE UPPER TRIANGULAR MATRIX Q.B.Z (SEE BRIEF DESCRIPTION); X: ; "ARRAY" X[1:N,1:N]; ENTRY: THE GIVEN MATRIX X; EXIT: THE TRANSFORMED MATRIX Q.X.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSH2COL = CP 34605 HSH2ROW3 = CP 34607 METHODE AND PERFORMANCE: THE REDUCTION OF THE MATRIX A TO UPPER HESSENBERG FORM WHILE PRESERVING THE TRIANGULARITY OF THE MATRIX B IS THE RESULT OF A NUMBER OF STEPS, WHICH DO THE FOLLOWING ACTIONS: INTRODUCING A ZERO ELEMENT IN A AND RESTORING THE DISTURBED ZERO IN B, WITHOUT DISTURBING THE ZERO INTRODUCED IN A. THIS IS DONE BY PRE-AND POSTMULTIPLICATIONS OF HOUSEHOLDER MATRICES. THE MATRIX X SHARES THE TRANSFORMATION. FOR FURTHER DETAILS SEE [1] EXAMPLE OF USE: THE PROCEDURE HESTGL3 IS USED IN QZI (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 11 SUBSECTION: HESTGL2: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HESTGL2(N,A,B); "VALUE" N; "INTEGER" N; "ARRAY" A,B; "CODE" 34604; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE GIVEN MATRICES; A: ; "ARRAY" A[1:N,1:N]; ENTRY: THE GIVEN MATRIX A; EXIT: THE UPPER HESSENBERG MATRIX Q.A.Z (SEE BRIEF DESCRIPTION); B: ; "ARRAY" B[1:N,1:N]; ENTRY: THE GIVEN UPPER TRIANGULAR MATRIX B EXIT: THE UPPER TRIANGULAR MATRIX Q.B.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSH2COL = CP 34605 HSH2ROW2 = CP 34608 METHODE AND PERFORMANCE: SEE HESTGL3, BUT HERE THE MATRIX X HAS BEEN LEFT OUT. EXAMPLE OF USE: THE PROCEDURE HESTGL2 IS USED IN QZIVAL (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 12 SUBSECTION HSH2COL: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH2COL(LA,LB,U,I,A1,A2,A,B); "VALUE" LA,LB,U,I,A1,A2; "INTEGER" LA,LB,U,I; "REAL" A1,A2; "ARRAY" A,B; "CODE" 34605; THE MEANING OF THE FORMAL PARAMETERS IS: LA: ; THE LOWER BOUND OF THE RUNNING COLUMN SUBSCRIPT OF A; LB: ; THE LOWER BOUND OF THE RUNNING COLUMN SUBSCRIPT OF B; U: ; THE UPPER BOUND OF THE RUNNING COLUMN SUBSCRIPTS OF A AND B. I: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPTS OF A AND B. I+1 IS THE UPPERBOUND. A1: ; THE I-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED. A2: ; THE (I+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A: ; "ARRAY" A[I:I+1,LA:U]; ENTRY THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX Q.A (SEE BRIEF DESCRIPTION); B: ; "ARRAY" B[I:I+1,LB:U]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX Q.B (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECMAT = CP 31070 METHOD AND PERFORMANCE: WHEN THE CALCULATED HOUSEHOLDER MATRIX Q PREMULTIPLIES A MATRIX M, ONLY ROWS I AND I+1 ARE CHANGED. IF THE ELEMENTS I AND I+1 IN A COLUMN OF M ARE ZERO, THEY REMAIN ZERO IN Q.M. THEREFORE ONLY THE SUBMATRICES A[I:I+1,LA:U] AND B[I:I+1,LB:U] ARE CHANGED, WHERE Q.A AND Q.B ARE OVERWRITTEN IN A RESP B. EXAMPLE OF USE: THE PROCEDURE HSH2COL IS USED IN QZI AND QZIVAL, (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 13 SUBSECTION HSH3COL: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH3COL(LA,LB,U,I,A1,A2,A3,A,B); "VALUE" LA,LB,U,I,A1,A2,A3;"INTEGER" LA,LB,I,U;"REAL" A1,A2,A3; "ARRAY" A,B; "CODE" 34606; THE MEANING OF THE FORMAL PARAMETERS IS: LA: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A; LB: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF B; U: ; THE UPPERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A AND B; I: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF A AND B, I+2 IS THE UPPERBOUND; A1: ; THE I-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A2: ; THE (I+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A3: ; THE (I+2)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED. A: ; "ARRAY" A[I:I+2,LA:U]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX Q.A (SEE BRIEF DESCRIPTION); B: ; "ARRAY" B[I:I+2,LB:U]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX Q.B (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECMAT = CP 31070; METHOD AND PERFORMANCE: WHEN THE CALCULATED HOUSEHOLDER MATRIX Q PREMULTIPLIES A MATRIX M, ONLY ROWS I, (I+1) AND (I+2) ARE CHANGED. IF THE ELEMENTS I, I+1 AND I+2 ARE ZERO, THEN THEY REMAIN ZERO IN Q.M. THEREFORE ONLY THE SUBMATRICES A[I:I+2,LA:U] AND B[I:I+2,LB:U] ARE CHANGED, WHERE Q.A AND Q.B ARE OVERWRITTEN IN A RESP B. EXAMPLE OF USE: THE PROCEDURE HSH3COL IS USED IN QZI AND QZIVAL (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 14 SUBSECTION HSH2ROW3: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH2ROW3(L,UA,UB,UX,J,A1,A2,A,B,X); "VALUE" L,UA,UB,UX,J,A1,A2; "INTEGER" L,UA,UB,UX,J; "REAL" A1,A2;"ARRAY" A,B,X; "CODE" 34607; THE MEANING OF THE FORMAL PARAMETERS IS: L: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF A,B AND X; UA: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF A; UB: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF B; UX: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF X; J: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A,B AND X; J+1 IS THE UPPERBOUND; A1: ; THE J-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A2: ; THE (J+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A: ; "ARRAY" A[L:UA,J:J+1]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX A.Z (SEE BRIEF DESCRIPTION); B: ; "ARRAY" B[L:UB,J:J+1]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX B.Z (SEE BRIEF DESCRIPTION); X: ; "ARRAY" X[L:UX,J:J+1]; ENTRY: THE GIVEN MATRIX X; EXIT: THE TRANSFORMED MATRIX X.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECTAM = CP 31073; METHOD AND PERFORMANCE: WHEN THE CALCULATED HOUSEHOLDER MATRIX Z POSTMULTIPLIES A MATRIX M, ONLY COLUMNS J AND J+1 ARE CHANGED. IF THE ELEMENTS J AND J+1 IN A ROW OF M ARE ZERO, THEN THEY REMAIN ZERO IN M.Z THEREFORE ONLY THE SUBMATRICES A[L:UA,J:J+1],B[L:UB,J:J+1] AND X[L:UX,J:J+1] ARE CHANGED, WHERE A.Z, B.Z AND X.Z ARE OVERWRITTEN IN RESP. A,B AND X. EXAMPLE OF USE: THE PROCEDURE HSH2ROW3 IS USED IN QZI (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 15 SUBSECTION HSH2ROW2: CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH2ROW2(LA,LB,UA,UB,J,A1,A2,A,B); "VALUE" LA,LB,UA, UB,J,A1,A2; "INTEGER" LA,LB,UA,UB,J; "REAL" A1,A2;"ARRAY" A,B; "CODE" 34608; THE MEANING OF THE FORMAL PARAMETERS IS: LA: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF A; LB: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF B; UA: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF A; UB: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF B; J: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A AND B; J+1 IS THE UPPERBOUND; A1: < ARITHMETIC EXPRESSION>; THE J-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A2: ; THE (J+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A: ; "REAL" "ARRAY" A[LA:UA,J:J+1]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX A.Z (SEE BRIEF DESCRIPTION); B: ; "REAL" "ARRAY" B[LB:UB,J:J+1]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX B.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECTAM = CP 31073; METHOD AND PERFORMANCE: SEE HSH2ROW3, BUT HERE THE MATRIX X HAS BEEN LEFT OUT. EXAMPLE OF USE: THE PROCEDURE HSH2ROW2 IS USED IN QZIVAL, (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 16 SUBSECTION: HSH3ROW3. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH3ROW3(L,U,UX,J,A1,A2,A3,A,B,X); "VALUE" L,U,UX,J,A1,A2,A3; "INTEGER" L,U,UX,J; "REAL" A1,A2,A3; "ARRAY" A,B,X; "CODE" 34609; THE MEANING OF THE FORMAL PARAMETERS IS: L: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF A AND B AND X; U: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF A AND B; UX: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF X; J: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A,B AND X; J+2 IS THE UPPERBOUND; A1: ; THE J-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A2: ; THE (J+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A3: ; THE (J+2)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A: ; "REAL" "ARRAY" A[L:U,J:J+2]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX A.Z (SEE BRIEF DESCRIPTION); B: ; "REAL" "ARRAY" B[L:U,J:J+2]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX B.Z (SEE BRIEF DESCRIPTION); X: ; "REAL" "ARRAY" X[L:UX,J:J+2]; ENTRY: THE GIVEN MATRIX X; EXIT: THE TRANSFORMED MATRIX X.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECTAM = CP 31073; METHOD AND PERFORMANCE: WHEN THE CALCULATED HOUSEHOLDER MATRIX Z POSTMULTIPLIES A MATRIX M, ONLY COLUMNS J,J+1 AND J+2 ARE CHANGED. IF THE ELEMENTS J, J+1 AND J+2 IN A ROW OF M ARE ZERO, THEN THEY REMAIN ZERO IN M.Z. THEREFORE ONLY THE SUBMATRICES A[L:U,J:J+2], B[L:U,J:J+2] AND X[L:UX,J:J+2] ARE CHANGED, WHERE A.Z , B.Z AND X.Z ARE OVERWRITTEN ON RESP. A,B AND X; EXAMPLE OF USE: THE PROCEDURE HSH3ROW3 IS USED IN QZI (THIS SECTION). 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 17 SUBSECTION: HSH3ROW2. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IS: "PROCEDURE" HSH3ROW2(LA,LB,U,J,A1,A2,A3,A,B); "VALUE" LA,LB,U,J,A1,A2,A3; "INTEGER" LA,LB,U,J; "REAL" A1,A2,A3; "ARRAY" A,B; "CODE" 34610; THE MEANING OF THE FORMAL PARAMETERS IS: LA: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF A; LB: ; THE LOWERBOUND OF THE RUNNING ROW SUBSCRIPT OF B; U: ; THE UPPERBOUND OF THE RUNNING ROW SUBSCRIPT OF A AND B; J: ; THE LOWERBOUND OF THE RUNNING COLUMN SUBSCRIPT OF A AND B, J+2 IS THE UPPERBOUND; A1: ; THE J-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A2: ; THE (J+1)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A3: ; THE (J+2)-TH COMPONENT OF THE VECTOR TO BE TRANSFORMED; A: ; "REAL" "ARRAY" A[LA:U,J:J+2]; ENTRY: THE GIVEN MATRIX A; EXIT: THE TRANSFORMED MATRIX A.Z (SEE BRIEF DESCRIPTION); B: ; "REAL" "ARRAY" B[LB:U,J:J+2]; ENTRY: THE GIVEN MATRIX B; EXIT: THE TRANSFORMED MATRIX B.Z (SEE BRIEF DESCRIPTION); PROCEDURES USED: HSHVECTAM = CP 31073; METHOD AND PERFORMANCE: SEE HSH3ROW3, BUT HERE THE MATRIX X HAS BEEN LEFT OUT. EXAMPLE OF USE: HSH3ROW2 IS USED IN QZIVAL (THIS SECTION). 1SECTION : 3.4.1.2 (JANUARY 1979) PAGE 18 SOURCE TEXTS: 0"CODE" 34600; "PROCEDURE" QZIVAL(N,A,B,ALFR,ALFI,BETA,ITER,EM); "VALUE" N;"INTEGER" N;"ARRAY" A,B,ALFR,ALFI,BETA,EM; "INTEGER" "ARRAY" ITER; "BEGIN" "REAL" DWARF,EPS,EPSA,EPSB; "PROCEDURE" QZIT(N,A,B,EPS,EPSA,EPSB,ITER);"VALUE"N,EPS; "REAL" EPS,EPSA,EPSB;"INTEGER" N;"INTEGER" "ARRAY" ITER;"ARRAY" A,B; "BEGIN" "REAL" ANORM,BNORM,ANI,BNI,CONST,A10,A20,A30,B11, B22,B33,B44,A11,A12,A21,A22,A33,A34,A43,A44,B12,B34,OLD1,OLD2; "INTEGER" I,Q,M,M1,Q1,J,K,K1,K2,K3,KM1;"BOOLEAN" STATIONARY; ANORM:=BNORM:=0;"FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" BNI:=0;ITER[I]:=0;ANI:="IF" I>1"THEN"ABS(A[I,I-1])"ELSE" 0; "FOR" J:=I "STEP" 1 "UNTIL" N "DO" "BEGIN" ANI:=ANI+ABS(A[I,J]);BNI:=BNI+ABS(B[I,J]) "END";"IF" ANI>ANORM "THEN" ANORM:=ANI;"IF" BNI>BNORM"THEN" BNORM:=BNI "END";"IF" ANORM=0 "THEN" ANORM:=EPS;"IF"BNORM=0 "THEN" BNORM:=EPS; EPSA:=EPS*ANORM;EPSB:=EPS*BNORM; "FOR" M:=N,M "WHILE" M>=3 "DO" "BEGIN" "FOR" I:=M+1,I-1 "WHILE"("IF" I>1 "THEN"ABS(A[I,I-1])>EPSA "ELSE" "FALSE") "DO" Q:=I-1; "IF" Q>1 "THEN" A[Q,Q-1]:=0; L: "IF" Q>=M-1 "THEN" M:=Q-1 "ELSE" "BEGIN" "IF" ABS(B[Q,Q])<=EPSB "THEN" "BEGIN" B[Q,Q]:=0;Q1:=Q+1; HSH2COL(Q,Q,M,Q,A[Q,Q],A[Q1,Q],A,B);A[Q1,Q]:=0; Q:=Q1;"GOTO" L "END" "ELSE" M1:=M-1;Q1:=Q+1;CONST:=0.75;ITER[M]:=ITER[M]+1; STATIONARY:="IF" ITER[M]=1 "THEN" "TRUE" "ELSE" ABS(A[M,M-1])>=CONST*OLD1"AND"ABS(A[M-1,M-2])>=CONST*OLD2; "IF" ITER[M]>30"AND"STATIONARY "THEN" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" ITER[I]:=-1; "GOTO" OUT "END"; "IF" ITER[M]=10"AND"STATIONARY "THEN" "BEGIN" A10:=0;A20:=1;A30:=1.1605 "END" "ELSE" "BEGIN" B11:=B[Q,Q];B22:="IF" ABS(B[Q1,Q1])M "THEN" M "ELSE" K+3; KM1:="IF" K-10 "DO" "IF"("IF" M>1 "THEN" A[M,M-1]=0 "ELSE" "TRUE") "THEN" "BEGIN" ALFR[M]:=A[M,M];BETA[M]:=B[M,M];ALFI[M]:=0;M:=M-1 "END" "ELSE" "BEGIN" L:=M-1;"IF" ABS(B[L,L])<=EPSB "THEN" "BEGIN" B[L,L]:=0;HSH2COL(L,L,N,L,A[L,L],A[M,L],A,B); A[M,L]:=B[M,L]:=0;ALFR[L]:=A[L,L];ALFR[M]:=A[M,M]; BETA[L]:=B[L,L];BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END" "ELSE" "IF" ABS(B[M,M])<=EPSB "THEN" "BEGIN"B[M,M]:=0;HSH2ROW2(1,1,M,M,L,A[M,M],A[M,L],A,B); A[M,L]:=B[M,L]:=0;ALFR[L]:=A[L,L];ALFR[M]:=A[M,M]; BETA[L]:=B[L,L];BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END""ELSE" "BEGIN" AN:=ABS(A[L,L])+ABS(A[L,M])+ABS(A[M,L])+ABS(A[M,M]); BN:=ABS(B[L,L])+ABS(B[L,M])+ABS(B[M,M]); A11:=A[L,L]/AN;A12:=A[L,M]/AN;A21:=A[M,L]/AN;A22:=A[M,M]/AN; "COMMENT" 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 20 ; B11:=B[L,L]/BN;B12:=B[L,M]/BN;B22:=B[M,M]/BN; E:=A11/B11; C:=((A22-E*B22)/B22-(A21*B12)/(B11*B22))/2; D:=C*C+(A21*(A12-E*B12))/(B11*B22); "IF" D>=0 "THEN" "BEGIN"E:=E+("IF"C<0"THEN"C-SQRT(D)"ELSE"C+SQRT(D)); A11:=A11-E*B11;A12:=A12-E*B12;A22:=A22-E*B22; "IF" ABS(A11)+ABS(A12)>=ABS(A21)+ABS(A22) "THEN" HSH2ROW2(1,1,M,M,L,A12,A11,A,B)"ELSE" HSH2ROW2(1,1,M,M,L,A22,A21,A,B); "IF"AN>=ABS(E)*BN"THEN" HSH2COL(L,L,N,L,B[L,L],B[M,L],A,B) "ELSE" HSH2COL(L,L,N,L,A[L,L],A[M,L],A,B); A[M,L]:=B[M,L]:=0; ALFR[L]:=A[L,L];ALFR[M]:=A[M,M];BETA[L]:=B[L,L]; BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END""ELSE" "BEGIN" ER:=E+C;EI:=SQRT(-D);A11R:=A11-ER*B11;A11I:=EI*B11; A12R:=A12-ER*B12;A12I:=EI*B12;A21R:=A21;A21I:=0; A22R:=A22-ER*B22;A22I:=EI*B22; "IF"ABS(A11R)+ABS(A11I)+ABS(A12R)+ABS(A12I)>= ABS(A21R)+ABS(A22R)+ABS(A22I)"THEN" CHSH2(A12R,A12I,-A11R,-A11I,CZ,SZR,SZI)"ELSE" CHSH2(A22R,A22I,-A21R,-A21I,CZ,SZR,SZI); "IF"AN>=(ABS(ER)+ABS(EI))*BN"THEN" CHSH2(CZ*B11+SZR*B12,SZI*B12,SZR*B22,SZI*B22,CQ,SQR,SQI) "ELSE"CHSH2(CZ*A11+SZR*A12,SZI*A12,CZ*A21+SZR*A22,SZI*A22, CQ,SQR,SQI);SSR:=SQR*SZR+SQI*SZI;SSI:=SQR*SZI-SQI*SZR; TR:=CQ*CZ*A11+CQ*SZR*A12+SQR*CZ*A21+SSR*A22; TI:=CQ*SZI*A12-SQI*CZ*A21+SSI*A22; BDR:=CQ*CZ*B11+CQ*SZR*B12+SSR*B22; BDI:=CQ*SZI*B12+SSI*B22; R:=SQRT(BDR*BDR+BDI*BDI);BETA[L]:=BN*R; ALFR[L]:=AN*(TR*BDR+TI*BDI)/R; ALFI[L]:=AN*(TR*BDI-TI*BDR)/R; TR:=SSR*A11-SQR*CZ*A12-CQ*SZR*A21+CQ*CZ*A22; TI:=-SSI*A11-SQI*CZ*A12+CQ*SZI*A21; BDR:=SSR*B11-SQR*CZ*B12+CQ*CZ*B22; BDI:=-SSI*B11-SQI*CZ*B12; R:=SQRT(BDR*BDR+BDI*BDI);BETA[M]:=BN*R; ALFR[M]:=AN*(TR*BDR+TI*BDI)/R; ALFI[M]:=AN*(TR*BDI-TI*BDR)/R; "END" "END";M:=M-2 "END" "END" QZVAL; DWARF:=EM[0];EPS:=EM[1]; HSHDECMUL(N,A,B,DWARF); HESTGL2(N,A,B); QZIT(N,A,B,EPS,EPSA,EPSB,ITER); QZVAL(N,A,B,EPSA,EPSB,ALFR,ALFI,BETA); "END" QZIVAL 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 21 ; "EOP" 0"CODE" 34601; "PROCEDURE" QZI(N,A,B,X,ALFR,ALFI,BETA,ITER,EM); "VALUE" N;"INTEGER" N;"ARRAY"A,B,X,ALFR,ALFI,BETA,EM; "INTEGER" "ARRAY" ITER; "BEGIN" "REAL" DWARF,EPS,EPSA,EPSB; "PROCEDURE" QZIT(N,A,B,X,EPS,EPSA,EPSB,ITER);"VALUE"N,EPS; "REAL" EPS,EPSA,EPSB;"INTEGER" N;"INTEGER" "ARRAY" ITER;"ARRAY" A,B,X; "BEGIN" "REAL" ANORM,BNORM,ANI,BNI,CONST,A10,A20,A30,B11, B22,B33,B44,A11,A12,A21,A22,A33,A34,A43,A44,B12,B34,OLD1,OLD2; "INTEGER" I,Q,M,M1,Q1,J,K,K1,K2,K3,KM1;"BOOLEAN" STATIONARY; ANORM:=BNORM:=0;"FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" BNI:=0;ITER[I]:=0;ANI:="IF" I>1"THEN"ABS(A[I,I-1])"ELSE" 0; "FOR" J:=I "STEP" 1 "UNTIL" N "DO" "BEGIN" ANI:=ANI+ABS(A[I,J]);BNI:=BNI+ABS(B[I,J]) "END";"IF" ANI>ANORM "THEN" ANORM:=ANI;"IF" BNI>BNORM"THEN" BNORM:=BNI "END";"IF" ANORM=0 "THEN" ANORM:=EPS;"IF"BNORM=0 "THEN" BNORM:=EPS; EPSA:=EPS*ANORM;EPSB:=EPS*BNORM; "FOR" M:=N,M "WHILE" M>=3 "DO" "BEGIN" "FOR" I:=M+1,I-1 "WHILE"("IF" I>1 "THEN"ABS(A[I,I-1])>EPSA "ELSE" "FALSE") "DO" Q:=I-1; "IF" Q>1 "THEN" A[Q,Q-1]:=0; L: "IF" Q>=M-1 "THEN" M:=Q-1 "ELSE" "BEGIN" "IF" ABS(B[Q,Q])<=EPSB "THEN" "BEGIN" B[Q,Q]:=0;Q1:=Q+1; HSH2COL(Q,Q,N,Q,A[Q,Q],A[Q1,Q],A,B);A[Q1,Q]:=0; Q:=Q1;"GOTO" L "END" "ELSE" M1:=M-1;Q1:=Q+1;CONST:=0.75;ITER[M]:=ITER[M]+1; STATIONARY:="IF" ITER[M]=1 "THEN" "TRUE" "ELSE" ABS(A[M,M-1])>=CONST*OLD1"AND"ABS(A[M-1,M-2])>=CONST*OLD2; "IF" ITER[M]>30"AND"STATIONARY "THEN" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" ITER[I]:=-1; "GOTO" OUT "END"; "IF" ITER[M]=10"AND"STATIONARY "THEN" "BEGIN" A10:=0;A20:=1;A30:=1.1605 "END" "ELSE" "BEGIN" B11:=B[Q,Q];B22:="IF" ABS(B[Q1,Q1])M "THEN" M "ELSE" K+3; KM1:="IF" K-10 "DO" "IF"("IF" M>1 "THEN" A[M,M-1]=0 "ELSE" "TRUE")"THEN" "BEGIN" ALFR[M]:=A[M,M];BETA[M]:=B[M,M];ALFI[M]:=0;M:=M-1 "END" "ELSE" "BEGIN" L:=M-1;"IF" ABS(B[L,L])<=EPSB "THEN" "BEGIN" B[L,L]:=0;HSH2COL(L,L,N,L,A[L,L],A[M,L],A,B); A[M,L]:=B[M,L]:=0;ALFR[L]:=A[L,L];ALFR[M]:=A[M,M]; BETA[L]:=B[L,L];BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END" "ELSE" "IF" ABS(B[M,M])<=EPSB "THEN" "BEGIN"B[M,M]:=0;HSH2ROW3(1,M,M,N,L,A[M,M],A[M,L],A,B,X); A[M,L]:=B[M,L]:=0;ALFR[L]:=A[L,L];ALFR[M]:=A[M,M]; BETA[L]:=B[L,L];BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END""ELSE" "BEGIN" AN:=ABS(A[L,L])+ABS(A[L,M])+ABS(A[M,L])+ABS(A[M,M]); "COMMENT" 1SECTION : 3.4.1.2 (FEBRUARY 1979) PAGE 23 ; BN:=ABS(B[L,L])+ABS(B[L,M])+ABS(B[M,M]); A11:=A[L,L]/AN;A12:=A[L,M]/AN;A21:=A[M,L]/AN;A22:=A[M,M]/AN; B11:=B[L,L]/BN;B12:=B[L,M]/BN;B22:=B[M,M]/BN; E:=A11/B11; C:=((A22-E*B22)/B22-(A21*B12)/(B11*B22))/2; D:=C*C+(A21*(A12-E*B12))/(B11*B22); "IF" D>=0 "THEN" "BEGIN"E:=E+("IF"C<0"THEN"C-SQRT(D)"ELSE"C+SQRT(D)); A11:=A11-E*B11;A12:=A12-E*B12;A22:=A22-E*B22; "IF" ABS(A11)+ABS(A12)>=ABS(A21)+ABS(A22) "THEN" HSH2ROW3(1,M,M,N,L,A12,A11,A,B,X)"ELSE" HSH2ROW3(1,M,M,N,L,A22,A21,A,B,X); "IF"AN>=ABS(E)*BN"THEN" HSH2COL(L,L,N,L,B[L,L],B[M,L],A,B) "ELSE" HSH2COL(L,L,N,L,A[L,L],A[M,L],A,B); A[M,L]:=B[M,L]:=0; ALFR[L]:=A[L,L];ALFR[M]:=A[M,M];BETA[L]:=B[L,L]; BETA[M]:=B[M,M];ALFI[M]:=ALFI[L]:=0; "END""ELSE" "BEGIN" ER:=E+C;EI:=SQRT(-D);A11R:=A11-ER*B11;A11I:=EI*B11; A12R:=A12-ER*B12;A12I:=EI*B12;A21R:=A21;A21I:=0; A22R:=A22-ER*B22;A22I:=EI*B22; "IF"ABS(A11R)+ABS(A11I)+ABS(A12R)+ABS(A12I)>= ABS(A21R)+ABS(A22R)+ABS(A22I)"THEN" CHSH2(A12R,A12I,-A11R,-A11I,CZ,SZR,SZI)"ELSE" CHSH2(A22R,A22I,-A21R,-A21I,CZ,SZR,SZI); "IF"AN>=(ABS(ER)+ABS(EI))*BN"THEN" CHSH2(CZ*B11+SZR*B12,SZI*B12,SZR*B22,SZI*B22,CQ,SQR,SQI) "ELSE"CHSH2(CZ*A11+SZR*A12,SZI*A12,CZ*A21+SZR*A22,SZI*A22, CQ,SQR,SQI);SSR:=SQR*SZR+SQI*SZI;SSI:=SQR*SZI-SQI*SZR; TR:=CQ*CZ*A11+CQ*SZR*A12+SQR*CZ*A21+SSR*A22; TI:=CQ*SZI*A12-SQI*CZ*A21+SSI*A22; BDR:=CQ*CZ*B11+CQ*SZR*B12+SSR*B22; BDI:=CQ*SZI*B12+SSI*B22; R:=SQRT(BDR*BDR+BDI*BDI);BETA[L]:=BN*R; ALFR[L]:=AN*(TR*BDR+TI*BDI)/R; ALFI[L]:=AN*(TR*BDI-TI*BDR)/R; TR:=SSR*A11-SQR*CZ*A12-CQ*SZR*A21+CQ*CZ*A22; TI:=-SSI*A11-SQI*CZ*A12+CQ*SZI*A21; BDR:=SSR*B11-SQR*CZ*B12+CQ*CZ*B22; BDI:=-SSI*B11-SQI*CZ*B12; R:=SQRT(BDR*BDR+BDI*BDI);BETA[M]:=BN*R; ALFR[M]:=AN*(TR*BDR+TI*BDI)/R; ALFI[M]:=AN*(TR*BDI-TI*BDR)/R; "END" "END";M:=M-2 "END" "END" QZVAL 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 24 ; "PROCEDURE" QZVEC(N,A,B,X,EPSA,EPSB,ALFR,ALFI,BETA);"VALUE"N,EPSA,EPSB; "INTEGER" N;"REAL" EPSA,EPSB;"ARRAY" A,B,ALFR,ALFI,BETA,X; "BEGIN""INTEGER" M,MR,MI,L,L1,J,K;"REAL" BETM,ALFM,SL,SK,D,TKK,TKL,TLK, TLL,ALMI,ALMR,TR,TI,SLR,SLI,SKR,SKI,DR,DI,TKKR,TKKI,TKLR,TKLI,TLKR, TLKI,TLLR,TLLI,S,R; "FOR" M:=N "STEP" -1 "UNTIL" 1 "DO" "IF" ALFI[M]=0 "THEN" "BEGIN" "COMMENT" M-TH REAL VECTOR; ALFM:=ALFR[M];BETM:=BETA[M];B[M,M]:=1;L1:=M; "FOR" L:=M-1 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" SL:=0; "FOR" J:=L1 "STEP" 1 "UNTIL" M "DO" SL:=SL+(BETM*A[L,J]-ALFM*B[L,J])*B[J,M]; "IF"("IF"L^=1"THEN"BETM*A[L,L-1]=0"ELSE""TRUE")"THEN" "BEGIN""COMMENT" 1-1 BLOCK; D:=BETM*A[L,L]-ALFM*B[L,L]; "IF" D=0 "THEN" D:=(EPSA+EPSB)/2;B[L,M]:=-SL/D "END""ELSE" "BEGIN" "COMMENT" 2-2 BLOCK;K:=L-1;SK:=0; "FOR"J:=L1 "STEP" 1 "UNTIL" M "DO" SK:=SK+(BETM*A[K,J]-ALFM*B[K,J])*B[J,M]; TKK:=BETM*A[K,K]-ALFM*B[K,K]; TKL:=BETM*A[K,L]-ALFM*B[K,L]; TLK:=BETM*A[L,K]; TLL:=BETM*A[L,L]-ALFM*B[L,L]; D:=TKK*TLL-TKL*TLK;"IF" D=0 "THEN" D:=(EPSA+EPSB)/2; B[L,M]:=(TLK*SK-TKK*SL)/D; B[K,M]:="IF"ABS(TKK)>=ABS(TLK)"THEN"-(SK+TKL*B[L,M])/TKK "ELSE" -(SL+TLL*B[L,M])/TLK;L:=L-1 "END";L1:=L "END" "END""ELSE" "BEGIN" "COMMENT" COMPLEX VECTOR; ALMR:=ALFR[M-1];ALMI:=ALFI[M-1];BETM:=BETA[M-1];MR:=M-1;MI:=M; B[M-1,MR]:=ALMI*B[M,M]/(BETM*A[M,M-1]); B[M-1,MI]:=(BETM*A[M,M]-ALMR*B[M,M])/(BETM*A[M,M-1]); B[M,MR]:=0;B[M,MI]:=-1;L1:=M-1; "FOR" L:=M-2 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" SLR:=SLI:=0; "FOR" J:=L1 "STEP" 1 "UNTIL" M "DO" "BEGIN" TR:=BETM*A[L,J]-ALMR*B[L,J]; TI:=-ALMI*B[L,J]; SLR:=SLR+TR*B[J,MR]-TI*B[J,MI]; SLI:=SLI+TR*B[J,MI]+TI*B[J,MR] "END"; "COMMENT" 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 25 ; "IF"("IF"L^=1"THEN"BETM*A[L,L-1]=0"ELSE""TRUE")"THEN" "BEGIN"DR:=BETM*A[L,L]-ALMR*B[L,L]; DI:=-ALMI*B[L,L]; COMDIV(-SLR,-SLI,DR,DI,B[L,MR],B[L,MI]); "END""ELSE" "BEGIN" K:=L-1;SKR:=SKI:=0; "FOR" J:=L1 "STEP" 1 "UNTIL" M "DO" "BEGIN" TR:=BETM*A[K,J]-ALMR*B[K,J]; TI:=-ALMI*B[K,J]; SKR:=SKR+TR*B[J,MR]-TI*B[J,MI]; SKI:=SKI+TR*B[J,MI]+TI*B[J,MR] "END"; TKKR:=BETM*A[K,K]-ALMR*B[K,K]; TKKI:=-ALMI*B[K,K]; TKLR:=BETM*A[K,L]-ALMR*B[K,L]; TKLI:=-ALMI*B[K,L]; TLKR:=BETM*A[L,K];TLKI:=0; TLLR:=BETM*A[L,L]-ALMR*B[L,L]; TLLI:=-ALMI*B[L,L]; DR:=TKKR*TLLR-TKKI*TLLI-TKLR*TLKR; DI:=TKKR*TLLI+TKKI*TLLR-TKLI*TLKR; "IF"DR=0"AND"DI=0"THEN"DR:=(EPSA+EPSB)/2; COMDIV(TLKR*SKR-TKKR*SLR+TKKI*SLI,TLKR*SKI-TKKR*SLI- TKKI*SLR,DR,DI,B[L,MR],B[L,MI]); "IF"ABS(TKKR)+ABS(TKKI)>=ABS(TLKR)"THEN" COMDIV(-SKR-TKLR*B[L,MR]+TKLI*B[L,MI],-SKI-TKLR*B[L,MI] -TKLI*B[L,MR],TKKR,TKKI,B[K,MR],B[K,MI])"ELSE" COMDIV(-SLR-TLLR*B[L,MR]+TLLI*B[L,MI],-SLI-TLLR*B[L,MI] -TLLI*B[L,MR],TLKR,TLKI,B[K,MR],B[K,MI]);L:=L-1 "END";L1:=L "END";M:=M-1 "END"; "FOR" M:=N "STEP" -1 "UNTIL" 1 "DO" "FOR" K:=1 "STEP" 1 "UNTIL" N "DO" X[K,M]:=MATMAT(1,M,K,M,X,B); "FOR" M:=N "STEP" -1 "UNTIL" 1 "DO" "BEGIN" S:=0;"IF" ALFI[M]=0 "THEN" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" R:=ABS(X[K,M]); "IF" R>=S "THEN""BEGIN" S:=R;D:=X[K,M] "END" "END";"FOR" K:=1 "STEP" 1 "UNTIL" N "DO" X[K,M]:=X[K,M]/D "END""ELSE" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" R:=ABS(X[K,M-1])+ABS(X[K,M]); R:=R*SQRT((X[K,M-1]/R)**2+(X[K,M]/R)**2); "IF" R>=S"THEN" "BEGIN" S:=R;DR:=X[K,M-1];DI:=X[K,M] "END" "END";"FOR" K:=1 "STEP" 1 "UNTIL" N "DO" COMDIV(X[K,M-1],X[K,M],DR,DI,X[K,M-1],X[K,M]);M:=M-1 "END" "END" "END" QZVEC; "COMMENT" 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 26 ; DWARF:=EM[0];EPS:=EM[1]; HSHDECMUL(N,A,B,DWARF); HESTGL3(N,A,B,X); QZIT(N,A,B,X,EPS,EPSA,EPSB,ITER); QZVAL(N,A,B,X,EPSA,EPSB,ALFR,ALFI,BETA); QZVEC(N,A,B,X,EPSA,EPSB,ALFR,ALFI,BETA) "END" QZI; "EOP" 0"CODE" 34602; "PROCEDURE" HSHDECMUL(N,A,B,DWARF);"VALUE"N,DWARF;"INTEGER"N; "REAL" DWARF;"ARRAY" A,B; "BEGIN" "ARRAY" V[1:N];"INTEGER" J,K,K1,N1;"REAL" R,T,C; K:=1;N1:=N+1; "FOR" K1:=2 "STEP" 1 "UNTIL" N1 "DO" "BEGIN" R:=TAMMAT(K1,N,K,K,B,B); "IF" R>DWARF "THEN" "BEGIN" R:="IF" B[K,K]<0 "THEN" -SQRT(R+B[K,K]*B[K,K]) "ELSE" SQRT(R+B[K,K]*B[K,K]);T:=B[K,K]+R;C:=-T/R; B[K,K]:=-R;V[K]:=1; "FOR" J:=K1 "STEP" 1 "UNTIL" N "DO" V[J]:=B[J,K]/T; HSHVECMAT(K,N,K1,N,C,V,B);HSHVECMAT(K,N,1,N,C,V,A) "END";K:=K1 "END" "END" HSHDECMUL; "EOP" 0"CODE" 34603; "PROCEDURE" HESTGL3(N,A,B,X);"VALUE" N;"INTEGER" N;"ARRAY" A,B,X; "BEGIN" "INTEGER" NM1,K,L,K1,L1; "IF" N>2 "THEN" "BEGIN" "FOR" K:=2 "STEP" 1 "UNTIL" N "DO" "FOR" L:=1 "STEP" 1 "UNTIL" K-1 "DO" B[K,L]:=0; NM1:=N-1;K:=1; "FOR" K1:= 2 "STEP" 1 "UNTIL" NM1 "DO" "BEGIN" L1:=N; "FOR" L:=N-1 "STEP" -1 "UNTIL" K1 "DO" "BEGIN" HSH2COL(K,L,N,L,A[L,K],A[L1,K],A,B);A[L1,K]:=0; HSH2ROW3(1,N,L1,N,L,B[L1,L1],B[L1,L],A,B,X); B[L1,L]:=0;L1:=L "END";K:=K1 "END" "END" "END" HESTGL3 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 27 ; "EOP" 0"CODE" 34604; "PROCEDURE" HESTGL2(N,A,B);"VALUE" N;"INTEGER" N;"ARRAY" A,B; "BEGIN" "INTEGER" NM1,K,L,K1,L1; "IF" N>2 "THEN" "BEGIN" "FOR" K:=2 "STEP" 1 "UNTIL" N "DO" "FOR" L:=1 "STEP" 1 "UNTIL" K-1 "DO" B[K,L]:=0; NM1:=N-1;K:=1; "FOR" K1:= 2 "STEP" 1 "UNTIL" NM1 "DO" "BEGIN" L1:=N; "FOR" L:=N-1 "STEP" -1 "UNTIL" K1 "DO" "BEGIN" HSH2COL(K,L,N,L,A[L,K],A[L1,K],A,B);A[L1,K]:=0; HSH2ROW2(1,1,N,L1,L,B[L1,L1],B[L1,L],A,B); B[L1,L]:=0;L1:=L "END";K:=K1 "END" "END" "END" HESTGL2; "EOP" 0"CODE" 34605; "PROCEDURE" HSH2COL(LA,LB,U,I,A1,A2,A,B);"VALUE" LA,LB,U,I,A1,A2; "INTEGER" LA,LB,U,I;"REAL" A1,A2;"ARRAY" A,B; "IF"A2^=0 "THEN" "BEGIN" "REAL" R,T,C;"ARRAY" V[I:I+1]; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2) "ELSE" SQRT(A1*A1+A2*A2); T:=A1+R;C:=-T/R;V[I]:=1;V[I+1]:=A2/T; HSHVECMAT(I,I+1,LA,U,C,V,A);HSHVECMAT(I,I+1,LB,U,C,V,B) "END" HSH2COL; "EOP" 0"CODE" 34606; "PROCEDURE" HSH3COL(LA,LB,U,I,A1,A2,A3,A,B); "VALUE"LA,LB,U,I,A1,A2,A3;"INTEGER"LA,LB,I,U;"REAL"A1,A2,A3;"ARRAY"A,B; "IF" A2^=0 "OR" A3^=0 "THEN" "BEGIN" "REAL" R,T,C;"ARRAY" V[I:I+2]; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2+A3*A3) "ELSE" SQRT(A1*A1+A2*A2+A3*A3); T:=A1+R;C:=-T/R;V[I]:=1;V[I+1]:=A2/T;V[I+2]:=A3/T; HSHVECMAT(I,I+2,LA,U,C,V,A);HSHVECMAT(I,I+2,LB,U,C,V,B) "END" HSH3COL 1SECTION : 3.4.1.2 (JANUARY 1976) PAGE 28 ; "EOP" 0"CODE" 34607; "PROCEDURE" HSH2ROW3(L,UA,UB,UX,J,A1,A2,A,B,X);"VALUE" L,UA,UB,UX, J,A1,A2;"INTEGER" L,UA,UB,UX,J;"REAL" A1,A2;"ARRAY" A,B,X; "IF"A2^=0 "THEN" "BEGIN""REAL" R,T,C;"INTEGER" K;"ARRAY" V[J:J+1]; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2) "ELSE" SQRT(A1*A1+A2*A2); T:=A1+R;C:=-T/R;V[J+1]:=1;V[J]:=A2/T; HSHVECTAM(L,UA,J,J+1,C,V,A);HSHVECTAM(L,UB,J,J+1,C,V,B); HSHVECTAM(1,UX,J,J+1,C,V,X) "END" HSH2ROW3; "EOP" 0"CODE" 34608; "PROCEDURE" HSH2ROW2(LA,LB,UA,UB,J,A1,A2,A,B);"VALUE"LA,LB,UA,UB, J,A1,A2;"INTEGER" LA,LB,UA,UB,J;"REAL" A1,A2;"ARRAY" A,B; "IF"A2^=0 "THEN" "BEGIN" "REAL" R,T,C;"INTEGER" K;"ARRAY" V[J:J+1]; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2) "ELSE" SQRT(A1*A1+A2*A2); T:=A1+R;C:=-T/R;V[J+1]:=1;V[J]:=A2/T; HSHVECTAM(LA,UA,J,J+1,C,V,A);HSHVECTAM(LB,UB,J,J+1,C,V,B) "END" HSH2ROW2; "EOP" 0"CODE" 34609; "PROCEDURE" HSH3ROW3(L,U,UX,J,A1,A2,A3,A,B,X); "VALUE"L,U,UX,J,A1,A2,A3;"INTEGER"L,J,U,UX;"REAL"A1,A2,A3;"ARRAY"A,B,X; "IF" A2^=0 "OR" A3^=0 "THEN" "BEGIN" "REAL" R,T,C;"ARRAY" V[J:J+2];"INTEGER" K; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2+A3*A3) "ELSE" SQRT(A1*A1+A2*A2+A3*A3); T:=A1+R;C:=-T/R;V[J+2]:=1;V[J+1]:=A2/T;V[J]:=A3/T; HSHVECTAM(L,U,J,J+2,C,V,A);HSHVECTAM(L,U,J,J+2,C,V,B); HSHVECTAM(L,UX,J,J+2,C,V,X) "END" HSH3ROW3; "EOP" 0"CODE" 34610; "PROCEDURE" HSH3ROW2(LA,LB,U,J,A1,A2,A3,A,B); "VALUE"LA,LB,U,J,A1,A2,A3;"INTEGER"LA,LB,U,J;"REAL"A1,A2,A3;"ARRAY"A,B; "IF" A2^=0 "OR" A3^=0 "THEN" "BEGIN" "REAL" R,T,C;"ARRAY" V[J:J+2]; R:="IF" A1<0 "THEN" -SQRT(A1*A1+A2*A2+A3*A3) "ELSE" SQRT(A1*A1+A2*A2+A3*A3); T:=A1+R;C:=-T/R;V[J+2]:=1;V[J+1]:=A2/T;V[J]:=A3/T; HSHVECTAM(LA,U,J,J+2,C,V,A);HSHVECTAM(LB,U,J,J+2,C,V,B) "END" HSH3ROW2; "EOP" 1SECTION : 3.5.1.1 (DECEMBER 1975) PAGE 1 AUTHORS : G.H.GOLUB AND C.REINSCH CONTRIBUTOR : D.T.WINTER INSTITUTE : MATHEMATICAL CENTRE RECEIVED : 731217 BRIEF DESCRIPTION : THIS SECTION CONTAINS TWO PROCEDURES, QRISNGVALBID AND QRISNGVALDECBID. BOTH PROCEDURES CALCULATE THE SINGULAR VALUES OF A BIDIAGONAL MATRIX. MOREOVER, THE SECOND PROCEDURE CALCULATES THE THE SINGULAR VALUES DECOMPOSITION OF A FULL MATRIX OF WHICH THE BIDIAGONAL AND THE PRE- AND POSTMULTIPLYING MATRICES, AS CALCULATED BY HSHREABID ARE GIVEN. KEYWORDS : SINGULAR VALUES QR ITERATION BIDIAGONAL MATRICES 1SECTION : 3.5.1.1 (DECEMBER 1975) PAGE 2 SUBSECTION : QRISNGVALBID CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "INTEGER" "PROCEDURE" QRISNGVALBID(D, B, N, EM); "VALUE" N; "INTEGER" N; "ARRAY" D, B, EM; "CODE" 34270; THE MEANING OF THE FORMAL PARAMETERS IS: D: ; "ARRAY" D[1:N]; ENTRY: THE DIAGONAL OF THE BIDIAGONAL MATRIX; EXIT: THE SINGULAR VALUES; B: ; "ARRAY" B[1:N]; ENTRY: THE SUPER DIAGONAL OF THE BIDIAGONAL MATRIX,IN B[1:N-1]; N: ; THE LENGTH OF B AND D; EM: ; "ARRAY" EM[1:7]; ENTRY: EM[1]: THE INFINITY NORM OF THE MATRIX; EM[2]: THE RELATIVE PRECISION IN THE SINGULAR VALUES; EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED; EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE; EXIT: EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6]. MOREOVER : QRISNGVALBID:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. A NUMBER NOT EQUAL TO ZERO IF THE NUMBER OF ITERATIONS EXCEEDS EM[4]. PROCEDURES USED : NONE REQUIRED CENTRAL MEMORY : NO AUXILIARY ARRAYS ARE DECLARED RUNNING TIME : THE RUNNING TIME DEPENDS STRONGLY UPON THE PROPERTIES OF THE MATRIX METHOD AND PERFORMANCE : THE METHOD IS DESCRIBED IN DETAIL IN [1]. THIS PROCEDURE IS A REWRITING OF PART OF THE PROCEDURE SVD PUBLISHED THERE BY G.H.GOLUB AND C.REINSCH. LANGUAGE : ALGOL 60. 1SECTION : 3.5.1.1 (DECEMBER 1975) PAGE 3 SUBSECTION : QRISNGVALDECBID CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "INTEGER" "PROCEDURE" QRISNGVALDECBID(D, B, M, N, U, V, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" D, B, U, V, EM; "CODE" 34271; THE MEANING OF THE FORMAL PARAMETERS IS : D: ; "ARRAY" D[1:N]; ENTRY: THE DIAGONAL OF THE BIDIAGONAL MATRIX; EXIT: THE SINGULAR VALUES; B: ; "ARRAY" B[1:N]; ENTRY: THE SUPER DIAGONAL OF THE BIDIAGONAL MATRIX,IN B[1:N-1]; M: ; THE NUMBER OF ROWS OF THE MATRIX U; N: ; THE LENGTH OF B AND D, THE NUMBER OF COLUMNS OF U AND THE NUMBER OF COLUMNS AND ROWS OF V; U: ; "ARRAY" U[1:M,1:N]; ENTRY: THE PREMULTIPLYING MATRIX AS PRODUCED BY PRETFMMAT (SECTION 3.2.2.1.1); EXIT: THE PREMULTIPLYING MATRIX U OF THE SINGULAR VALUES DECOMPOSITION U * S * V'; V: ; "ARRAY" V[1:N,1:N]; ENTRY: THE TRANSPOSE OF THE POSTMULTIPLYING MATRIX AS PRODUCED BY PSTTFMMAT (SECTION 3.2.2.1.1); EXIT: THE TRANSPOSE OF THE POSTMULTIPLYING MATRIX V OF THE SINGULAR VALUES DECOMPOSITION; EM: ; "ARRAY" EM[1:7]; ENTRY: EM[1]: THE INFINITY NORM OF THE MATRIX; EM[2]: THE RELATIVE PRECISION IN THE SINGULAR VALUES; EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED; EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE; EXIT: EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6]. MOREOVER : QRISNGVALDECBID:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. A NUMBER NOT EQUAL TO ZERO IF THE NUMBER OF ITERATIONS EXCEEDS EM[4]. 1SECTION : 3.5.1.1 (DECEMBER 1975) PAGE 4 PROCEDURES USED : ROTCOL = CP34040 REQUIRED CENTRAL MEMORY : NO AUXILIARY ARRAYS ARE DECLARED RUNNING TIME : THE RUNNING TIME DEPENDS STRONGLY UPON THE PROPERTIES OF THE MATRIX METHOD AND PERFORMANCE : THE METHOD IS DESCRIBED IN DETAIL IN [1]. THIS PROCEDURE IS A REWRITING OF PART OF THE PROCEDURE SVD PUBLISHED THERE BY G.H.GOLUB AND C.REINSCH. LANGUAGE : ALGOL 60 REFERENCES : [1] WILKINSON, J.H. AND C.REINSCH HANDBOOK OF AUTOMATIC COMPUTATION, VOL. 2 LINEAR ALGEBRA HEIDELBERG (1971) EXAMPLE OF USE : FOR AN EXAMPLE OF USE ONE IS REFERRED TO SECTION 3.5.1.2 SOURCE TEXT(S): 0"CODE" 34270; "INTEGER" "PROCEDURE" QRISNGVALBID(D, B, N, EM); "VALUE" N; "INTEGER" N; "ARRAY" D, B, EM; "BEGIN" "INTEGER" N1, K, K1, I, I1, COUNT, MAX, RNK; "REAL" TOL, BMAX, Z, X, Y, G, H, F, C, S, MIN; TOL:= EM[2] * EM[1]; COUNT:= 0; BMAX:= 0; MAX:= EM[4]; MIN:= EM[6]; RNK:= N; IN: K:= N; N1:= N - 1; NEXT: K:= K - 1; "IF" K > 0 "THEN" "BEGIN" "IF" ABS(B[K]) >= TOL "THEN" "BEGIN" "IF" ABS(D[K]) >= TOL "THEN" "GOTO" NEXT; C:= 0; S:= 1; "FOR" I:= K "STEP" 1 "UNTIL" N1 "DO" "BEGIN" F:= S * B[I]; B[I]:= C * B[I]; I1:= I + 1; "IF" ABS(F) < TOL "THEN" "GOTO" NEGLECT; G:= D[I1]; D[I1]:= H:= SQRT(F * F + G * G); C:= G / H; S:= - F / H "END" 1SECTION : 3.5.1.1 (JULY 1974) PAGE 5 ; NEGLECT: "END" "ELSE" "IF" ABS(B[K]) > BMAX "THEN" BMAX:= ABS(B[K]) "END"; "IF" K = N1 "THEN" "BEGIN" "IF" D[N] < 0 "THEN" D[N]:= - D[N]; "IF" D[N] <= MIN "THEN" RNK:= RNK - 1; N:= N1 "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" END; K1:= K + 1; Z:= D[N]; X:= D[K1]; Y:= D[N1]; G:= "IF" N1 = 1 "THEN" 0 "ELSE" B[N1 - 1]; H:= B[N1]; F:= ((Y - Z) * (Y + Z) + (G - H) * (G + H)) / (2 * H * Y); G:= SQRT(F * F + 1); F:= ((X - Z) * (X + Z) + H * (Y / ("IF" F < 0 "THEN" F - G "ELSE" F + G) - H)) / X; C:= S:= 1; "FOR" I:= K1 + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" I1:= I - 1; G:= B[I1]; Y:= D[I]; H:= S * G; G:= C * G; Z:= SQRT(F * F + H * H); C:= F / Z; S:= H / Z; "IF" I1 ^= K1 "THEN" B[I1 - 1]:= Z; F:= X * C + G * S; G:= G * C - X * S; H:= Y * S; Y:= Y * C; D[I1]:= Z:= SQRT(F * F + H * H); C:= F / Z; S:= H / Z; F:= C * G + S * Y; X:= C * Y - S * G "END"; B[N1]:= F; D[N]:= X "END"; "IF" N > 0 "THEN" "GOTO" IN; END: EM[3]:= BMAX; EM[5]:= COUNT; EM[7]:= RNK; QRISNGVALBID:= N "END" QRISNGVALBID; "EOP" 0"CODE" 34271; "INTEGER" "PROCEDURE" QRISNGVALDECBID(D, B, M, N, U, V, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" D, B, U, V, EM; "BEGIN" "INTEGER" N0, N1, K, K1, I, I1, COUNT, MAX, RNK; "REAL" TOL, BMAX, Z, X, Y, G, H, F, C, S, MIN; "COMMENT" 1SECTION : 3.5.1.1 (JULY 1974) PAGE 6 ; TOL:= EM[2] * EM[1]; COUNT:= 0; BMAX:= 0; MAX:= EM[4]; MIN:= EM[6]; RNK:= N0:= N; IN: K:= N; N1:= N - 1; NEXT: K:= K - 1; "IF" K > 0 "THEN" "BEGIN" "IF" ABS(B[K]) >= TOL "THEN" "BEGIN" "IF" ABS(D[K]) >= TOL "THEN" "GOTO" NEXT; C:= 0; S:= 1; "FOR" I:= K "STEP" 1 "UNTIL" N1 "DO" "BEGIN" F:= S * B[I]; B[I]:= C * B[I]; I1:= I + 1; "IF" ABS(F) < TOL "THEN" "GOTO" NEGLECT; G:= D[I1]; D[I1]:= H:= SQRT(F * F + G * G); C:= G / H; S:= - F / H; ROTCOL(1, M, K, I1, U, C, S) "END"; NEGLECT: "END" "ELSE" "IF" ABS(B[K]) > BMAX "THEN" BMAX:= ABS(B[K]) "END"; "IF" K = N1 "THEN" "BEGIN" "IF" D[N] < 0 "THEN" "BEGIN" D[N]:= - D[N]; "FOR" I:= 1 "STEP" 1 "UNTIL" N0 "DO" V[I,N]:= - V[I,N] "END"; "IF" D[N] <= MIN "THEN" RNK:= RNK - 1; N:= N1 "END" "ELSE" "BEGIN" COUNT:= COUNT + 1; "IF" COUNT > MAX "THEN" "GOTO" END; K1:= K + 1; Z:= D[N]; X:= D[K1]; Y:= D[N1]; G:= "IF" N1 = 1 "THEN" 0 "ELSE" B[N1 - 1]; H:= B[N1]; F:= ((Y - Z) * (Y + Z) + (G - H) * (G + H)) / (2 * H * Y); G:= SQRT(F * F + 1); F:= ((X - Z) * (X + Z) + H * (Y / ("IF" F < 0 "THEN" F - G "ELSE" F + G) - H)) / X; C:= S:= 1; "FOR" I:= K1 + 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" I1:= I - 1; G:= B[I1]; Y:= D[I]; H:= S * G; G:= C * G; Z:= SQRT(F * F + H * H); C:= F / Z; S:= H / Z; "IF" I1 ^= K1 "THEN" B[I1 - 1]:= Z; F:= X * C + G * S; G:= G * C - X * S; H:= Y * S; Y:= Y * C; ROTCOL(1, N0, I1, I, V, C, S); D[I1]:= Z:= SQRT(F * F + H * H); C:= F / Z; S:= H / Z; F:= C * G + S * Y; X:= C * Y - S * G; ROTCOL(1, M, I1, I, U, C, S) "END"; B[N1]:= F; D[N]:= X "END"; "IF" N > 0 "THEN" "GOTO" IN; END: EM[3]:= BMAX; EM[5]:= COUNT; EM[7]:= RNK; QRISNGVALDECBID:= N "END" QRISNGVALDECBID; "EOP" 1SECTION : 3.5.1.2 (JULY 1974) PAGE 1 AUTHOR : D.T.WINTER INSTITUTE : MATHEMATICAL CENTRE RECEIVED : 731217 BRIEF DESCRIPTION : THIS SECTION CONTAINS TWO PROCEDURES, QRISNGVAL AND QRISNGVALDEC. QRISNGVAL CALCULATES THE SINGULAR VALUES OF A GIVEN MATRIX. QRISNGVALDEC CALCULATES THE SINGULAR VALUES DECOMPOSITION U * S * V', WITH U AND V ORTHOGONAL AND S POSITIVE DIAGONAL. KEYWORDS : SINGULAR VALUES QR ITERATION 1SECTION : 3.5.1.2 (DECEMBER 1975) PAGE 2 SUBSECTION : QRISNGVAL CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "INTEGER" "PROCEDURE" QRISNGVAL(A, M, N, VAL, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" A, VAL, EM; "CODE" 34272; THE MEANING OF THE FORMAL PARAMETERS IS : A: ; "ARRAY" A[1:M,1:N]; ENTRY: THE INPUT MATRIX; EXIT: DATA CONCERNING THE TRANSFORMATION TO BIDIAGONAL FORM; M: ; THE NUMBER OF ROWS OF A; N: ; THE NUMBER OF COLUMNS OF A, N SHOULD SATISFY N <= M; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE SINGULAR VALUES; EM: ; "ARRAY" EM[0:7]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE PRECISION IN THE SINGULAR VALUES; EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED; EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE; EXIT: EM[1]: THE INFINITY NORM OF THE MATRIX; EM[3]: THE MAXIMAL NEGLECTED SUPERDIAGONAL ELEMENT; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6]. MOREOVER : QRISNGVAL:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. A NUMBER NOT EQUAL TO ZERO IF THE NUMBER OF ITERATIONS EXCEEDS EM[4]. PROCEDURES USED : HSHREABID = CP34260 QRISNGVALBID = CP34270 REQUIRED CENTRAL MEMORY : AN AUXILIARY ARRAY OF N REALS IS DECLARED RUNNING TIME : THE RUNNING TIME DEPENDS UPON THE PROPERTIES OF THE MATRIX, HOWEVER THE PROCESS OF BIDIAGONALIZATION DOMINATES, AND ITS RUNNING TIME IS PROPORTIONAL TO (M + N) * N * N METHOD AND PERFORMANCE : THE MATRIX IS FIRST TRANSFORMED TO BIDIAGONAL FORM BY THE PROCEDURE HSHREABID (SECTION 3.2.2.1.1) , AND THEN THE SINGULAR VALUES ARE CALCULATED BY QRISNGVALBID (SECTION 3.5.1.1). LANGUAGE : ALGOL 60 1SECTION : 3.5.1.2 (DECEMBER 1975) PAGE 3 SUBSECTION : QRISNGVALDEC CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "INTEGER" "PROCEDURE" QRISNGVALDEC(A, M, N, VAL, V, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" A, VAL, V, EM; "CODE" 34273; THE MEANING OF THE FORMAL PARAMETERS IS: A: ; "ARRAY" A[1:M,1:N]; ENTRY: THE GIVEN MATRIX; EXIT: THE MATRIX U IN THE SINGULAR VALUES DECOMPOSITION U * S * V'; M: ; THE NUMBER OF ROWS OF A; N: ; THE NUMBER OF COLUMNS OF A, N SHOULD SATISFY N <= M; VAL: ; "ARRAY" VAL[1:N]; EXIT: THE SINGULAR VALUES; V: ; "ARRAY" V[1:N,1:N]; EXIT: THE MATRIX V OF THE SINGULAR VALUES DECOMPOSITION (A = U . S . V' ); EM: ; "ARRAY" EM[0:7]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE PRECISION IN THE SINGULAR VALUES; EM[4]: THE MAXIMAL NUMBER OF ITERATIONS TO BE PERFORMED; EM[6]: THE MINIMAL NON-NEGLECTABLE SINGULAR VALUE; EXIT: EM[1]: THE INFINITY NORM OF THE MATRIX; EM[3]: THE MAXIMAL NEGLECTED SUPER DIAGONAL ELEMENT; EM[5]: THE NUMBER OF ITERATIONS PERFORMED; EM[7]: THE NUMERICAL RANK OF THE MATRIX, I.E. THE NUMBER OF SINGULAR VALUES GREATER THAN OR EQUAL TO EM[6]. MOREOVER : QRISNGVALDEC:= THE NUMBER OF SINGULAR VALUES NOT FOUND, I.E. A NUMBER NOT EQUAL TO ZERO IF THE NUMBER OF ITERATIONS EXCEEDS EM[4]. PROCEDURES USED : HSHREABID = CP34260 PSTTFMMAT = CP34261 PRETFMMAT = CP34262 QRISNGVALDECBID = CP34271 REQUIRED CENTRAL MEMORY : AN AUXILIARY ARRAY OF N ELEMENTS IS DECLARED 1SECTION : 3.5.1.2 (DECEMBER 1975) PAGE 4 RUNNING TIME : THE RUNNING TIME DEPENDS UPON THE PROPERTIES OF THE MATRIX, HOWEVER THE PROCESS OF BIDIAGONALIZATION DOMINATES, AND ITS RUNNING TIME IS PROPORTIONAL TO (M + N) * N * N METHOD AND PERFORMANCE: THE MATRIX IS FIRST TRANSFORMED TO BIDIAGONAL FORM BY THE PROCEDURE HSHREABID (SECTION 3.2.2.1.1), THE TWO TRANSFORMING MATRICES ARE CALCULATED BY THE PROCEDURES PSTTFMMAT AND PRETFMMAT (SECTIONS 3.2.2.1.2 AND 3.2.2.1.3 RESPECTIVELY), AND FINALLY THE SINGULAR VALUES DECOMPOSITION IS CALCULATED BY QRISNGVALDECBID (SECTION 3.5.1.1). LANGUAGE : ALGOL 60 REFERENCES : WILKINSON, J.H. AND C.REINSCH HANDBOOK OF AUTOMATIC COMPUTATION, VOL. 2 LINEAR ALGEBRA HEIDELBERG (1971) EXAMPLE OF USE : AS THE PROCEDURE QRISNGVALDEC CALCULATES THE SINGULAR VALUES OF A MATRIX IN EXACTLY THE SAME WAY AS QRISNGVAL, WE GIVE HER ONLY AN EXAMPLE OF USE OF THE PROCEDURE QRISNGVALDEC. FIRST WE GIVE A PROGRAM, AND THEN THE RESULTS OF THIS PROGRAM: "BEGIN" "ARRAY" A[1:6,1:5], V[1:5,1:5], VAL[1:5], EM[0:7]; "INTEGER" I, J; "FOR" I:= 1 "STEP" 1 "UNTIL" 6 "DO" "FOR" J:= 1 "STEP" 1 "UNTIL" 5 "DO" A[I,J]:= 1 / (I + J - 1); EM[0]:= "-14; EM[2]:= "-12; EM[4]:= 25; EM[6]:= "-10; I:= QRISNGVALDEC(A, 6, 5, VAL, V, EM); OUTPUT(61, "("3B, "("NUMBER SINGULAR VALUES NOT FOUND : ")", 3ZD, /, 3B, "("INFINITY NORM : ")", N, /, 3B, "("MAX NEGLECTED SUBDIAGONAL ELEMENT : ")", N, /, 3B, "("NUMBER ITERATIONS : ")", 3ZD, /, 3B, "("NUMERICAL RANK : ")", 3ZD, /")", I, EM[1], EM[3], EM[5], EM[7]); OUTPUT(61, "("/, 3B, "("SINGULAR VALUES : ")", /")"); "FOR" I:= 1 "STEP" 1 "UNTIL" 5 "DO" OUTPUT(61, "("/, 3B, N")", VAL[I]); OUTPUT(61, "("/, /, 3B, "("MATRIX U, FIRST 3 COLUMNS")", /")"); "FOR" I:= 1 "STEP" 1 "UNTIL" 6 "DO" OUTPUT(61, "("/, 3B, 3(N)")", A[I,1], A[I,2], A[I,3]); OUTPUT(61, "("/, /, 13B, "("LAST 2 COLUMNS")", /")"); "FOR" I:= 1 "STEP" 1 "UNTIL" 6 "DO" OUTPUT(61, "("/, 13B, 2(N)")", A[I,4], A[I,5]) "END" 1SECTION : 3.5.1.2 (JULY 1974) PAGE 5 NUMBER SINGULAR VALUES NOT FOUND : 0 INFINITY NORM : +2.2833333333334"+000 MAX NEGLECTED SUBDIAGONAL ELEMENT : +5.7786437871158"-014 NUMBER ITERATIONS : 5 NUMERICAL RANK : 5 SINGULAR VALUES : +1.5921172587262"+000 +2.2449595426097"-001 +1.3610556101029"-002 +4.3245382038374"-004 +6.4001947134260"-006 MATRIX U, FIRST 3 COLUMNS -7.5497918208386"-001 +6.1011090790645"-001 -2.3287173869184"-001 -4.3909273679284"-001 -2.2602102994174"-001 +7.0245315582712"-001 -3.1703146681544"-001 -3.7306964696148"-001 +2.1607293656979"-001 -2.4999458583084"-001 -3.9557817833576"-001 -1.4665595223684"-001 -2.0704999076883"-001 -3.8483260608872"-001 -3.6803786187007"-001 -1.7699734614538"-001 -3.6458192866515"-001 -4.9868122801331"-001 LAST 2 COLUMNS +5.8625326935176"-002 -1.0184205426735"-002 -4.8169088124009"-001 +1.7189132301455"-001 +5.4982292571999"-001 -5.9788920283495"-001 +4.0633053815463"-001 +4.5989617524697"-001 -6.1755991033503"-002 +4.3029765325422"-001 -5.4158416488948"-001 -4.6499203623570"-001 1SECTION : 3.5.1.2 (JULY 1974) PAGE 6 SOURCE TEXT(S): 0"CODE" 34272; "INTEGER" "PROCEDURE" QRISNGVAL(A, M, N, VAL, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" A, VAL, EM; "BEGIN" "ARRAY" B[1:N]; HSHREABID(A, M, N, VAL, B, EM); QRISNGVAL:= QRISNGVALBID(VAL, B, N, EM) "END" QRISNGVAL; "EOP" 0"CODE" 34273; "INTEGER" "PROCEDURE" QRISNGVALDEC(A, M, N, VAL, V, EM); "VALUE" M, N; "INTEGER" M, N; "ARRAY" A, VAL, V, EM; "BEGIN" "ARRAY" B[1:N]; HSHREABID(A, M, N, VAL, B, EM); PSTTFMMAT(A, N, V, B); PRETFMMAT(A, M, N, VAL); QRISNGVALDEC:= QRISNGVALDECBID(VAL, B, M, N, A, V, EM) "END" QRISNGVALDEC; "EOP" 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 1 AUTHORS: TH.J. DEKKER AND TH.H.P. REYMER CONTRIBUTOR: TH.H.P. REYMER INSTITUTE: UNIVERSITY OF AMSTERDAM RECEIVED: 770427; BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO PROCEDURES: A) ZERPOL CALCULATES ALL ROOTS OF A POLYNOMIAL WITH REAL COEFFICIENTS BY MEANS OF LAGUERRE'S METHOD; B) BOUNDS CALCULATES UPPERBOUNDS FOR THE ABSOLUTE ERROR IN GIVEN APPROXIMATED ZEROS OF A POLYNOMIAL WITH REAL COEFFICIENTS; KEYWORDS: ZEROS; REAL POLYNOMIAL; LAGUERRE'S METHOD; ERROR BOUNDS; SUBSECTION: ZERPOL CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "INTEGER" "PROCEDURE" ZERPOL(N, A, EM, RE, IM, D); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM, D; "CODE" 34501 ; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: THE DEGREE OF THE POLYNOMIAL; A: ; "ARRAY" A[0 : N]; ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL, IN SUCH A WAY THAT P(Z) = (..(A[N]*Z + A[N-1])*Z +..+ A[1])*Z + A[0]; 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 2 EM: ; "ARRAY" EM[0 : 4]; ENTRY: EM[0]: MACHINE PRECISION; EM[1]: THE MAXIMAL NUMBER OF ITERATIONS ALLOWED FOR EACH ZERO; EXIT: EM[2]: FAIL INDICATION; 0 SUCCESSFUL CALL; 1 UPON ENTRY DEGREE N <= 0; 2 UPON ENTRY LEADING COEFFICIENT A[N] = 0; 3 NUMBER OF ITERATIONS EXCEEDED EM[1]; EM[3]: NUMBER OF NEW STARTS IN THE LAST ITERATION; EM[4]: TOTAL NUMBER OF ITERATIONS PERFORMED; FOR THE CD CYBER 70 SYSTEM SUITABLE VALUES ARE: EM[0]:= "-14; EM[1]:= 40; IF, UPON EXIT, EM[2] = 3 AND EM[3] < 5 THEN IT MAY BE USEFUL TO START AGAIN WITH A HIGHER VALUE OF EM[1]; RE, IM: ; "ARRAY" RE, IM[1 : N]; EXIT: THE REAL AND IMAGINARY PARTS OF THE ZEROS OF THE POLYNOMIAL; THE MEMBERS OF EACH NONREAL COMPLEX CONJUGATE PAIR ARE CONSECUTIVE; D: ; "ARRAY" D[0 : N]; EXIT: IF THE CALL IS UNSUCCESSFUL AND ONLY N-K ZEROS HAVE BEEN FOUND, THEN D[0 : K] CONTAINS THE COEFFICIENTS OF THE (DEFLATED) POLYNOMIAL; MOREOVER, THEN THE ZEROS FOUND ARE DELIVERED IN RE, IM[ K + 1 : N], WHEREAS THE REMAINING PARTS OF RE AND IM CONTAIN NO INFORMATION; ZERPOL:= THE NUMBER, K, OF ZEROS NOT FOUND; PROCEDURES USED: DWARF = CP30003; GIANT = CP30004; COMABS = CP34340; COMSQRT = CP34343; REQUIRED CENTRAL MEMORY: TOTAL SIZE OF LOCAL ARRAYS IS N + 16 REAL LOCATIONS; RUNNING TIME: ROUGHLY PROPORTIONAL TO N**2; 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 3 METHOD AND PERFORMANCE: THE PROCEDURE USES LAGUERRE'S METHOD TO FIND ZEROS OF THE GIVEN POLYNOMIAL (SEE [2]); WHEN A ZERO HAS BEEN FOUND, A COMPOSITE DEFLATION TECHNIQUE IS USED TO OBTAIN A NEW POLYNOMIAL OF LOWER DEGREE (SEE [1], [3], [4]); IF CONVERGENCE IS NOT APPARENT, SEVERAL RESTARTS, THE NUMBER OF WHICH DEPENDS ON EM[1] BUT HAS A MAXIMUM OF 5, ARE MADE IN THE NEIGHBOURHOOD OF THE ABSOLUTE LARGEST ZERO; THE ACCURACY OF THE CALCULATED ZEROS STRONGLY DEPENDS ON THE POLYNOMIAL; A ROUGH INDICATION FOR THE ERROR IN A CALCULATED ZERO Z FOLLOWS FROM P(Z) / DP(Z), WHERE P DENOTES THE GIVEN POLYNOMIAL AND DP ITS FIRST DERIVATIVE (SEE E.G. [5]); TO FIND A TRUE UPPERBOUND FOR THESE ERRORS, ONE CAN USE PROCEDURE BOUNDS (SEE NEXT SUBSECTION); FOR A MORE DETAILED DESCRIPTION OF THE PROCEDURE AND TEST RESULTS SEE [4]; REFERENCES: [1] D.A. ADAMS, A STOPPING CRITERION FOR POLYNOMIAL ROOT FINDING, CACM 10, NO 10, PP. 655-658, OCTOBER 1967; [2] T.J. DEKKER, NEWTON-LAGUERRE ITERATION, MATHEMATISCH CENTRUM MR82, 1966; [3] G. PETERS AND J.H. WILKINSON, PRACTICAL PROBLEMS ARISING IN THE SOLUTION OF POLYNOMIAL EQUATIONS, J. INST. MATHS APPLICS 1971, NO. 8, PP. 16-35; [4] TH.H.P. REYMER, BEREKENING VAN NULPUNTEN VAN REELE POLYNOMEN EN FOUTGRENZEN VOOR DEZE NULPUNTEN, DOCTORAAL SCRIPTIE UVA, APRIL 1977; [5] J.H. WILKINSON, ROUNDING ERRORS IN ALGEBRAIC PROCESSES, PRENTICE HALL, 1963; EXAMPLE OF USE: FOR AN EXAMPLE OF USE SEE PROCEDURE BOUNDS (NEXT SUBSECTION); 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 4 SUBSECTION: BOUNDS CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" BOUNDS(N,A,RE,IM,RELE,ABSE,RECENTRE,IMCENTRE,BOUND); "VALUE" N, RELE, ABSE; "INTEGER" N; "REAL" RELE, ABSE; "ARRAY" A, RE, IM, RECENTRE, IMCENTRE, BOUND; "CODE" 34502 ; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: DEGREE OF THE POLYNOMIAL; A: ; "ARRAY" A[0 : N]; ENTRY: THE COEFFICIENTS OF THE POLYNOMIAL OF WHICH RE[J] + I * IM[J] ARE THE APPROXIMATED ZEROS, IN SUCH A WAY THAT P(Z) = (..(A[N]*Z + A[N-1])*Z +..+A[1])*Z + A[0] RE, IM: ; "ARRAY" RE, IM[1 : N]; ENTRY: REAL AND IMAGINARY PARTS OF APPROXIMATED ZEROS OF A POLYNOMIAL, SUCH THAT THE MEMBERS OF EACH NONREAL COMPLEX CONJUGATE PAIR ARE CONSECUTIVE; EXIT: A PERMUTATION OF THE INPUT DATA; RELE: ; ENTRY: RELATIVE ERROR IN THE NON-VANISHING COEFFICIENTS A[J] OF THE GIVEN POLYNOMIAL; ABSE: ; ENTRY: ABSOLUTE ERROR IN THE VANISHING COEFFICIENTS A[J] GIVEN POLYNOMIAL; IF THERE ARE NO VANISHING COEFFICIENTS, ABSE SHOULD BE ZERO; RECENTRE, IMCENTRE: ; "ARRAY" RECENTRE, IMCENTRE[1 : N]; EXIT: REAL AND IMAGINARY PARTS OF THE CENTERS OF DISKS IN WHICH SOME NUMBER OF ZEROS OF THE POLYNOMIAL GIVEN BY A ARE SITUATED; THE NUMBER OF IDENTICAL CENTERS DENOTES THE NUMBER OF ZEROS IN THAT DISK; BOUND: ; "ARRAY" BOUND[1 : N]; EXIT: RADIUS OF THE DISKS WHOSE CENTERS ARE GIVEN CORRESPONDINGLY IN RECENTRE AND IMCENTRE; PROCEDURES USED: ARREB = CP30002; GIANT = CP30004; 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 5 REQUIRED CENTRAL MEMORY: TOTAL SIZE OF LOCAL ARRAYS IS AT MOST 7 * N REAL LOCATIONS; RUNNING TIME: APPROXIMATELY OF ORDER N**2; METHOD AND PERFORMANCE: FROM THE APPROXIMATED ZEROS A POLYNOMIAL IS RECONSTRUCTED AND COMPARED WITH THE GIVEN POLYNOMIAL; SUBSEQUENTLY, THE PROCEDURE CALCULATES DISKS SUCH THAT THE NUMBER OF GIVEN APPROXIMATED ZEROS WITHIN EACH DISK EQUALS THE NUMBER OF ZEROS OF THE GIVEN POLYNOMIAL WITHIN THAT DISK; UPON EXIT EVERY TWO NON-IDENTICAL DISKS ARE DISJOINT; FOR A MORE DETAILED DESCRIPTION SEE [1], [2]; REFERENCES: [1] G. PETERS AND J.H. WILKINSON, PRACTICAL PROBLEMS ARISING IN THE SOLUTION OF POLYNOMIAL EQUATIONS, J.INST.MATHS APPLICS 1971, NO 8, PP. 16-35; [2] TH.H.P. REYMER, BEREKENING VAN NULPUNTEN VAN REELE POLYNOMEN EN FOUTGRENZEN VOOR DEZE NULPUNTEN, DOCTORAAL SCRIPTIE UVA, APRIL 1977; EXAMPLE OF USE: "BEGIN" "INTEGER" I, J; "ARRAY" A, D[0:7], RE, IM[1:7], EM[0:4]; A[7]:= 1; A[6]:= -3; A[5]:= -3; A[4]:= 25; A[3]:= -46; A[2]:= 38; A[1]:= -12; A[0]:= 0; EM[0]:= "-14; EM[1]:= 40; I:= ZERPOL(7, A, EM, RE, IM, D); OUTPUT(61,"(""("COEFFICIENTS OF POLYNOMIAL:")",//")"); "FOR" J:=7 "STEP" -1 "UNTIL" 0 "DO" OUTPUT(61,"("-2ZD3B")",A[J]); OUTPUT(61,"("//,"("NUMBER NOT FOUND ZEROS ")",3ZD,/, "("FAIL INDICATION ")",3ZD,/,"("NUMBER NEW STARTS ")",3ZD,/, "("NUMBER OF ITERATIONS ")",3ZD,/")",I,EM[2],EM[3],EM[4]); OUTPUT(61,"("/,"("ZEROS: ")",/")"); "FOR" J:= I+1 "STEP" 1 "UNTIL" 7 "DO" "IF" IM[J] = 0 "THEN" OUTPUT(61,"("/,N")",RE[J]) "ELSE" OUTPUT(61,"("/,2(N)")",RE[J],IM[J]); 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 6 "IF" I = 0 "THEN" "BEGIN" "ARRAY" RECENTRE, IMCENTRE, BOUND[1:7]; BOUNDS(7, A, RE, IM, 0, 0, RECENTRE, IMCENTRE, BOUND); OUTPUT(61,"("2/,"("REAL AND IMAG. PART OF CENTRE + RADIUS")",/")"); "FOR" J:= 1 "STEP" 1 "UNTIL" 7 "DO" OUTPUT(61,"("/,3(N)")",RECENTRE[J],IMCENTRE[J],BOUND[J]) "END" "END" RESULTS : COEFFICIENTS OF POLYNOMIAL: 1 -3 -3 25 -46 38 -12 0 NUMBER NOT FOUND ZEROS 0 FAIL INDICATION 0 NUMBER NEW STARTS 0 NUMBER OF ITERATIONS 11 ZEROS: +2.0000000000000"+000 -3.0000000000000"+000 +1.0000000000000"+000 -1.0000000000000"+000 +1.0000000000000"+000 +1.0000000000000"+000 +1.0000000083024"+000 +9.9999999169752"-001 +0.0000000000000"+000 REAL AND IMAG. PART OF CENTRE + RADIUS +2.0000000000000"+000 +0.0000000000000"+000 +1.3238111117716"-011 -3.0000000000000"+000 +0.0000000000000"+000 +3.8857510604494"-013 +1.0000000000000"+000 -1.0000000000000"+000 +4.0912729775463"-012 +1.0000000000000"+000 +1.0000000000000"+000 +4.0912729775463"-012 +9.9999999999998"-001 +0.0000000000000"+000 +2.2533888428865"-006 +9.9999999999998"-001 +0.0000000000000"+000 +2.2533888428865"-006 +0.0000000000000"+000 +0.0000000000000"+000 +0.0000000000000"+000 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 7 SOURCE TEXT(S) : "CODE" 34501; "INTEGER""PROCEDURE" ZERPOL(N, A, EM, RE, IM, D); "VALUE" N; "INTEGER" N; "ARRAY" A, EM, RE, IM, D; "BEGIN" "INTEGER" I, TOTIT, IT, FAIL, START, UP, MAX, GIEX, ITMAX; "REAL" X, Y, NEWF, OLDF, MAXRAD, AE, TOL, H1, H2, LN2; "ARRAY" F[0 : 5], TRIES[1 : 10]; "BOOLEAN""PROCEDURE" FUNCTION; "BEGIN" "INTEGER" K, M1, M2; "REAL" P, Q, QSQRT, F01, F02, F03, F11, F12, F13, F21, F22, F23, STOP; IT:= IT + 1; P:= 2 * X; Q:= -(X * X + Y * Y); QSQRT:= SQRT(-Q); F01:= F11:= F21:= D[0]; F02:= F12:= F22:= 0; M1:= N - 4; M2:= N - 2; STOP:= ABS(F01) * 0.8; "FOR" K:= 1 "STEP" 1 "UNTIL" M1 "DO" "BEGIN" F03:= F02; F02:= F01; F01:= D[K] + P * F02 + Q * F03; F13:= F12; F12:= F11; F11:= F01 + P * F12 + Q * F13; F23:= F22; F22:= F21; F21:= F11 + P * F22 + Q * F23; STOP:= QSQRT * STOP + ABS(F01) "END"; "IF" M1 < 0 "THEN" M1:= 0; "FOR" K:= M1 + 1 "STEP" 1 "UNTIL" M2 "DO" "BEGIN" F03:= F02; F02:= F01; F01:= D[K] + P * F02 + Q * F03; F13:= F12; F12:= F11; F11:= F01 + P * F12 + Q * F13; STOP:= QSQRT * STOP + ABS(F01) "END"; "IF" N = 3 "THEN" F21:= 0; F03:= F02; F02:= F01; F01:= D[N - 1] + P * F02 + Q * F03; F[0]:= D[N] + X * F01 + Q * F02; F[1]:= Y * F01; F[2]:= F01 - 2 * F12 * Y * Y; F[3]:= 2 * Y * (- X * F12 + F11); F[4]:= 2 * (- X * F12 + F11) - 8 * Y * Y * (- X * F22 + F21); F[5]:= Y * (6 * F12 - 8 * Y * Y * F22); STOP:= QSQRT * (QSQRT * STOP + ABS(F01)) + ABS(F[0]); NEWF:= F02:= COMABS(F[0], F[1]); FUNCTION:= F02 < (2 * ABS(X * F01) - 8 * (ABS(F[0]) + ABS(F01) * QSQRT) + 10 * STOP) * TOL * (1 + TOL) ** (4 * N + 3) "END" OF FUNCTION; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 8 ; "BOOLEAN""PROCEDURE" CONTROL; "IF" IT > ITMAX "THEN" "BEGIN" TOTIT:= TOTIT + IT; FAIL:= 3; "GOTO" EXIT "END" "ELSE" "IF" IT = 0 "THEN" "BEGIN" "INTEGER" I, H; "REAL" H1, SIDE; MAXRAD:= 0; MAX:= (GIEX - LN(ABS(D[0])) / LN2) / N; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" H1:= "IF" D[I] = 0 "THEN" 0 "ELSE" EXP(LN(ABS(D[I] / D[0])) / I); "IF" H1 > MAXRAD "THEN" MAXRAD:= H1 "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "IF" D[I] ^= 0 "THEN" "BEGIN" H:= (GIEX - LN(ABS(D[I])) / LN2) / (N - I); "IF" H < MAX "THEN" MAX:= H "END"; MAX:= MAX * LN2 / LN(N); SIDE:= - D[1] / D[0]; SIDE:= "IF" ABS(SIDE) < TOL "THEN" 0 "ELSE" SIGN(SIDE); "IF" SIDE = 0 "THEN" "BEGIN" TRIES[7]:= TRIES[2]:= MAXRAD; TRIES[9]:= -MAXRAD; TRIES[6]:= TRIES[4]:= TRIES[3]:= MAXRAD / SQRT(2); TRIES[5]:= -TRIES[3]; TRIES[10]:= TRIES[8]:= TRIES[1]:= 0 "END" "ELSE" "BEGIN" TRIES[8]:= TRIES[4]:= MAXRAD/ SQRT(2); TRIES[1]:= SIDE * MAXRAD; TRIES[3]:= TRIES[4] * SIDE; TRIES[6]:= MAXRAD; TRIES[7]:= -TRIES[3]; TRIES[9]:= -TRIES[1]; TRIES[2]:= TRIES[5]:= TRIES[10]:= 0 "END"; "IF" COMABS(X, Y) > 2 * MAXRAD "THEN" X:= Y:= 0; CONTROL:= "FALSE" "END" "ELSE" "BEGIN" "IF" IT > 1 & NEWF >= OLDF "THEN" "BEGIN" UP:= UP+ 1; "IF" UP = 5 & START < 5 "THEN" "BEGIN" START:= START + 1; UP:= 0; X:= TRIES[2 * START - 1]; Y:= TRIES[2 * START]; CONTROL:= "FALSE" "END" "ELSE" CONTROL:= "TRUE" "END" "ELSE" CONTROL:= "TRUE" "END" OF CONTROL; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 9 ; "PROCEDURE" DEFLATION; "IF" X = 0 & Y = 0 "THEN" N:= N - 1 "ELSE" "BEGIN" "INTEGER" I, SPLIT; "REAL" H1, H2; "ARRAY" B[0 : N - 1]; "IF" Y = 0 "THEN" "BEGIN" N:= N - 1; B[N]:= -D[N + 1] / X; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" B[N - I]:= (B[N - I + 1] - D[N - I + 1]) / X; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" D[I]:= D[I] + D[I - 1] * X "END" "ELSE" "BEGIN" H1:= - 2 * X; H2:= X * X + Y * Y; N:= N - 2; B[N]:= D[N + 2] / H2; B[N - 1]:= (D[N + 1] - H1 * B[N]) / H2; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" B[N - I]:= (D[N - I + 2] - H1 * B[N - I + 1] - B[N - I + 2])/H2; D[1]:= D[1] - H1 * D[0]; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" D[I]:= D[I] - H1 * D[I-1] - H2 * D[I-2] "END"; SPLIT:= N; H2:= ABS(D[N] - B[N]) / (ABS(D[N]) + ABS(B[N])); "FOR" I:= N - 1 "STEP" -1 "UNTIL" 0 "DO" "BEGIN" H1:= ABS(D[I]) + ABS(B[I]); "IF" H1 > TOL "THEN" "BEGIN" H1:= ABS(D[I] - B[I]) / H1; "IF" H1 < H2 "THEN" "BEGIN" H2:= H1; SPLIT:= I "END" "END" "END"; "FOR" I := SPLIT + 1 "STEP" 1 "UNTIL" N "DO" D[I]:= B[I]; D[SPLIT]:= (D[SPLIT] + B[SPLIT]) / 2 "END" OF DEFLATION; "PROCEDURE" LAGUERRE; "BEGIN" "INTEGER" M; "REAL" S1RE, S1IM, S2RE, S2IM, DX, DY, H1, H2, H3, H4, H5, H6; "IF" ABS(F[0]) > ABS(F[1]) "THEN" "BEGIN" H1:= F[0]; H6:= F[1] / H1; H2:= F[2] + H6 * F[3]; H3:= F[3] - H6 * F[2]; H4:= F[4] + H6 * F[5]; H5:= F[5] - H6 * F[4]; H6:= H6 * F[1] + H1 "END" "ELSE" "BEGIN" H1:= F[1]; H6:= F[0] / H1; H2:= H6 * F[2] + F[3]; H3:= H6 * F[3] - F[2]; H4:= H6 * F[4] + F[5]; H5:= H6 * F[5] - F[4]; H6:= H6 * F[0] + F[1] "END"; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 10 ; S1RE:= H2 / H6; S1IM:= H3 / H6; H2:= S1RE * S1RE - S1IM * S1IM; H3:= 2 * S1RE * S1IM; S2RE:= H2 - H4 / H6; S2IM:= H3 - H5 / H6; H1:= S2RE * S2RE + S2IM * S2IM; H1:= "IF" H1 ^= 0 "THEN" (S2RE * H2 + S2IM * H3) / H1 "ELSE" 1; M:= "IF" H1 >= N - 1 "THEN" ("IF" N > 1 "THEN" N - 1 "ELSE" 1) "ELSE" "IF" H1 > 1 "THEN" H1 "ELSE" 1; H1:= (N - M) / M; COMSQRT(H1 * (N * S2RE - H2), H1 * (N * S2IM - H3), H2, H3); "IF" S1RE * H2 + S1IM * H3 < 0 "THEN" "BEGIN" H2:= - H2; H3:= - H3 "END"; H2:= S1RE + H2; H3:= S1IM + H3; H1:= H2 * H2 + H3 * H3; "IF" H1 = 0 "THEN" "BEGIN" DX:= -N; DY:= N "END" "ELSE" "BEGIN" DX:= - N * H2 / H1; DY:= N * H3 / H1 "END"; H1:= ABS(X) * TOL + AE; H2:= ABS(Y) * TOL+ AE; "IF" ABS(DX) < H1 & ABS(DY) < H2 "THEN" "BEGIN" DX:= "IF" DX = 0 "THEN" H1 "ELSE" SIGN(DX) * H1; DY:= "IF" DY = 0 "THEN" H2 "ELSE" SIGN(DY) * H2 "END"; X:= X + DX; Y:= Y + DY; "IF" COMABS(X, Y) > 2 * MAXRAD "THEN" "BEGIN" H1:= "IF" ABS(X) > ABS(Y) "THEN" ABS(X) "ELSE" ABS(Y); H2:= LN(H1) / LN2 + 1 - MAX; "IF" H2 > 0 "THEN" "BEGIN" H2:= 2 ** H2; X:= X / H2; Y:= Y / H2 "END" "END" "END" OF LAGUERRE; TOTIT:= IT:= FAIL:= UP:= START:= 0; LN2:= LN(2); NEWF:= GIANT; AE:= DWARF; GIEX:= LN(NEWF) / LN2 - 40; TOL:= EM[0]; ITMAX:= EM[1]; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I]:= A[N-I]; "IF" N <= 0 "THEN" "BEGIN" FAIL:= 1; "GOTO" EXIT "END" "ELSE" "IF" D[0] = 0 "THEN" "BEGIN" FAIL:= 2; "GOTO" EXIT "END"; "FOR" I:= 1 "WHILE" D[N] = 0 & N > 0 "DO" "BEGIN" RE[N]:= IM[N]:= 0; N:= N - 1 "END"; X:= Y:= 0; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 11 ; "FOR" I:= 1 "WHILE" N > 2 "DO" "BEGIN" "IF" CONTROL "THEN" LAGUERRE; OLDF:= NEWF; "IF" FUNCTION "THEN" "BEGIN" "IF" Y ^= 0 & ABS(Y) < .1 "THEN" "BEGIN" "REAL" H; H:= Y; Y:= 0; "IF" ^ FUNCTION "THEN" Y:= H "END"; RE[N]:= X; IM[N]:= Y; "IF" Y ^= 0 "THEN" "BEGIN" RE[N - 1]:= X; IM[N - 1]:= -Y "END"; DEFLATION; TOTIT:= TOTIT + IT; UP:= START:= IT:= 0 "END" "END"; "IF" N = 1 "THEN" "BEGIN" RE[1]:= - D[1] / D[0]; IM[1]:= 0 "END" "ELSE" "BEGIN" "REAL" H1, H2; H1:= - 0.5 * D[1] / D[0]; H2:= H1 * H1 - D[2] / D[0]; "IF" H2 >= 0 "THEN" "BEGIN" RE[2]:= "IF" H1 < 0 "THEN" H1 - SQRT(H2) "ELSE" H1 + SQRT(H2); RE[1]:= D[2] / (D[0] * RE[2]); IM[2]:= IM[1]:= 0 "END" "ELSE" "BEGIN" RE[2]:= RE[1]:= H1; IM[2]:= SQRT(-H2); IM[1]:= -IM[2] "END" "END"; N:= 0; EXIT: EM[2]:= FAIL; EM[3]:= START; EM[4]:= TOTIT; "FOR" I:= (N-1) "DIV" 2 "STEP" -1 "UNTIL" 0 "DO" "BEGIN" TOL := D[I]; D[I]:= D[N-I]; D[N-I]:= TOL "END"; ZERPOL:= N "END" OF ZERPOL; "EOP" "CODE" 34502; "PROCEDURE" BOUNDS(N,A,RE,IM,RELE,ABSE,RECENTRE,IMCENTRE,BOUND); "VALUE" N, RELE, ABSE; "INTEGER" N; "REAL" RELE, ABSE; "ARRAY" RE, IM, A, RECENTRE, IMCENTRE, BOUND; "BEGIN" "INTEGER" I, J, K, L, INDEX1, INDEX2; "BOOLEAN" GOON; "REAL" H, MIN, RECENT, IMCENT, GIA, XK, YK, ZK, CORR; "ARRAY" RC, C, RCE[0:N], CLUST[1:N]; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 12 ; "REAL""PROCEDURE" G(RAD, RECENT, IMCENT, K, M); "VALUE" RAD, RECENT, IMCENT, K, M; "REAL" RAD, RECENT, IMCENT; "INTEGER" K, M; "BEGIN" "REAL" S, H1, H2; "INTEGER" I; S:= SQRT(RECENT * RECENT + IMCENT * IMCENT) + RAD; H1:= RC[1]; H2:= RC[0]; "FOR" I:= 2 "STEP" 1 "UNTIL" N "DO" H1:= H1*S + RC[I]; "FOR" I:= 1 "STEP" 1 "UNTIL" M-1, M+K "STEP" 1 "UNTIL" N "DO" H2:= H2 * ABS(SQRT((RE[I]-RECENT)**2 + (IM[I]-IMCENT)**2) - RAD); G:= "IF" H1=0 "THEN" 0 "ELSE" "IF" H2=0 "THEN" -10 "ELSE" H1 / H2 "END"; "PROCEDURE" KCLUSTER(K, M); "VALUE" K, M; "INTEGER" K, M; "BEGIN" "INTEGER" I, J, STOP, L; "BOOLEAN" NONZERO; "REAL" RECENT, IMCENT, D, PROD, RAD, GR, R; "ARRAY" DIST[M: M+K-1]; RECENT:= RE[M]; IMCENT:= IM[M]; STOP:= M+K-1; L:= SIGN(IMCENT); NONZERO:= L ^= 0; "FOR" I:= M+1 "STEP" 1 "UNTIL" STOP "DO" "BEGIN" RECENT:= RECENT+RE[I]; "IF" NONZERO "THEN" "BEGIN" NONZERO:= L = SIGN(IM[I]); IMCENT:= IMCENT+IM[I] "END" "END"; RECENT:= RECENT/K; IMCENT:= "IF" NONZERO "THEN" IMCENT/K "ELSE" 0; D:= 0; RAD:= 0; "FOR" I:= M "STEP" 1 "UNTIL" STOP "DO" "BEGIN" RECENTRE[I]:= RECENT; IMCENTRE[I]:= IMCENT; DIST[I]:= SQRT((RE[I] -RECENT)**2 + (IM[I]-IMCENT)**2); "IF" D < DIST[I] "THEN" D:= DIST[I] "END"; GR:= ABS(G(0, RECENT, IMCENT, K, M)); "IF" GR > 0 "THEN" "BEGIN" "FOR" J:= 1, 1 "WHILE" PROD <= GR "DO" "BEGIN" R:= RAD; RAD:= D + EXP(LN(1.1*GR)/K); "IF" RAD = R "THEN" RAD:= EXP(LN(1.1)/K) * RAD; GR:= G(RAD, RECENT, IMCENT, K, M); PROD:= 1; "FOR" I:= M "STEP" 1 "UNTIL" STOP "DO" PROD:= PROD*(RAD-DIST[I]) "END" "END"; "FOR" I:= M "STEP" 1 "UNTIL" STOP "DO" "BEGIN" BOUND[I]:= RAD; CLUST[I]:= K "END"; "END"; "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1979) PAGE 13 ; "PROCEDURE" SHIFT(INDEX, NEW); "VALUE" INDEX, NEW; "INTEGER" INDEX, NEW; "BEGIN" "INTEGER" J, PLACE, CLUSTIN; "REAL" BOUNDIN, IMCENT, RECENT; "REAL""ARRAY" WA1, WA2[1:CLUST[INDEX]]; CLUSTIN:= CLUST[INDEX]; BOUNDIN:= BOUND[INDEX]; IMCENT:= IMCENTRE[INDEX]; RECENT:= RECENTRE[INDEX]; "FOR" J:= 1 "STEP" 1 "UNTIL" CLUSTIN "DO" "BEGIN" PLACE:=INDEX+J-1; WA1[J]:= RE[PLACE]; WA2[J]:= IM[PLACE]; "END"; "FOR" J:= INDEX-1 "STEP" -1 "UNTIL" NEW "DO" "BEGIN" PLACE:= J+CLUSTIN; RE[PLACE]:= RE[J]; IM[PLACE]:= IM[J]; CLUST[PLACE]:= CLUST[J]; BOUND[PLACE]:= BOUND[J]; RECENTRE[PLACE]:= RECENTRE[J]; IMCENTRE[PLACE]:= IMCENTRE[J] "END"; "FOR" J:= NEW+CLUSTIN-1 "STEP" -1 "UNTIL" NEW "DO" "BEGIN" PLACE:= J+1-NEW; RE[J]:= WA1[PLACE]; IM[J]:= WA2[PLACE]; BOUND[J]:= BOUNDIN; CLUST[J]:= CLUSTIN; RECENTRE[J]:= RECENT; IMCENTRE[J]:= IMCENT "END" "END"; GIA:= GIANT; RC[0]:= C[0]:= A[N]; RCE[0]:= ABS(C[0]); K:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" RC[I]:= RCE[I]:= 0 ; C[I]:= A[N-I] "END"; "FOR" I:= 0 "WHILE" K < N "DO" "BEGIN" K:= K+1; XK:= RE[K]; YK:= IM[K]; ZK:= XK*XK+YK*YK; "FOR" J:= K "STEP" -1 "UNTIL" 1 "DO" RCE[J]:= RCE[J]+RCE[J-1]*SQRT(ZK); "IF" YK = 0 "THEN" "BEGIN" "FOR" J:= K "STEP" -1 "UNTIL" 1 "DO" RC[J]:= RC[J]-XK*RC[J-1] "END" "ELSE" "BEGIN" K:= K+1; "IF" K <= N & XK = RE[K] & YK = -IM[K] "THEN" "BEGIN" XK:= -2*XK; "FOR" J:= K "STEP" -1 "UNTIL" 1 "DO" RCE[J]:= RCE[J]+RCE[J-1]*SQRT(ZK); "FOR" J:= K "STEP" -1 "UNTIL" 2 "DO" RC[J]:= RC[J]+XK*RC[J-1]+ZK*RC[J-2]; RC[1]:= RC[1]+XK*RC[0] "END" "END" "END"; RC[0]:= RCE[0]; CORR:= 1.06*ARREB; "FOR" I:= 1 "STEP" 1 "UNTIL" N-1 "DO" RC[I]:= ABS(RC[I]-C[I])+RCE[I]*CORR*(N+I-2)+RELE*ABS(C[I])+ABSE; RC[N]:= ABS(RC[N]-C[N])+RCE[N]*CORR*(N-1)+RELE*ABS(C[N])+ABSE; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" KCLUSTER(1, I); "COMMENT" 1SECTION : 3.6.1 (DECEMBER 1978) PAGE 14 ; GOON:= "TRUE"; "FOR" L:= 1 "WHILE" GOON "DO" "BEGIN" INDEX1:= INDEX2:= 0; MIN:= GIANT; I:= N-CLUST[N]+1; "FOR" I:= I "WHILE" I >= 2 "DO" "BEGIN" J:= I; RECENT:= RECENTRE[I]; IMCENT:= IMCENTRE[I]; "FOR" J:= J "WHILE" J >= 2 "DO" "BEGIN" J:= J-CLUST[J-1]; H:= SQRT((RECENT-RECENTRE[J])**2 + (IMCENT-IMCENTRE[J])**2); "IF" H < BOUND[I] + BOUND[J] & H <= MIN "THEN" "BEGIN" INDEX1:= J; INDEX2:= I; MIN:= H "END" "END"; I:= I-CLUST[I-1] "END"; "IF" INDEX1 = 0 "THEN" GOON:= "FALSE" "ELSE" "BEGIN" "IF" IMCENTRE[INDEX1] = 0 "THEN" "BEGIN" "IF" IMCENTRE[INDEX2] ^= 0 "THEN" CLUST[INDEX2]:= 2*CLUST[INDEX2] "END" "ELSE" "IF" IMCENTRE[INDEX2] = 0 "THEN" CLUST[INDEX1]:= 2*CLUST[INDEX1]; K:= INDEX1+CLUST[INDEX1]; "IF" K ^= INDEX2 "THEN" SHIFT(INDEX2, K); K:= CLUST[INDEX1]+CLUST[K]; KCLUSTER(K, INDEX1) "END" "END" "END"; "EOP" 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 1 AUTHORS: M. BAKKER, P.J. HARINGHUIZEN AND C.G. VAN DER LAAN. CONTRIBUTORS: M. BAKKER, P.J. HARINGHUIZEN, C.G. VAN DER LAAN AND M. VOORINTHOLT. INSTITUTE: MATHEMATICAL CENTRE AND RIJKSUNIVERSITEIT GRONINGEN. RECEIVED: 780601. BRIEF DESCRIPTION: THIS SECTION CONTAINS FIVE PROCEDURES FOR CALCULATING ZEROS OF ORTHOGONAL POLYNOMIALS WHICH ARE GIVEN BY THE COEFFICIENTS OF THEIR RECURRENCE RELATION: ALLZERORTPOL: CALCULATES ALL ZEROS, LUPZERORTPOL: CALCULATES A NUMBER OF ADJACENT UPPER OR LOWER ZEROS, SELZERORTPOL: CALCULATES A NUMBER OF ADJACENT ZEROS. IT IS EFFICIENT TO USE ALLZERORTPOL IF MORE THAN 50 PERCENT OF EXTREME ZEROS OR MORE THAN 25 PERCENT OF SELECTED ZEROS ARE WANTED. ALLJACZER : CALCULATES THE ZEROS OF THE N-TH JACOBIAN POLYNOMIAL. ALLLAGZER : CALCULATES THE ZEROS OF THE N-TH LAGUERRE POLYNOMIAL. KEYWORDS: ZEROS, ORTHOGONAL POLYNOMIALS, CHRISTOFFEL ABSCISSAS. REFERENCES: ABRAMOWITZ, M. AND I.A. STEGUN (1964): HANDBOOK OF MATHEMATICAL FUNCTIONS. DOVER PUBLICATIONS INC. GOLUB, G.H. AND J.H. WELSCH (1969): CALCULATION OF GAUSS QUADRATURE RULES. MATH. COMP. VOL. 23, P.221-230. LANCZOS, C. (1957): APPLIED ANALYSIS. PRENTICE HALL. 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 2 STOER, J. (1972): EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK 1. HEIDELBERG TASCHENBUECHER 105, SPRINGER. WILKINSON,J AND REINSCH,C. : HANDBOOK OF AUTOMATIC COMPUTATION. VOL. 2. LINEAR ALGEBRA HEIDELBERG (1971). SUBSECTION: ALLZERORTPOL. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" ALLZERORTPOL (N, B, C, ZER, EM); "VALUE" N; "INTEGER" N; "ARRAY" B, C, ZER, EM; "CODE" 31362; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: THE DEGREE OF THE ORTHOGONAL POLYNOMIAL OF WHICH THE ZEROS ARE TO BE CALCULATED; B, C: ; "ARRAY" B, C [0:N-1]; ENTRY: THE ELEMENTS B[I] AND C[I], I = 0, 1, ... , N-1, CONTAIN THE COEFFICIENTS OF THE RECURRENCE RELATION P[I+1](X) = (X - B[I]) * P[I](X) - C[I] * P[I-1](X) I = 0, 1, ... , N-1; ASSUMED IS C[0]=0, WHILE THE CONTENTS OF THE ARRAYS ARE PRESERVED; ZER: ; "ARRAY" ZER[1:N]; EXIT: THE ZEROS OF THE N-TH DEGREE ORTHOGONAL POLYNOMIAL; (B MAY BE USED FOR ZER, BUT THEN THIS RECURRENCE COEFFICIENTS ARE OVERWRITTEN BY THE ZEROS AND THE ORIGINAL CONTENTS OF B ARE NOT PRESERVED.) EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION; EM[2]: THE RELATIVE TOLERANCE OF THE ZEROS; EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS, E.G. 5 * N; EXIT: EM[1]: THE MAXIMUM OF ABS(B[0])+1,C[I]+ABS(B[I])+1, (I=1,...N-2),C[N-1]+ABS(B[N-1]); EM[3]: INFORMATION CONCERNING THE PROCESS USED; I.E. THE MAXIMUM ABSOLUTE VALUE OF THE CODIAGONAL ELEMENTS NEGLECTED, (SEE ALSO SECTION 3.3.1.1.1.); EM[5]: THE NUMBER OF ITERATIONS PERFORMED. 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 3 PROCEDURES USED: QRIVALSYMTRI = CP34160, DUPVEC = CP31030. METHOD AND PERFORMANCE : SEE SELZERORTPOL (THIS SECTION). EXAMPLE OF USE: AS A FORMAL TEST OF THE PROCEDURE WE CALCULATE THE ZEROS OF THE CHEBYSHEV POLYNOMIAL (OF THE FIRST KIND) OF THE THIRD DEGREE. THE RECURRENCE COEFFICIENTS ARE: B[I] = 0, I = 0, 1, ....; C[0] = 0, C[1] = .5, C[I] = .25 , I = 2, 3, ..... (IT IS RECOMMENDED TO STORE THE ELEMENTS OF THE ARRAYS B AND C IN REVERSED ORDER IF THESE ELEMENTS ARE STRONGLY INCREASING). "BEGIN" "ARRAY" B, C[0:3], ZER[1:3], EM[0:5]; EM[0]:= EM[2]:= "-14; EM[4]:=15; B[2]:=B[1]:=B[0]:=0; C[0]:= 0; C[1]:= .5; C[2]:= .25; ALLZERORTPOL (3, B, C, ZER, EM); OUTPUT(61,"(""("THE THREE ZEROS:")",/,3(/ZD5B,N),2/, "("EM[1]:")",5BD.2D"+2D, /, "("EM[3]:")",5BD.2D"+2D, /,"("EM[5]:")",5ZD")", 1,ZER[1],2,ZER[2],3,ZER[3],EM[1],EM[3],EM[5]) "END" THE THREE ZEROS: 1 -8.6602540378444"-001 2 +8.6602540378444"-001 3 -1.0000000000000"-014 EM[1]: 1.50"+00 EM[3]: 7.07"-15 EM[5]: 1 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 4 SUBSECTION: LUPZERORTPOL. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" LUPZERORTPOL (N, M, B, C, ZER, EM); "VALUE" N, M; "INTEGER" N, M; "ARRAY" B, C, ZER, EM; "CODE" 31363; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: THE DEGREE OF THE ORTHOGONAL POLYNOMIAL OF WHICH THE ZEROS ARE TO BE CALCULATED; M: ; ENTRY: THE NUMBER OF ZEROS TO BE CALCULATED; B, C: ; "ARRAY" B, C [0:N-1]; ENTRY: THE ELEMENTS B[I] AND C[I], I = 0, 1, ... , N-1, CONTAIN THE COEFFICIENTS OF THE RECURRENCE RELATION P[I+1](X) = (X - B[I]) * P[I](X) - C[I] * P[I-1](X) I = 0, 1, ... , N-1; ASSUMED IS C[0]=0, WHILE THE CONTENTS OF THE ARRAYS ARE NOT PRESERVED; ZER: ; "ARRAY" ZER[1:M]; EXIT: THE M LOWEST ZEROS ARE DELIVERED; IF HOWEVER THE ARRAY B[0:N-1] CONTAINED THE OPPOSIT VALUES OF THE CORRESPONDING RECURRENCE COEFFICIENTS THEN THE OPPOSITE VALUES OF THE M UPPER ZEROS ARE DELIVERED. IN EITHER CASE, ZER[I]; "ARRAY" EM[0:6]; ENTRY: EM[0]: THE MACHINE PRECISION. EM[2]: THE RELATIVE TOLERANCE OF THE ZEROS; EM[4]: THE MAXIMUM ALLOWED NUMBER OF ITERATIONS, E.G. 15 * M; EM[6]: IF ALL ZEROS ARE KNOWN TO BE POSITIVE THEN 1 ELSE 0; EXIT: EM[1]: THE MAXIMUM OF ABS(B[0]) + 1 , C[I]+ABS(B[I])+1,(I=1,...N-2), C[N-1]+ABS(B[N-1]); EM[3]: INFORMATION CONCERNING THE PROCESS USED, I.E. THE MAXIMUM ABSOLUTE VALUE OF THE THEORETICAL ERRORS OF THE ZEROS (SEE WILKINSON AND REINSCH, 1971, P.263); EM[5]: THE NUMBER OF ITERATIONS PERFORMED. 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 5 PROCEDURES USED: DUPVEC = CP31030, INFNRMVEC = CP31061. METHOD AND PERFORMANCE : SEE SELZERORTPOL (THIS SECTION). EXAMPLE OF USE: WE CALCULATE THE TWO LOWER AND THE TWO UPPER ZEROS OF THE LAGUERRE POLYNOMIAL OF THE THIRD DEGREE. THE RECURRENCE COEFFICIENTS ARE OBTAINED FROM [1], P.782: B[I] = - A2I / A3I = 2I + 1; C[I] = A4I / (A3I * A3(I-1)) * A1(I-1) = I * I, I = 0, 1, 2. (IT IS RECOMMENDED TO STORE THE ELEMENTS OF THE ARRAYS B AND C IN REVERSED ORDER IF THESE ELEMENTS ARE STRONGLY DECREASING). "BEGIN" "ARRAY" B, C[0:3], ZER[1:2], EM[0:6]; "INTEGER" I; EM[0]:= EM[2]:= "-14; EM[4]:= 45; EM[6]:= 1; "FOR" I:= 0, 1, 2 "DO" "BEGIN" B[I]:= 2 * I + 1; C[I]:= I * I "END"; LUPZERORTPOL (3, 2, B, C, ZER, EM); OUTPUT(61,"(""("THE TWO LOWER ZEROS:")",/,2(/ZD5B,N),2/, "("EM[1]:")",5BD.2D"+2D, /, "("EM[3]:")",5BD.2D"+2D, /,"("EM[5]:")",5ZD")", 1,ZER[1],2,ZER[2],EM[1],EM[3],EM[5]); EM[6]:= 0; "FOR" I:= 0, 1, 2 "DO" "BEGIN" B[I]:= - 2 * I - 1; C[I]:= I * I "END"; LUPZERORTPOL (3, 2, B, C, ZER, EM); OUTPUT(61,"("3/,"("THE TWO UPPER ZEROS:")",/,2(/ZD5B,N),2/, "("EM[1]:")",5BD.2D"+2D, /, "("EM[3]:")",5BD.2D"+2D, /,"("EM[5]:")",5ZD")", 1,-ZER[1],2,-ZER[2],EM[1],EM[3],EM[5]); "END" THE TWO LOWER ZEROS: 1 +4.1577455678348"-001 2 +2.2942803602791"+000 EM[1]: 9.00"+00 EM[3]: 5.72"-16 EM[5]: 12 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 6 THE TWO UPPER ZEROS: 1 +6.2899450829375"+000 2 +2.2942803602791"+000 EM[1]: 9.00"+00 EM[3]: 4.70"-20 EM[5]: 14 SUBSECTION: SELZERORTPOL. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" SELZERORTPOL (N, N1, N2, B, C, ZER, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" B, C, ZER, EM; "CODE" 31364; THE MEANING OF THE FORMAL PARAMETERS IS : N,N1,N2:; ENTRY: N IS THE DEGREE OF THE ORTHOGONAL POLYNOMIAL OF WHICH THE N1-ST UP TO AND INCLUDING N2-ND ZEROS ARE TO BE CALCULATED(ZER[N1]>=ZER[N2]); B, C: ; "ARRAY" B,C [0 : N-1]; ENTRY: THE ELEMENTS B[I] AND C[I], I = 0, 1, ... , N-1, CONTAIN THE COEFFICIENTS OF THE RECURRENCE RELATION P[I+1](X) = (X - B[I]) * P[I](X) - C[I] * P[I-1](X) I = 0, 1, ... , N-1, ASSUMED IS C[0]=0, WHILE THE CONTENTS OF THE ARRAYS IS PRESERVED; ZER: ; "ARRAY" ZER [N1:N2]; EXIT: THE N2-N1+1 CALCULATED ZEROS IN DECREASING ORDER. EM: ; "ARRAY" EM[0:5]; ENTRY: EM[0]: THE MACHINE PRECISION. EM[2]: THE RELATIVE TOLERANCE OF THE ZEROS; EXIT: EM[1]: THE MAXIMUM OF ABS(B[0]) + 1, C[I]+ABS(B[I])+1 (I=1,...N-2) AND C[N-1]+ABS(B[N-1]); EM[5]: THE NUMBER OF ITERATIONS PERFORMED. PROCEDURES USED: VALSYMTRI = CP34151. 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 7 METHOD AND PERFORMANCE: THE ZEROS OF AN ORTHOGONAL POLYNOMIAL ARE THE EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX (SEE GOLUB AND WELSCH (1969), LANCZOS (1957),P.375,376, STOER (1972),P.120). THE ORTHOGONAL POLYNOMIAL IS DEFINED BY A LINEAR THREE-TERM HOMOGENEOUS RECURRENCE RELATION. EXAMPLE OF USE : WE CALCULATE THE THIRD ZERO OF THE LEGENDRE POLYNOMIAL OF THE FOURTH DEGREE. THE RECURRENCE COEFFICIENTS ARE OBTAINED FROM ABRAMOWITZ AND STEGUN (1964),P.782: B[I] = 0, I = 0, 1, ....; C[I] = A4I / (A3I * A3(I-1)) * A1(I-1) = I * I / ( 4 * I * I - 1), I = 0, 1, ..... (IT IS RECOMMENDED TO STORE THE ELEMENTS OF THE ARRAYS B AND C IN REVERSED ORDER IF THESE ELEMENTS ARE STRONGLY DECREASING). "BEGIN" "ARRAY" B, C[0:4], ZER[3:3], EM[0:5]; "INTEGER" I; EM[0]:= EM[2]:= "-14; "FOR" I:= 0, 1, 2, 3 "DO" "BEGIN" B[I]:= 0; C[I]:= I * I / (4 * I * I - 1) "END"; SELZERORTPOL (4, 3, 3, B, C, ZER, EM); OUTPUT(61,"(""("THE THIRD ZERO:")",2/,ZD5B,N,2/, "("EM[1]:")",5BD.2D"+2D, /, "("EM[5]:")",5ZD")",3,ZER[3],EM[1],EM[5]) "END" THE THIRD ZERO: 3 -3.3998104358486"-001 EM[1]: 1.33"+00 EM[5]: 12 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 8 SUBSECTION: ALL JAC ZER. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" ALL JAC ZER(N, ALFA, BETA, ZER); "VALUE" N, ALFA, BETA; "INTEGER" N; "REAL" ALFA, BETA; "ARRAY" ZER; "CODE" 31370; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE UPPER BOUND OF THE ARRAY ZER; N >= 1; ALFA,BETA: ; THE PARAMETERS OF THE JACOBI POLYNOMIAL, SEE ABRAMOWITZ AND STEGUN (1964); ALFA, BETA > - 1; ZER: ; "ARRAY" ZER[1 : N]; EXIT: ZER[1], ..., ZER[N] ARE THE ZEROS OF THE N-TH JACOBI POLYNOMIAL WITH PARAMETERS ALFA AND BETA. PROCEDURES USED: ALL ZER ORT POL = CP 31362. REQUIRED CENTRAL MEMORY: IF ALFA = BETA THEN TWO AUXILIARY ARRAYS OF N//2 REALS ARE USED, OTHERWISE TWO AUXILIARY ARRAYS OF N REALS ARE DECLARED. METHOD AND PERFORMANCE: THE JACOBI POLYNOMIALS ARE A SPECIAL CASE OF ORTHOGONAL POLYNOMIALS (SEE ABRAMOWITZ AND STEGUN (1964)); ALL JAC ZER COMPUTES THE COEFFICIENTS OF THE THREE-TERM RECURRENCE RELATION AND CALLS THE PROCEDURE ALL ZER ORT POL TO COMPUTE THE ZEROS; IF ALFA=BETA, THE POLYNOMIALS ARE ODD OR EVEN, HENCE ONLY THE POSITIVE ZEROS ARE CALCULATED; THIS IS DONE BY MEANS OF THE FORMULAS P(2*M, ALFA, ALFA, X) = C(M)*P(M, ALFA, -0.5, 2*X*X - 1), P(2*M - 1, ALFA, ALFA, X) = D(M)*P(M, ALFA, +0.5, 2*X*X - 1)*X (SEE ABRAMOWITZ AND STEGUN (1964), FORMULAS 22.5.20 - 22.5.27). 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 9 EXAMPLE OF USE: THE PROGRAM "BEGIN" "ARRAY" X[1:3]; ALL JAC ZER(3,-.5,-.5,X); OUTPUT(61,"("3(4B-D.13D"-ZD)")",X[1],X[2],X[3]) "END" DELIVERS THE FOLOWING RESULTS: -8.6602540378444"-1 0.0000000000000" 0 8.6602540378444"-1 SUBSECTION: ALL LAG ZER. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" ALL LAG ZER(N, ALFA, ZER); "VALUE" N, ALFA; "INTEGER" N; "REAL" ALFA; "ARRAY" ZER; "CODE" 31371; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE UPPER BOUND OF THE ARRAY ZER; N >= 1; ALFA: ; THE PARAMETER OF THE LAGUERRE POLYNOMIAL, SEE ABRAMOWITZ AND STEGUN (1964); ALFA > -1; ZER: ; "ARRAY" ZER[1 : N]; EXIT: ZER[1], ..., ZER[N] ARE THE ZEROS OF THE N-TH LAGUERRE POLYNOMIAL WITH PARAMETER ALFA. PROCEDURES USED: ALL ZER ORT POL = CP 31362. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF N REALS ARE USED. 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 10 METHOD AND PERFORMANCE: THE LAGUERRE POLYNOMIALS ARE A SPECIAL CASE OF ORTHOGONAL POLYNOMIALS (SEE ABRAMOWITZ AND STEGUN (1964)); ALL LAG ZER COMPUTES THE COEFFICIENTS OF THE THREE-TERM RECURRENCE RELATION AND CALLS THE PROCEDURE ALL ZER ORT POL TO COMPUTE THE ZEROS. EXAMPLE OF USE: "BEGIN" "ARRAY" X[1:3]; ALL LAG ZER(3,-.5,X); OUTPUT(61,"("3(4B-D.13D"-ZD)")",X[1],X[2],X[3]) "END" DELIVERS THE FOLOWING RESULTS: 5.5253437422633" 0. 1.7844927485432" 0 1.9016350919350" -1 SOURCE TEXT(S) : 0"CODE"31362; "PROCEDURE" ALLZERORTPOL (N, B, C, ZER, EM); "VALUE" N; "INTEGER" N; "ARRAY" B, C, ZER, EM; "BEGIN" "INTEGER"I;"REAL"NRM;"ARRAY"BB[1:N]; "PROCEDURE" DUPCEV (L, U, SHIFT, A, B); "VALUE"L,U,SHIFT;"INTEGER"L,U,SHIFT;"ARRAY"A,B; "FOR" U:=U "STEP" -1 "UNTIL" L "DO" A[U]:=B[U+SHIFT]; NRM:=ABS(B[0]); "FOR"I:=1"STEP"1"UNTIL"N-2"DO""IF"C[I]+ABS(B[I])>NRM"THEN" NRM:=C[I]+ABS(B[I]); "IF"N>1"THEN"NRM:="IF"NRM+1>=C[N-1]+ABS(B[N-1])"THEN"NRM+1"ELSE" C[N-1]+ABS(B[N-1]); EM[1]:=NRM; DUPCEV(1,N,-1,ZER,B); DUPVEC(1,N-1,0,BB,C);BB[N]:=0; QRIVALSYMTRI(ZER,BB,N,EM) "END" ALLZERORTPOL 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 11 ; "EOP" "CODE"31363; "PROCEDURE" LUPZERORTPOL (N, M, B, C, ZER, EM); "VALUE" N, M; "INTEGER" N, M; "ARRAY" B, C, ZER, EM; "BEGIN" "PROCEDURE" RATQR(N,M,POSDEF,DLAM,EPS)TRANS:(D,B2); "VALUE" N,M,POSDEF,DLAM,EPS; "INTEGER" N,M; "BOOLEAN" POSDEF; "REAL" DLAM,EPS; "ARRAY" D,B2; "COMMENT" QR ALGORITHM FOR THE COMPUTATION OF THE LOWEST EIGENVALUES OF A SYMMETRIC TRIDIAGONAL MATRIX. A RATIONAL VARIANT OF THE QR TRANSFORMATION IS USED, CONSISTING OF TWO SUCCESSIVE QD STEPS PER ITERATION. A SHIFT OF THE SPECTRUM AFTER EACH ITERATION GIVES AN ACCELERATED RATE OF CONVERGENCE. A NEWTON CORRECTION,DERIVED FROM THE CHARACTERISTIC POLYNOMIAL,IS USED AS SHIFT. RATQR IS IMPLEMENTED BY REINSCH AND BAUER, SEE WILKINSON AND REINSCH ,1971, CONTR. II-6. FORMATS: D,B2[1:N]; "BEGIN" "INTEGER" I,J,K,T; "REAL" DELTA,E,EP,ERR,P,Q,QP,R,S,TOT; "COMMENT" LOWER BOUND FOR EIGENVALUES FROM GERSHGORIN, INITIAL SHIFT; B2[1]:= ERR:= Q:= S:= 0; TOT:= D[1]; "FOR" I:= N "STEP" -1 "UNTIL" 1 "DO" "BEGIN" P:= Q; Q:= SQRT(B2[I]); E:= D[I]-P-Q; "IF" E < TOT "THEN" TOT:= E "END" I; "IF" POSDEF & TOT < 0 "THEN" TOT:= 0 "ELSE" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" D[I]:= D[I]-TOT; T:= 0; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" NEXT QR TRANSFORMATION: T:= T + 1; TOT:= TOT + S; DELTA:= D[N]-S; I:= N; E:= ABS(EPS*TOT); "IF" DLAM < E "THEN" DLAM:= E; "IF" DELTA < = DLAM "THEN" "GOTO" CONVERGENCE; E:= B2[N]/DELTA; QP:= DELTA+E; P:= 1; "FOR" I:= N-1 "STEP" -1 "UNTIL" K "DO" "BEGIN" Q:= D[I]-S-E; R:= Q/QP; P:= P*R+1; EP:= E*R; D[I+1]:= QP+EP; DELTA:= Q-EP; "IF" DELTA < = DLAM "THEN" "GOTO" CONVERGENCE; E:= B2[I]/Q;QP:= DELTA+E; B2[I+1]:= QP*EP "END" I; D[K]:= QP; S:= QP/P; "COMMENT" 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 12 ; "IF" TOT+S > TOT "THEN" "GOTO" NEXT QR TRANSFORMATION; "COMMENT" IRREGULAR END OF ITERATION, DEFLATE MINIMUM DIAGONAL ELEMENT; S:= 0; I:= K; DELTA:= QP; "FOR" J:= K+1 "STEP" 1 "UNTIL" N "DO" "IF" D[J] < DELTA "THEN" "BEGIN" I:= J; DELTA:= D[J] "END"; CONVERGENCE: "IF" I < N "THEN" B2[I+1]:= B2[I]*E/QP; "FOR" J:= I-1 "STEP" -1 "UNTIL" K "DO" "BEGIN" D[J+1]:= D[J]-S; B2[J+1]:= B2[J] "END" J; D[K]:= TOT; B2[K]:= ERR:= ERR+ABS(DELTA) "END" K; EM[5]:=T;EM[3]:=INFNRMVEC(1,M,T,B2); "END" RATQR; "PROCEDURE" DUPCEV (L, U, SHIFT, A, B); "VALUE"L,U,SHIFT;"INTEGER"L,U,SHIFT;"ARRAY"A,B; "FOR" U:=U "STEP" -1 "UNTIL" L "DO" A[U]:=B[U+SHIFT]; "INTEGER" I;"REAL"NRM; NRM:=ABS(B[0]); "FOR"I:=1"STEP"1"UNTIL"N-2"DO""IF"C[I]+ABS(B[I])>NRM"THEN" NRM:=C[I]+ABS(B[I]); "IF"N>1"THEN"NRM:="IF"NRM+1>=C[N-1]+ABS(B[N-1])"THEN"NRM+1"ELSE" C[N-1]+ABS(B[N-1]); EM[1]:=NRM; DUPCEV(1,N,-1,B,B); DUPCEV(2,N,-1,C,C); RATQR (N, M, EM[6] = 1, EM[2], EM[0], B, C); DUPVEC (1, M, 0, ZER, B) "END" LUPZERORTPOL; "EOP" "CODE"31364; "PROCEDURE" SELZERORTPOL (N, N1, N2, B, C, ZER, EM); "VALUE" N, N1, N2; "INTEGER" N, N1, N2; "ARRAY" B, C, ZER, EM; "BEGIN" "INTEGER"I;"REAL"NRM;"ARRAY"D[1:N]; "PROCEDURE" DUPCEV (L, U, SHIFT, A, B); "VALUE"L,U,SHIFT;"INTEGER"L,U,SHIFT;"ARRAY"A,B; "FOR" U:=U "STEP" -1 "UNTIL" L "DO" A[U]:=B[U+SHIFT]; NRM:=ABS(B[0]); "FOR"I:=N-2"STEP"-1"UNTIL"1"DO""IF"C[I]+ABS(B[I])>NRM"THEN" NRM:=C[I]+ABS(B[I]); "COMMENT" 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 13 ; "IF"N>1"THEN"NRM:="IF"NRM+1>=C[N-1]+ABS(B[N-1])"THEN"NRM+1"ELSE" C[N-1]+ABS(B[N-1]); EM[1]:=NRM; DUPCEV(1,N,-1,D,B); VALSYMTRI (D, C, N, N1, N2, ZER, EM); EM[5]:=EM[3] "END" SELZERORTPOL; "EOP" "CODE" 31370; "PROCEDURE" ALL JAC ZER(N, ALFA, BETA, ZER); "VALUE" N, ALFA, BETA ; "INTEGER" N; "REAL" ALFA, BETA ; "ARRAY" ZER; "IF" ALFA = BETA "THEN" "BEGIN" "INTEGER" I, M; "ARRAY" A, B[0:N//2], EM[0:5]; "REAL" MIN, GAMMA, SUM, ZERI; M:= N//2; "IF" N ^= 2*M "THEN" "BEGIN" GAMMA:= + 0.5; ZER[M + 1]:= 0 "END" "ELSE" GAMMA:= - 0.5; MIN:= 0.25 - ALFA*ALFA; SUM:= ALFA + GAMMA + 2; A[0]:= (GAMMA - ALFA)/SUM; A[1]:= MIN/SUM/(SUM + 2); B[1]:= 4*(1 + ALFA)*(1 + GAMMA)/SUM/SUM/(SUM + 1); "FOR" I:= 2 "STEP" 1 "UNTIL" M - 1 "DO" "BEGIN" SUM:= I + I + ALFA + GAMMA; A[I]:= MIN/SUM/(SUM + 2); SUM := SUM*SUM; B[I]:= 4*I*(I + ALFA + GAMMA)*(I + ALFA)*(I + GAMMA)/ SUM/(SUM - 1) "END"; EM[0]:=ARREB; EM[2]:="-10; EM[4]:= 6*M; ALL ZER ORT POL (M, A, B, ZER, EM); "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" ZER[I]:= ZERI:= - SQRT((1 + ZER[I])/2); ZER[N + 1 - I]:= - ZERI "END" "END" "ELSE" "BEGIN" "INTEGER" I; "REAL" SUM, MIN; "ARRAY" A, B[0:N], EM[0:5]; "COMMENT" 1SECTION : 3.6.2 (DECEMBER 1978) PAGE 14 ; MIN:= (BETA - ALFA)*(BETA + ALFA); SUM:= ALFA + BETA + 2; B[0]:= 0; A[0]:= (BETA - ALFA)/SUM; A[1]:= MIN/SUM/(SUM + 2); B[1]:= 4*(1 + ALFA)*(1 + BETA)/SUM/SUM/(SUM + 1); "FOR" I:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" SUM:= I + I + ALFA + BETA; A[I]:= MIN/SUM/(SUM + 2); SUM:= SUM*SUM; B[I]:= 4*I*(I + ALFA + BETA)*(I + ALFA)*(I + BETA)/ (SUM - 1)/SUM "END"; EM[0]:=ARREB; EM[2]:= 1.0"-8; EM[4]:= 6*N; ALL ZER ORT POL(N, A, B, ZER, EM) "END" ALL JAC ZER; "EOP" "CODE" 31371; "PROCEDURE" ALL LAG ZER(N, ALFA, ZER); "VALUE" N, ALFA ; "INTEGER"N; "REAL" ALFA ; "ARRAY" ZER; "BEGIN" "INTEGER" I; "ARRAY" A, B[0:N], EM[0:5]; B[0]:= 0; A[N - 1]:= N + N + ALFA - 1; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" A[I - 1]:= I + I + ALFA - 1; B[I]:= I*(I + ALFA) "END"; EM[0]:=ARREB; EM[2]:= "-10;EM[4]:= 6*N; ALL ZER ORT POL(N, A, B, ZER, EM) "END" ALL LAG ZER; "EOP" 1SECTION : 3.6.3 (JULY 1974) PAGE 1 AUTHOR: C.G. VAN DER LAAN. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730815. BRIEF DESCRIPTION: COMKWD CALCULATES THE ROOTS OF A QUADRATIC EQUATION WITH COMPLEX COEFFICIENTS. KEYWORDS: ZEROS,QUADRATIC EQUATION,POLYNOMIAL EQUATION,COMPLEX COEFFICIENTS. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE"COMKWD(PR,PI,QR,QI,GR,GI,KR,KI); "VALUE"PR,PI,QR,QI;"REAL"PR,PI,QR,QI,GR,GI,KR,KI; "CODE" 34345; THE MEANING OF THE FORMAL PARAMETERS IS: PR,PI,QR,QI:; ENTRY:PR,QR ARE THE REAL PARTS AND PI,QI ARE THE IMAGINARY PARTS OF THE COEFFICIENTS OF THE QUADRATIC EQUATION: X**2-2*(PR+I*PI)*X-(QR+I*QI)=0; GR,GI,KR,KI:; EXIT:THE REAL PARTS AND THE IMAGINARY PARTS OF THE DINOMIAL ARE DELIVERED IN GR,KR AND GI,KI, RESPECTIVELY;MOREOVER,THE MODULUS OF GR+I*GI IS GREATER OR EQUAL THE MODULUS OF KR+I*KI. PROCEDURES USED: COMMUL=CP34341; COMDIV=CP34342; COMSQRT=CP34343. LANGUAGE: ALGOL 60. 1SECTION : 3.6.3 (JULY 1974) PAGE 2 EXAMPLE OF USE: "BEGIN""REAL"GR,GI,KR,KI; COMKWD(-.1,.3,.11,.02,GR,GI,KR,KI); OUTPUT(61,"(""("X**2-2(-.1+.3*I)*X-( .11+.02*I) HAS ROOTS")",/, -D.DD,+D.DD,"("*I")",/, -D.DD,+D.DD,"("*I")"")",GR,GI,KR,KI) "END" X**2-2(-.1+.3*I)*X-( .11+.02*I) HAS ROOTS -0.30+0.40*I 0.10+0.20*I SOURCE TEXT(S): 0"CODE"34345; "PROCEDURE" COMKWD(PR,PI,QR,QI,GR,GI,KR,KI); "VALUE" PR,PI,QR,QI;"REAL" PR,PI,QR,QI,GR,GI,KR,KI; "BEGIN" "IF" QR=0 & QI = 0 "THEN" "BEGIN" KR:=KI:=0 ;GR:=PR*2;GI:=PI*2 "END" "ELSE" "IF" PR=0 & PI= 0 "THEN" "BEGIN" COMSQRT(QR,QI,GR,GI);KR:=-GR;KI:=-GI "END" "ELSE" "BEGIN" "REAL" HR,HI; "IF" ABS(PR) > 1 "OR" ABS(PI) >1 "THEN" "BEGIN" COMDIV(QR,QI,PR,PI,HR,HI); COMDIV(HR,HI,PR,PI,HR,HI); COMSQRT(1+HR,HI,HR,HI); COMMUL(PR,PI,HR+1,HI,GR,GI); "END" "ELSE" "BEGIN" COMSQRT(QR+(PR+PI)*(PR-PI),QI+ PR*PI*2,HR,HI); "IF" PR * HR + PI * HI > 0 "THEN" "BEGIN" GR := PR + HR;GI := PI + HI "END" "ELSE" "BEGIN" GR := PR - HR;GI:= PI - HI "END"; "END"; COMDIV(-QR,-QI,GR,GI,KR,KI); "END" "END" COMKWD; "EOP" 1SECTION : 4.1 (JULY 1974) PAGE 1 AUTHOR : J.W. DANIEL. REVISOR : J. KOK. INSTITUTE : MATHEMATICAL CENTRE. RECEIVED : 730528 (EULER). 730917 (SUMPOSSERIES). BRIEF DESCRIPTION : THIS SECTION CONTAINS TWO PROCEDURES FOR THE SUMMATION OF CONVERGENT INFINITE SERIES: EULER PERFORMS THE SUMMATION OF AN ALTERNATING SERIES. SUMPOSSERIES PERFORMS THE SUMMATION OF A CONVERGENT SERIES WITH POSITIVE MONOTONOUSLY DECREASING TERMS USING THE VAN WIJNGAARDEN TRANSFORMATION OF THE SERIES TO AN ALTERNATING SERIES. KEYWORDS : SUMMATION, SERIES, VAN WIJNGAARDEN TRANSFORMATION. SUBSECTION : EULER. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "REAL""PROCEDURE" EULER(AI, I, EPS, TIM); "VALUE" EPS, TIM; "INTEGER" I, TIM; "REAL" AI, EPS; "CODE" 32010; EULER : DELIVERS THE COMPUTED SUM OF THE INFINITE SERIES SUM( A[I], I:= 0,1,... ) . THE MEANING OF THE FORMAL PARAMETERS IS: AI : ; THE SUMMAND, THIS EXPRESSION WILL BE DEPENDENT ON THE JENSEN PARAMETER I; AI IS THE I-TH TERM OF THE SERIES (I >= 0). I : ; JENSEN PARAMETER. EPS,TIM: ; THE SUMMATION IS CONTINUED UNTIL TIM SUCCESSIVE TERMS OF THE TRANSFORMED SERIES ARE IN ABSOLUTE VALUE LESS THAN EPS. 1SECTION : 4.1 (DECEMBER 1975) PAGE 2 PROCEDURES USED : NONE. REQUIRED CENTRAL MEMORY : EXECUTION FIELD LENGTH : 25. LANGUAGE : ALGOL 60. METHOD AND PERFORMANCE : EULER PERFORMS THE SUMMATION OF AN ALTERNATING SEQUENCE BY APPLYING EULER'S TRANSFORMATION. BY THIS TRANSFORMATION THE SEQUENCE OF TERMS IS REPLACED BY THE SEQUENCE OF MEANS OF TWO SUCCESSIVE TERMS. IF NECESSARY THE NEW SEQUENCE IS AGAIN TRANSFORMED BY EULER'S TRANSFORMATION. THE SUMMATION STOPS WHEN TIM SUCCESSIVE TERMS OF THE (ONCE OR SEVERAL TIMES TRANSFORMED) SEQUENCE ARE IN ABSOLUTE VALUE LESS THAN EPS. REFERENCES : P.NAUR, ED. : REVISED REPORT ON THE ALGORITHMIC LANGUAGE ALGOL 60. COPENHAGEN (1964). EXAMPLE OF USE : THE PROGRAM : "BEGIN""INTEGER" K; OUTPUT(61, "("+.8D"+2D")", EULER((- 1) ** K / (K + 1) ** 2, K, "- 6, 100)) "END" DELIVERS : +.82246703"+00. 1SECTION : 4.1 (JULY 1974) PAGE 3 SUBSECTION : SUMPOSSERIES. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE IS : "REAL""PROCEDURE" SUMPOSSERIES(AI, I, MAXADDUP, MAXZERO, MAXRECURS, MACHEXP, TIM); "VALUE" MAXADDUP, MAXZERO, MAXRECURS, MACHEXP, TIM; "REAL" AI, I, MAXZERO; "INTEGER" MAXADDUP, MAXRECURS, MACHEXP, TIM; "CODE" 32020; SUMPOSSERIES : DELIVERS THE COMPUTED SUM OF THE INFINITE SERIES A[I] , I:= 1,2,.... . THE MEANING OF THE FORMAL PARAMETERS IS: AI : ; THE SUMMAND, THIS EXPRESSION SHOULD BE DEPENDENT ON THE JENSEN PARAMETER I; AI IS THE I-TH TERM OF THE SERIES (I >= 1). I : ; JENSEN PARAMETER. MAXADDUP : ; UPPER LIMIT FOR THE NUMBER OF STRAIGHTFORWARD ADDITIONS . MAXZERO, TIM : ; TOLERANCES EPS AND TIM NEEDED FOR A CALL OF THE PROCEDURE EULER (THIS SECTION). MAXZERO IS ALSO USED AS A TOLERANCE FOR MAXADDUP STRAIGHTFORWARD ADDITIONS. MAXRECURS : ; UPPER LIMIT FOR THE RECURSION DEPTH OF THE VAN WIJNGAARDEN TRANSFORMATIONS. MACHEXP : ; IN ORDER TO AVOID OVERFLOW AND EVALUATION OF THOSE TERMS WHICH CAN BE NEGLECTED, MACHEXP HAS TO BE THE LARGEST ADMISSIBLE VALUE X FOR WHICH TERMS WITH INDEX J = K * (2 ** X) CAN BE COMPUTED (K IS SMALL). OTHERWISE OVERFLOW MIGHT OCCUR IN COMPUTING A VALUE FOR THE JENSEN PARAMETER I, WHICH CAN BE AN UNUSUALLY HIGH POWER OF 2. PROCEDURES USED : EULER = CP32010. REQUIRED CENTRAL MEMORY : EXECUTION FIELD LENGTH : ABOUT 1000 * RECURSION DEPTH. LANGUAGE : ALGOL 60. 1SECTION : 4.1 (JULY 1974) PAGE 4 METHOD AND PERFORMANCE : WHEN THE TERMS AI WITH INDICES MAXADDUP + 1, ... , MAXADDUP + TIM ARE ALL LESS THAN MAXZERO, CONVERGENCE IS ASSUMED AND SUMPOSSERIES DELIVERS THE SUM OF THE SERIES BY STRAIGHTFORWARD ADDITION UNTIL TIM SERIAL TERMS ARE LESS THAN MAXZERO. OTHERWISE THE VAN WIJNGAARDEN TRANSFORMATION IS APPLIED, YIELDING AN ALTERNATING SERIES WHICH IS SUMMED UP WITH EULER'S METHOD. SINCE THE TERMS OF THIS ALTERNATING SERIES ARE THEMSELVES INFINITE SERIES WITH POSITIVE TERMS, THE HERE DESCRIBED PROCESS IS RECURSIVELY CALLED FOR THE SUMMATION OF EACH TERM THAT IS WANTED BY EULER'S METHOD. HOWEVER, ONLY STRAIGHTFORWARD ADDITION IS APPLIED IF THE ALLOWED RECURSION LEVEL (SPECIFIED BY MAXRECURS) HAS BEEN REACHED. IN THE RECURSION THE PROCESS ASKS FOR TERMS AI WITH INDICES OF THE TYPE J * (2 ** K), IN WHICH K CAN BE VERY LARGE. IN ORDER TO AVOID OVERFLOW AN UPPER BOUND FOR K MUST BE GIVEN IN MACHEXP. IF K EXCEEDS THIS BOUND THE CORRESPONDING TERM IS TAKEN TO BE ZERO. REFERENCES : [1] DANIEL, J.W. : SUMMATION OF A SERIES OF POSITIVE TERMS BY CONDENSATION TRANSFORMATIONS. MATH. OF COMP. V.23, P.91-96 (1969). [2] WIJNGAARDEN, A. VAN : COURSE SCIENTIFIC COMPUTING B, PROCESS ANALYSIS (DUTCH) MATHEMATISCH CENTRUM CR-18 (1965). EXAMPLE OF USE : THE PROGRAM : "BEGIN""COMMENT" 730808, EXAMPLE OF THE USE OF SUMPOSSERIES; "REAL" I; OUTPUT(61, "("/, +.12D"+DD")", SUMPOSSERIES(1 / I ** 2, I, 100, "- 7, 8, 1068, 10)) "END" DELIVERS : +.164493406604"+01 NUMBER OF TERMS USED : 462, RECURSION DEPTH : 1. 1SECTION : 4.1 (JULY 1974) PAGE 5 SOURCE TEXT(S) : 0"CODE" 32010; "REAL""PROCEDURE" EULER(AI, I, EPS, TIM); "VALUE" EPS, TIM; "INTEGER" I, TIM; "REAL" AI, EPS; "BEGIN""INTEGER" K, N, T; "REAL" MN, MP, DS, SUM; "ARRAY" M[0:15]; N:= T:= 0; I:= 0; M[0]:= AI; SUM:= M[0] / 2; NEXT TERM: I:= I + 1; MN:= AI; "FOR" K:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" MP:= (MN + M[K]) / 2; M[K]:= MN; MN:= MP "END"; "IF" ABS(MN) < ABS(M[N]) & N < 15 "THEN" "BEGIN" DS:= MN / 2; N:= N + 1; M[N]:= MN "END" "ELSE" DS:= MN; SUM:= SUM + DS; T:= "IF" ABS(DS) < EPS "THEN" T + 1 "ELSE" 0; "IF" T < TIM "THEN" "GO TO" NEXT TERM; EULER:= SUM "END" EULER; "EOP" 0"CODE" 32020; "REAL" "PROCEDURE" SUMPOSSERIES(AI, I, MAXADDUP, MAXZERO,MAXRECURS, MACHEXP, TIM); "VALUE" MAXADDUP, MAXZERO, MAXRECURS, MACHEXP, TIM; "REAL" AI, I, MAXZERO; "INTEGER" MAXADDUP, MAXRECURS, MACHEXP, TIM; "BEGIN" "INTEGER" RECURS, VL, VL2, VL4; "REAL" "PROCEDURE" SUMUP(AI, I); "REAL" AI, I; "BEGIN" "INTEGER" J; "REAL" SUM, NEXTTERM; I:= MAXADDUP + 1; J:= 1; CHECK ADD: "IF" AI <= MAXZERO "THEN" "BEGIN""IF" J < TIM "THEN" "BEGIN" J:= J + 1; I:= I + 1; "GO TO" CHECK ADD "END" "END""ELSE" "IF" RECURS ^= MAXRECURS "THEN""GO TO" TRANSFORMSERIES; SUM:= 0; I:= 0; J:= 0; ADD LOOP: I:= I + 1; NEXTTERM:= AI; J:= "IF" NEXTTERM <= MAXZERO "THEN" J + 1 "ELSE" 0; SUM:= SUM + NEXTTERM; "IF" J < TIM "THEN""GO TO" ADD LOOP; SUMUP:= SUM; "GO TO" GOTSUM; TRANSFORMSERIES: "BEGIN""BOOLEAN" JODD; "INTEGER" J2; "ARRAY" V[1:VL]; "REAL""PROCEDURE" BJK(J, K); "VALUE" J, K; "REAL" K; "INTEGER" J; "BEGIN""REAL" COEFF; "IF" K > MACHEXP "THEN" BJK:= 0 "ELSE" "BEGIN" COEFF:= 2 ** (K - 1); I:= J * COEFF; BJK:= COEFF * AI "END" "END" BJK 1SECTION : 4.1 (JULY 1974) PAGE 6 ; "REAL""PROCEDURE" VJ(J); "VALUE" J; "INTEGER" J; "BEGIN""REAL" TEMP, K; "IF" JODD "THEN" "BEGIN" JODD:= "FALSE"; RECURS:= RECURS + 1; TEMP:= VJ:= SUMUP(BJK(J, K), K); RECURS:= RECURS - 1; "IF" J <= VL "THEN" V[J]:= TEMP "ELSE" "IF" J <= VL2 "THEN" V[J - VL]:= TEMP "END""ELSE" "BEGIN" JODD:= "TRUE"; "IF" J > VL4 "THEN" "BEGIN" RECURS:= RECURS + 1; VJ:= - SUMUP(BJK(J, K), K); RECURS:= RECURS - 1 "END""ELSE" "BEGIN" J2:= J2 + 1; I:= J2; "IF" J > VL2 "THEN" VJ:= - (V[J2 - VL] - AI) / 2 "ELSE" "BEGIN" TEMP:= V["IF" J <= VL "THEN" J "ELSE" J - VL]:= (V[J2] - AI) / 2; VJ:= - TEMP "END" "END" "END" "END" VJ; J2:= 0; JODD:= "TRUE"; SUMUP:= EULER(VJ(J + 1), J, MAXZERO, TIM) "END" TRANSFORMSERIES; GOTSUM: "END" SUMUP; RECURS:= 0; VL:= 1000; VL2:= 2 * VL; VL4:= 2 * VL2; SUMPOSSERIES:= SUMUP(AI, I) "END" SUMPOSSERIES; "EOP" 1SECTION : 4.2.1 (JULY 1974) PAGE 1 SECTION 4.2.1 CONTAINS TWO ALTERNATIVE PROCEDURES FOR THE COMPUTATION OF A DEFINITE INTEGRAL. A. THE PROCEDURE QADRAT USES HIGH ORDER INTEGRATION RULES (UP TO 16-TH ORDER) AND IS APPROPRIATE FOR THE EVALUATION OVER A FINITE INTERVAL. B. THE PROCEDURE INTEGRAL USES A 5-TH ORDER METHOD AND CAN ALSO BE USED TO CALCULATE THE INTEGRAL OVER A NUMBER OF CONSECUTIVE INTERVALS. MOREOVER THE PROCEDURE CAN BE USED FOR THE COMPUTATION OF THE DEFINITE INTEGRAL OVER AN INFINITE INTERVAL. FOR A COMPARISON OF A NUMBER OF PROCEDURES THAT EVALUATE DEFINITE INTEGRALS : SEE REF[2]. REFERENCES : [1] T.J.DEKKER AND C.J.ROOTHART. INTRODUCTION TO NUMERICAL ANALYSIS. (DUTCH). MATH. CENTRE REPORT CR 244/74, AMSTERDAM. [2] C.J.ROOTHART AND H. FIOLET. QUADRATURE PROCEDURES. MATH. CENTRE REPORT MR 137/72, AMSTERDAM. 1SECTION : 4.2.1.A (JULY 1974) PAGE 1 AUTHORS: C.J.ROOTHART. CONTRIBUTORS: P.W.HEMKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730530. BRIEF DESCRIPTION: QADRAT COMPUTES THE DEFINITE INTEGRAL OF A FUNCTION OF ONE VARIABLE OVER A FINITE INTERVAL. KEYWORDS: INTEGRATION, QUADRATURE, DEFINITE INTEGRAL. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "REAL" "PROCEDURE" QADRAT (X, A, B, FX, E); "VALUE" A, B; "REAL" X, A, B, FX; "ARRAY" E; "CODE" 32070; QADRAT: DELIVERS THE COMPUTED VALUE OF THE DEFINITE INTEGRAL FROM A TO B OF THE FUNCTION F(X); THE MEANING OF THE FORMAL PARAMETERS IS: X: ; INTEGRATION VARIABLE; X CAN BE USED AS A JENSEN-PARAMETER DURING THE EVALUATIONS OF FX; A,B: ; (A,B) DENOTES THE INTERVAL OF INTEGRATION; B < A IS ALLOWED; FX: ; FX DENOTES THE INTEGRAND F(X). THIS EXPRESSION WILL BE DEPENDENT ON THE JENSEN-PARAMETER X. E: ; "ARRAY" E[1:3]; ENTRY: E[1]: THE RELATIVE ACCURACY REQUIRED; E[2]: THE ABSOLUTE ACCURACY REQUIRED; EXIT: E[3]: THE NUMBER OF ELEMENTARY INTEGRATIONS WITH H < ABS(B-A) * E[1]. 1SECTION : 4.2.1.A (JULY 1974) PAGE 2 PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 16 + 9 * RECURSION DEPTH. RUNNING TIME: DEPENDS STRONGLY ON THE DEFINITE INTEGRAL TO COMPUTE. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE REF[1]. REFERENCES: [1].C.J.ROOTHART AND H.FIOLET. QUADRATURE PROCEDURES. MATH.CENTRE, AMSTERDAM. REPORT MR 137/72. EXAMPLE OF USE: "BEGIN" "ARRAY" E[1:3]; "REAL" T,Q; EXECUTION FIELD LENGTH: CIRCA 16 + 26 * RECURSION DEPTH. E[1]:= E[2]:= "-9; Q:= QADRAT(T,0,3.141592653589 ,SIN(T),E); OUTPUT(61,"("/,+.15D"+3D,3B,3ZD,/")",Q,E[3]); "END" DELIVERS: +.200000000000033"+001 0 1SECTION : 4.2.1.A (NOVEMBER 1976) PAGE 3 SOURCE TEXT(S): 0"CODE" 32070; "REAL" "PROCEDURE" QADRAT(X, A, B, FX, E); "VALUE" A, B; "REAL" X, A, B, FX; "ARRAY" E; "BEGIN" "REAL" F0, F2, F3, F5, F6, F7, F9, F14, V, W, HMIN, HMAX, RE, AE; "REAL" "PROCEDURE" LINT(X0, XN, F0, F2, F3, F5, F6, F7, F9, F14); "REAL" X0, XN, F0, F2, F3, F5, F6, F7, F9, F14; "BEGIN" "REAL" H, XM, F1, F4, F8, F10, F11, F12, F13; XM:= (X0 + XN) / 2; H:= (XN - X0) / 32; X:= XM + 4 * H; F8:= FX; X:= XN - 4 * H; F11:= FX; X:= XN - 2 * H; F12:= FX; V:= 0.330580178199226 * F7 + 0.173485115707338 * (F6 + F8) + 0.321105426559972*(F5 + F9) + 0.135007708341042 * (F3 + F11) + 0.165714514228223*(F2 + F12) + 0.393971460638127"- 1 * (F0 + F14); X:= X0 + H; F1:= FX; X:= XN - H; F13:= FX; W:= 0.260652434656970 * F7 + 0.239063286684765 * (F6 + F8) + 0.263062635477467*(F5 + F9) + 0.218681931383057 * (F3 + F11) + 0.275789764664284"- 1 * (F2 + F12) + 0.105575010053846* (F1 + F13) + 0.157119426059518"- 1 * (F0 + F14); "IF" ABS(H) < HMIN "THEN" E[3]:= E[3] + 1; "IF" ABS(V - W) < ABS(W) * RE + AE "OR" ABS(H) < HMIN "THEN" LINT:= H * W "ELSE" "BEGIN" X:= X0 + 6 * H; F4:= FX; X:= XN - 6 * H; F10:= FX; V:= 0.245673430093324* F7 + 0.255786258286921* (F6 + F8) + 0.228526063690406*(F5 + F9) + 0.500557131525460"- 1*(F4 + F10) + 0.177946487736780*(F3 + F11)+0.584014599347449"- 1 * (F2 + F12) + 0.874830942871331"- 1 * (F1 + F13) + 0.189642078648079"- 1 * (F0 + F14); LINT:= "IF" ABS(V - W) < ABS(V) * RE + AE "THEN" H * V "ELSE" LINT(X0, XM, F0, F1, F2, F3, F4, F5, F6, F7) - LINT(XN, XM, F14, F13, F12, F11, F10, F9, F8, F7) "END" "END" LINT; HMAX:= (B - A) / 16; "IF" HMAX=0 "THEN" "BEGIN" QADRAT:= 0; "GOTO" RETURN "END"; RE:= E[1]; AE:= 2 * E[2] / ABS(B - A); E[3]:= 0; HMIN:= ABS(B - A) * RE; X:= A; F0:= FX; X:= A + HMAX; F2:= FX; X:= A + 2 * HMAX; F3:= FX; X:= A + 4 * HMAX; F5:= FX; X:= A + 6 * HMAX; F6:= FX; X:= A + 8 * HMAX; F7:= FX; X:= B - 4 * HMAX; F9:= FX; X:= B; F14:= FX; QADRAT:= LINT(A, B, F0, F2, F3, F5, F6, F7, F9, F14) * 16; RETURN: "END" QADRAT; "EOP" 1SECTION : 4.2.1.B (JULY 1974) PAGE 1 AUTHOR: H.N.GLORIE. CONTRIBUTOR: H.FIOLET. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730606. BRIEF DESCRIPTION: INTEGRAL CALCULATES THE DEFINITE INTEGRAL OF A FUNCTION OF ONE VARIABLE, OVER A FINITE OR INFINITE INTERVAL OR OVER A NUMBER OF CONSECUTIVE INTERVALS. KEYWORDS: DEFINITE INTEGRAL, INFINITE INTERVAL, SIMPSON RULE, RICHARDSON CORRECTION. 1SECTION : 4.2.1.B (JULY 1974) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "REAL" "PROCEDURE" INTEGRAL(X,A,B,FX,E,UA,UB); "VALUE" A,B;"REAL" X,A,B,FX;"ARRAY" E;"BOOLEAN" UA,UB; "CODE" 32051; INTEGRAL: DELIVERS THE COMPUTED VALUE OF THE DEFINITE INTEGRAL OF THE FUNCTION FROM A TO B; AFTER SUCCESSIVE CALLS OF THE PROCEDURE, THE INTEGRAL OVER THE TOTAL INTERVAL IS DELIVERED, I.E. THE VALUE OF A IN THE LAST CALL WITH UA="TRUE" IS THE STARTING POINT OF THE INTERVAL. THE MEANING OF THE FORMAL PARAMETERS IS: X: ; INTEGRATION VARIABLE; X IS USED AS JENSEN-PARAMETER FOR FX. A,B: ; THE INTERVAL OF INTEGRATION IS EITHER (A,B) OR (A,INFINITY) (SEE PARAMETER UB); B; THE INTEGRAND F(X); E: ; "ARRAY" E[1:6]; ENTRY : E[1]:THE RELATIVE ACCURACY REQUIRED; E[2]:THE ABSOLUTE ACCURACY REQUIRED; E[5]:ALTERNATIVE STARTING POINT (SEE PAR. UA); EXIT: E[3]:THE NUMBER OF SKIPPED INTEGRATION STEPS; E[4]:THE VALUE OF THE INTEGRAL FROM A TO B; E[5]:"IF" UB "THEN" B "ELSE" 0; E[6]:"IF" UB "THEN" F(B) "ELSE" 0; UA: ; DETERMINES THE STARTING POINT OF THE INTEGRATION; STARTING POINT:="IF" UA "THEN" A "ELSE" E[5]; UB: ; DETERMINES THE FINAL POINT OF THE INTEGRATION; FINAL POINT:="IF" UB "THEN" B "ELSE" "IF" B>A "THEN" +INFINITY "ELSE" -INFINITY; IN THE CASE UB="FALSE" , THE VALUE OF B IS STILL RELEVANT (SEE METHOD AND PERFORMANCE). PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 16 + 5 * RECURSION LEVEL. RUNNING TIME: DEPENDS STRONGLY UPON THE INTEGRAL TO BE COMPUTED. 1SECTION : 4.2.1.B (MARCH 1977) PAGE 3 LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: INTEGRAL USES THE SUBPROCEDURE QAD FOR THE CALCULATION OF THE DEFINITE INTEGRAL OVER A FINITE INTERVAL. THIS IS DONE BY MEANS OF SIMPSON 'S RULE WITH RICHARDSON CORRECTION.IF THE FOURTH DERIVATIVE IS TOO LARGE (AND THUS THE CORRECTION TERM) , THE TOTAL INTERVAL IS SPLIT INTO TWO EQUAL PARTS AND THE INTEGRATION PROCESS IS INVOKED RECURSIVELY. THIS IS DONE IN SUCH A WAY THAT THE TOTAL AMOUNT OF RICHARDSON CORRECTIONS IS SLIGHTLY SMALLER THAN OR EQUAL TO E[1] * ABS ( THE INTEGRAL FROM A TO B OF (FX) ) + E[2]. THE INTEGRATION OVER AN INFINITE INTERVAL REQUIRES TWO CALLS OF THE PROCEDURE QAD. IN THE FIRST CALL QAD COMPUTES THE DEFINITE INTEGRAL FROM A TO B . IN THE INTERVAL FROM B TO + OR - INFINITY THE INTEGRAND IS TRANSFORMED BY MEANS OF THE SUBSTITUTION Z=1/(X+1-B) TO THE DEFINITE INTEGRAL OF F(B-1+1/Z)/Z**2 FROM 0 TO 1. FOR THE INTEGRATION OF A DEFINITE INTEGRAL OVER A FINITE INTERVAL THE USE OF QADRAT IS RECOMMENDED ,ESPECIALLY WHEN HIGH ACCURACY IS REQUIRED. REFERENCES: [1] T.J.DEKKER AND C.J.ROOTHART. INTRODUCTION TO NUMERICAL ANALYSIS.(DUTCH) MATH.CENTR. REPORT CR 24/71, AMSTERDAM. [2] C.J.ROOTHART AND H.FIOLET. QUADRATURE PROCEDURES. MATH.CENTR. REPORT MR 137/72, AMSTERDAM. EXAMPLE OF USE: "BEGIN" "ARRAY" E[1:6];"REAL" A,B,X;"BOOLEAN" UA,UB; UA:="TRUE";E[1]:=E[2]:="-14; "FOR" B:=2,4,20,100 "DO" "BEGIN" UB:=B<50; A:=INTEGRAL(X,-1,-B,10/X**2,E,UA,UB); OUTPUT(61,"("N,B+DDB,N,2(B+DDDB),/")", A,E[3],E[4],E[5],E[6]); UA:="FALSE" "END" "END" DELIVERS: -4.9999999999999"+000 +00 -4.9999999999999 -002 +003 -7.4999999999998"+000 +00 -7.4999999999998 -004 +001 -9.5000000000000"+000 +00 -9.5000000000000 -020 +000 -9.9999999999998"+000 +01 -9.9999999999998 +000 +000 . 1SECTION : 4.2.1.B (MARCH 1977) PAGE 4 SOURCE TEXT(S): 0"CODE" 32051; "REAL" "PROCEDURE" INTEGRAL(X, A, B, FX, E, UA, UB); "VALUE" A,B;"REAL" X, A, B, FX; "ARRAY" E; "BOOLEAN" UA, UB; "BEGIN" "REAL" "PROCEDURE" TRANSF; "BEGIN" Z:= 1 / X; X:= Z + B1; TRANSF:= FX * Z * Z "END"; "REAL" "PROCEDURE" QAD(FX); "REAL" FX; "BEGIN" "REAL" T, V, SUM, HMIN; "PROCEDURE" INT; "BEGIN" "REAL" X3, X4, F3, F4, H; X4:= X2; X2:= X1; F4:= F2; F2:= F1; ANEW: X:= X1:= (X0 + X2) * .5; F1:= FX; X:= X3:= (X2 + X4) * .5; F3:= FX; H:= X4 - X0; V:= (4 * (F1 + F3) +2 * F2 + F0 + F4) * 15; T:= 6 * F2 -4 * (F1 + F3) + F0 + F4; "IF" ABS(T) < ABS(V) * RE + AE "THEN" SUM:=SUM + (V - T) * H "ELSE" "IF" ABS(H) < HMIN "THEN" E[3]:= E[3] +1 "ELSE" "BEGIN" INT; X2:= X3; F2:= F3; "GOTO" ANEW "END"; X0:= X4; F0:= F4 "END" INT; HMIN:= ABS(X0 - X2) * RE; X:= X1:= (X0 + X2) * .5; F1:=FX;SUM:= 0; INT; QAD:= SUM / 180 "END" QAD; "REAL" X0, X1, X2, F0, F1, F2, RE, AE, B1, Z; RE:= E[1]; "IF" UB "THEN" AE:= E[2] * 180 / ABS(B - A) "ELSE" AE:= E[2] * 90 / ABS(B - A); "IF" UA "THEN" "BEGIN" E[3]:= E[4]:= 0; X:= X0:= A; F0:= FX "END" "ELSE" "BEGIN" X:= X0:= A:= E[5]; F0:= E[6] "END"; E[5]:= X:= X2:= B; E[6]:= F2:= FX; E[4]:= E[4] + QAD(FX); "IF" ^UB "THEN" "BEGIN" "IF" A < B "THEN" "BEGIN" B1:= B -1 ; X0:= 1 "END" "ELSE" "BEGIN" B1:= B +1 ; X0:= -1 "END"; F0:= E[6]; E[5]:= X2:= 0; E[6]:= F2:= 0; AE:= E[2] * 90; E[4]:= E[4] - QAD(TRANSF) "END"; INTEGRAL:= E[4] "END" INTEGRAL; "EOP" 1SECTION : 4.2.2 (DECEMBER 1979) PAGE 1 AUTHOR : P.W. HEMKER. CONTRIBUTOR : F.GROEN. INSTITUTE : MATHEMATICAL CENTRE. RECEIVED : 740620. BRIEF DESCRIPTION : TRICUB COMPUTES THE DEFINITE INTEGRAL OF A FUNCTION OF TWO VARIABLES OVER A TRIANGULAR DOMAIN. KEYWORDS : INTEGRATION, QUADRATURE, MOREDIMENSIONAL QUADRATURE, CUBATURE, DEFINITE INTEGRAL. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "REAL" "PROCEDURE" TRICUB ( XI, YI, XJ, YJ, XK, YK, F, RE, AE ); "VALUE" XI, YI, XJ, YJ, XK, YK, RE, AE; "REAL" XI, YI, XJ, YJ, XK, YK, RE, AE; "REAL" "PROCEDURE" F; "CODE" 32075; TRICUB := THE COMPUTED VALUE OF THE DEFINITE INTEGRAL OF THE FUNCTION F ( X, Y ) OVER THE TRIANGULAR DOMAIN T WITH VERTICES ( XI, YI ), ( XJ, YJ ) AND ( XK, YK ). 1SECTION : 4.2.2 (DECEMBER 1979) PAGE 2 THE MEANING OF THE FORMAL PARAMETERS IS : XI, YI: ; ENTRY : THE COORDINATES OF THE FIRST VERTEX OF THE TRIANGULAR DOMAIN OF INTEGRATION; XJ, YJ: ; ENTRY : THE COORDINATES OF THE SECOND VERTEX OF THE TRIANGULAR DOMAIN OF INTEGRATION; XK, YK: ; ENTRY : THE COORDINATES OF THE THIRD VERTEX OF THE TRIANGULAR DOMAIN OF INTEGRATION; REMARK: THE ALGORITHM IS SYMMETRICAL IN THE VERTICES; THIS IMPLIES THAT THE RESULT OF THE PROCEDURE (ON ALL COUNTS) IS INVARIANT FOR ANY PERMUTATION OF THE VERTICES. F : ; THE HEADING OF THIS PROCEDURE READS: "REAL" "PROCEDURE" F ( X, Y ); "REAL" X, Y; THIS PROCEDURE DEFINES THE INTEGRAND; AE, RE: ; ENTRY: THE REQUIRED ABSOLUTE AND RELATIVE ERROR RESPECTIVELY. ONE SHOULD TAKE FOR "AE" AND "RE" VALUES WHICH ARE GREATER THAN THE ABSOLUTE AND RELATIVE ERROR IN THE COMPUTATION OF THE INTEGRAND F. PROCEDURES USED : NONE. REQUIRED CENTRAL MEMORY : THE PROCESS IS PROGRAMMED RECURSIVELY. AT EACH RECURSION LEVEL 43 REAL NUMBERS ARE USED. HOWEVER, FOR ANY PROPERLY CHOSEN VALUES OF RE AND AE THE RECURSION DEPTH WILL NOT EXCEED THE NUMBER OF BITS IN A REAL'S MANTISSA. RUNNING TIME : DEPENDS STRONGLY ON THE INTEGRAL TO COMPUTE. METHOD AND PERFORMANCE : A NESTED SEQUENCE OF CUBATURE RULES OF ORDER 2, 3, 4 AND 5 IS APPLIED. IF THE DIFFERENCE BETWEEN THE RESULT WITH THE 4-TH DEGREE RULE AND THE RESULT WITH THE 5-TH DEGREE RULE IS TOO LARGE, THEN THE TRIANGLE IS DIVIDED INTO FOUR CONGRUENT TRIANGLES. THIS PROCESS IS APPLIED RECURSIVELY IN ORDER TO OBTAIN AN ADAPTIVE CUBATURE ALGORITHM. 1SECTION : 4.2.2 (OCTOBER 1975) PAGE 3 REFERENCES : [1].P.W.HEMKER, A SEQUENCE OF NESTED CUBATURE RULES, MATH.CENTRE, AMSTERDAM, REPORT NW 3/73. EXAMPLE OF USE: THE FOLLOWING PROGRAM EVALUATES THE INTEGRAL OF F ( X, Y ) = COS ( X ) * COS ( Y ) OVER THE TRIANGLE T WITH VERTICES ( 0, 0 ), ( O, PI/2 ) AND ( PI/2, PI/2 ). ON EACH LINE ARE LISTED : A: THE REQUIRED RELATIVE AND ABSOLUTE PRECISION; B: THE COMPUTED VALUE OF THE INTEGRAL; C: THE NUMBER OF CALLS OF THE FUNCTION F, "BEGIN" "INTEGER" N,C,I,K; "REAL" PI,ACC,R,S; "REAL" "PROCEDURE" E(X,Y); "REAL" X,Y; "BEGIN" C:= C+1; "IF" C> 20000 "THEN" "GOTO" CC; E:= COS(X) * COS(Y); "END" E; PI:= 3.14159265359; "FOR" ACC:= "-1,"-2,"-3,"-4,"-5,"-6,"-7,"-8,"-9,"-10,"-11 "DO" "BEGIN" C:=0; OUTPUT(61,"("+.D"+ZD,2B,+.14D"+2ZD,2B,10ZD,/")", ACC, TRICUB(0,0,0,PI/2,PI/2,PI/2,E,ACC,ACC) ,C); "END"; CC: OUTPUT(61,"("*")"); "END" RESULTS: +.1" +0 +.50063973801970" +0 7 +.1" -1 +.50063973801970" +0 7 +.1" -2 +.50063973801970" +0 7 +.1" -3 +.49999110261504" +0 10 +.1" -4 +.49999848959226" +0 13 +.1" -5 +.49999848959226" +0 13 +.1" -6 +.49999997378240" +0 43 +.1" -7 +.49999999209792" +0 133 +.1" -8 +.49999999893172" +0 313 +.1" -9 +.49999999985571" +0 733 +.1"-10 +.49999999998692" +0 1723 1SECTION : 4.2.2 (DECEMBER 1979) PAGE 4 SOURCE TEXT(S): 0"CODE" 32075; "REAL" "PROCEDURE" TRICUB(XI,YI,XJ,YJ,XK,YK,G,RE,AE); "VALUE" XI,YI,XJ,YJ,XK,YK,RE,AE; "REAL" XI,YI,XJ,YJ,XK,YK,RE,AE; "REAL" "PROCEDURE" G; "BEGIN" "REAL" SURF,SURFMIN,XZ,YZ,GI,GJ,GK; "REAL" "PROCEDURE" INT(AX1,AY1,AF1,AX2,AY2,AF2,AX3,AY3,AF3, BX1,BY1,BF1,BX2,BY2,BF2,BX3,BY3,BF3, PX,PY,PF); "VALUE" BX1,BY1,BF1,BX2,BY2,BF2,BX3,BY3,BF3,PX,PY,PF; "REAL" BX1,BY1,BF1,BX2,BY2,BF2,BX3,BY3,BF3,PX,PY,PF, AX1,AY1,AF1,AX2,AY2,AF2,AX3,AY3,AF3; "BEGIN" "REAL" E,I3,I4,I5,A,B,C,SX1,SY1,SX2,SY2,SX3,SY3, CX1,CY1,CF1,CX2,CY2,CF2,CX3,CY3,CF3, DX1,DY1,DF1,DX2,DY2,DF2,DX3,DY3,DF3; A:= AF1 + AF2 + AF3; B:= BF1 + BF2 + BF3; I3:= 3 * A + 27 * PF + 8 * B; E:= ABS(I3) * RE + AE; "IF" SURF < SURFMIN "OR" ABS(5 * A + 45 * PF - I3) < E "THEN" INT:= I3 * SURF "ELSE" "BEGIN" CX1:= AX1 + PX; CY1:= AY1 + PY; CF1:= G(CX1,CY1); CX2:= AX2 + PX; CY2:= AY2 + PY; CF2:= G(CX2,CY2); CX3:= AX3 + PX; CY3:= AY3 + PY; CF3:= G(CX3,CY3); C:= CF1 + CF2 + CF3; I4:= A + 9 * PF + 4 * B + 12 * C; "IF" ABS(I3 - I4) < E "THEN" INT:= I4 * SURF "ELSE" "BEGIN" SX1:= .5 * BX1; SY1:= .5 * BY1; DX1:= AX1 + SX1; DY1:= AY1 + SY1; DF1:= G(DX1,DY1); SX2:= .5 * BX2; SY2:= .5 * BY2; DX2:= AX2 + SX2; DY2:= AY2 + SY2; DF2:= G(DX2,DY2); SX3:= .5 * BX3; SY3:= .5 * BY3; DX3:= AX3 + SX3; DY3:= AY3 + SY3; DF3:= G(DX3,DY3); I5:= (51 * A + 2187 * PF + 276 * B + 972 * C - 768 * (DF1 + DF2 + DF3))/63; "COMMENT" 1SECTION : 4.2.2 (OCTOBER 1975) PAGE 5 ; "IF" ABS(I4 - I5) < E "THEN" INT:= I5 * SURF "ELSE" "BEGIN" SURF:= .25 * SURF; INT:= INT(SX1,SY1,BF1,SX2,SY2,BF2,SX3,SY3,BF3, DX1,DY1,DF1,DX2,DY2,DF2,DX3,DY3,DF3, PX,PY,PF) + INT(AX1,AY1,AF1,SX3,SY3,BF3,SX2,SY2,BF2,DX1,DY1,DF1, AX1 + SX2,AY1 + SY2,G(AX1 + SX2,AY1 + SY2), AX1 + SX3,AY1 + SY3,G(AX1 + SX3,AY1 + SY3), .5 * CX1,.5 * CY1,CF1) + INT(AX2,AY2,AF2,SX3,SY3,BF3,SX1,SY1,BF1,DX2,DY2,DF2, AX2 + SX1,AY2 + SY1,G(AX2 + SX1,AY2 + SY1), AX2 + SX3,AY2 + SY3,G(AX2 + SX3,AY2 + SY3), .5 * CX2,.5 * CY2,CF2) + INT(AX3,AY3,AF3,SX1,SY1,BF1,SX2,SY2,BF2,DX3,DY3,DF3, AX3 + SX2,AY3 + SY2,G(AX3 + SX2,AY3 + SY2), AX3 + SX1,AY3 + SY1,G(AX3 + SX1,AY3 + SY1), .5 * CX3,.5 * CY3,CF3); SURF:= 4 * SURF "END" "END" "END" "END" INT; SURF:= 0.5 * ABS(XJ * YK - XK * YJ + XI * YJ - XJ * YI + XK * YI - XI * YK); SURFMIN:= SURF*RE; RE:= 30*RE; AE:= 30*AE/SURF; XZ:= (XI + XJ + XK)/3; YZ:= (YI + YJ + YK)/3; GI:= G(XI,YI); GJ:= G(XJ,YJ); GK:= G(XK,YK); XI:= XI*.5; YI:= YI*.5; XJ:= XJ*.5; YJ:= YJ*.5; XK:= XK*.5; YK:= YK*.5; TRICUB:= INT(XI,YI,GI,XJ,YJ,GJ,XK,YK,GK, XJ+XK,YJ+YK,G(XJ+XK,YJ+YK), XK+XI,YK+YI,G(XK+XI,YK+YI), XI+XJ,YI+YJ,G(XI+XJ,YI+YJ), .5 * XZ,.5 * YZ,G(XZ,YZ))/60 "END" TRICUB; "EOP" 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 1 AUTHOR: C.G. VAN DER LAAN. CONTRIBUTORS: C.G. VAN DER LAAN AND M. VOORINTHOLT. INSTITUTE: REKENCENTRUM RIJKSUNIVERSITEIT GRONINGEN. RECEIVED: 780601. BRIEF DESCRIPTION: THIS SECTION CONTAINS THE PROCEDURES GSSWTS, GSSWTSSYM AND RECCOF. RECCOF CALCULATES FROM A GIVEN WEIGHT FUNCTION ON [-1,1] THE RECURRENCE COEFFICIENTS OF THE CORRESPONDING ORTHOGONAL POLYNOMIALS; GSSWTS AND GSSWTSSYM CALCULATE FROM THE RECURRENCE COEFFICIENTS THE GAUSSIAN WEIGHTS OF THE CORRESPONDING WEIGHT FUNCTION. KEYWORDS: RECURRENCE COEFFICIENTS ORTHOGONAL POLYNOMIALS, GAUSSIAN WEIGHTS, GAUSSIAN QUADRATURE. SUBSECTION: RECCOF. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" RECCOF(N,M,X,WX,B,C,L,SYM); "VALUE" N,M,SYM; "INTEGER" N,M; "BOOLEAN" SYM; "REAL" X,WX; "ARRAY" B,C,L; "CODE" 31254; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: UPPER BOUND FOR THE INDICES OF THE ARRAYS B, C, L (N>=0); M: ; ENTRY: THE NUMBER OF POINTS USED IN THE GAUSS-CHEBYSHEV QUADRATURE RULE FOR CALCULATING THE APPROXIMATION OF THE INTEGRAL REPRESENTATIONS OF B[K],C[K] 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 2 SYM: ; ENTRY: "IF" SYM "THEN" WEIGHT FUNCTION ON [-1,1] IS EVEN "ELSE" WEIGHT FUNCTION ON [-1,1] IS NOT EVEN; X,WX: ; ENTRY: JENSEN VARIABLES WITH WX AN EXPRESSION OF X DENOTING THE WEIGHT FUNCTION ON [-1,1]; B,C,L: ; "ARRAY" B,C,L[0:N]; EXIT: THE APPROXIMATE RECURRENCE COEFFICIENTS FOR P[K+1](X) = (X-B[K])*P[K](X) - C[K]*P[K-1](X), K=0,1,2,...N, AND THE APPROXIMATE SQUARE LENGTHS X = +1 L[K] = INTEGRAL ( W(X) * P[K](X) ** 2 ) DX X = -1 PROCEDURES USED: ORTPOL = CP31044. RUNNING TIME: PROPORTIONAL TO M*N**2. METHOD AND PERFORMANCE: THE RECURRENCE COEFFICIENTS ARE REPRESENTED BY X = +1 B[K] = ( INTEGRAL ( W(X) * X * P[K](X) ** 2 ) DX ) / L[K], X = -1 C[K] = L[K] / L[K-1]], WHERE P[K](X) IS THE K-TH ORTHOGONAL POLYNOMIAL. THE INTEGRALS ARE APPROXIMATED BY THE M-POINTS GAUSS-CHEBYSHEV RULE AS X = +1 M INTEGRAL(F(X))DX := PI / M * SUM SIN(THETA[J])*F(COS(THETA[J])) X = -1 J=1 WITH THETA[J] = (2*J-1) * PI / (2*M) (SEE GAUTSCHI, 1968A). THE VALUE OF M IS TO BE SUPPLIED BY THE USER. REFERENCES: GAUTSCHI, W. (1968A): CONTRUCTION OF GAUSS-CHRISTOFFEL FORMULAS. MATH. COMP., 22,P.251-270. GAUTSCHI, W. (1968B): GAUSSIAN QUADRATURE FORMULAS. COMM. ACM. CALGO 331. 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 3 EXAMPLE OF USE: THE FOLLOWING PROGRAM DELIVERS AN APPROXIMATION FOR THE RECURSION COEFFICIENTS C[1] AND C[2], OF THE CHEBYSHEV POLYNOMIALS OF THE SECOND KIND; "BEGIN" "REAL" X; "ARRAY" B,C,L[0:2]; RECCOF(2,200,X,SQRT(1 - X**2),B,C,L,"TRUE"); OUTPUT(61,"("2/,2(3B,-ZD.3D)")",C[1],C[2]); "END"; RESULTS: 0.250 0.250 SUBSECTION: GSSWTS. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" GSSWTS(N,ZER,B,C,W); "VALUE" N; "INTEGER" N; "ARRAY" ZER,B,C,W; "CODE" 31253; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: THE NUMBER OF WEIGHTS TO BE COMPUTED; UPPER BOUND FOR THE ARRAYS Z AND W (N>=1); ZER: ; "ARRAY" ZER[1:N]; ENTRY: THE ZEROS OF THE N-TH DEGREE ORTHOGONAL POLYNOMIAL; B,C: ; "ARRAY" B[0:N-1], C[1:N-1]; ENTRY: THE RECURRENCE COEFFICIENTS; W: ; "ARRAY" W[1:N]; EXIT : THE GAUSSIAN WEIGHTS DIVIDED BY THE INTEGRAL OVER THE WEIGHT FUNCTION. PROCEDURES USED: ALLORTPOL = CP 31045. 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 4 METHOD AND PERFORMANCE: THE GAUSSIAN WEIGHTS OF AN N-POINTS RULE DIVIDED BY THE INTEGRAL OF THE WEIGHT FUNCTION MAY BE REPRESENTED AS W[K] = 1/(...((P[N-1](Z)**2/C[N-1]+P[N-2](Z)**2)/C[N-2]+... ...+P[1](Z)**2)/C[1]+1) , K=1,2,...N WITH Z= K-TH ZERO OF P[N](X). (GAUTSCHI, 1970). ALLZERORTPOL AND GSSWTS MAY BE USED TO GENERATE GAUSSIAN QUADRATURE RULES PROVIDED THE RECURRENCE COEFFICIENTS AND THE INTEGRAL OF THE WEIGHT FUNCTION ARE KNOWN. FOR EXAMPLE THE GAUSS-LAGUERRE QUADRATURE RULE APPLIED TO F MAY BE OBTAINED BY THE CALLS "FOR" K:=1 "STEP" 1 "UNTIL" N-1 "DO" "BEGIN" B[K]:=2*K+ALPHA+1; C[K]:=K*(K+ALPHA) "END"; B[0]:=ALPHA+1; ALLZERORTPOL(N,ZER,B,C); GSSWTS(N,ZER,B,C,W); GAUSSRULE:=0; "FOR" K:=1 "STEP" 1 "UNTIL" N "DO" GAUSSRULE:=GAUSSRULE+W[K]*F(ZER[K]); GAUSSRULE:=GAUSSRULE*GAMMA(ALPHA+1) GAUSSRULE CONTAINS THE VALUE OF THE APPOXIMATING GAUSS QUADRATURE RULE AND ZER[1:N],W[1:N] CONTAIN THE GAUSSIAN ABSCISSA AND WEIGHTS. 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 5 IN THE FOLLOWING TABEL WE SUMMARIZE CLASSICAL QUADRATURE RULES : WEIGHT : RECURRENCE COEFFICIENTS : INTEGRAL GAUSSIAN : FUNCTION :---------------:---------------: OF QUADRATURE : W(X) : B[K] : C[K] :WEIGHT FUNCTION -----------:-----------:---------------:---------------:--------------- : : : : LEGENDRE : 1 : 0 :K**2*(4*K**2-1): 2 : : : : CHEBYSHEV : 1/SQRT(1- : 0 : 1/2 , K=1 : PI (1-ST KIND): X**2) : : 1/4 , K>1 : : : : : CHEBYSHEV : SQRT(1- : 0 : 1/4 : PI/2 (2-ND KIND): X**2) : : : : : : : JACOBI : (1-X)** : -(ALPHA**2- : 4*(1+ALPHA)* : 2**(ALPHA+ : ALPHA*(1+ : BETA**2)/((2* : (1+BETA)/ : BETA+1)* : X)**BETA : K+ALPHA+BETA)*: ((ALPHA+BETA+ : GAMMA(ALPHA+ : : (2*K+ALPHA+ : 2)**2*(ALPHA+ : 1)*GAMMA(BETA+ : : BETA+2)) : BETA+3)) ,K=1 : 1)/GAMMA(ALPHA : : : : +BETA+2) : : : 4*K*(K+ALPHA)*: : : : (K+BETA)*(K+ : : : : ALPHA+BETA)/ : : : : ((2*K+ALPHA+ : : : : BETA)**2* : : : : ((2*K+ALPHA+ : : : : BETA)**2-1)) : : : : ,K>1 : : : : : : : :(ALPHA,BETA>-1): : : : : LAGUERRE : EXP(-X)* : 2*K+ALPHA+1 : K*(K+ALPHA) : GAMMA(ALPHA+1) : X**ALPHA : : : : : : : HERMITE : EXP(-X**2): 0 : K/2 : SQRT(PI) (THE INTEGRATION INTERVALS ARE: [-INFINITY,INFINITY] FOR HERMITE; [0,INFINITY] FOR LAGUERRE; [-1,1] FOR THE OTHERS.) FOR NON-CLASSICAL WEIGHT FUNCTIONS ON A FINITE INTERVAL THE RECURSION COEFFICIENTS (AND THE SQUARE LENGTHS OF THE CORRESPONDING ORTHOGONAL- POLYNOMIALS) CAN BE OBTAINED BY THE PROCEDURE RECCOF (THIS SECTION). 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 6 REFERENCES: GAUTSCHI, W. (1970): GENERATION OF GAUSSIAN QUADRATURE RULES AND ORTHOGONAL POLYNOMIALS. IN: COLLOQUIUM APPROXIMATIETHEORIE, MC SYLLABUS 14. EXAMPLE OF USE: SEE SUBSECTION GSSWTSSYM. SUBSECTION: GSSWTSSYM. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" GSSWTSSYM(N,ZER,C,W); "VALUE" N; "INTEGER" N; "ARRAY" ZER,C,W; "CODE" 31252; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; ENTRY: THE WEIGHTS OF AN N-POINTS GAUSS RULE ARE TO BE CALCULATED (BECAUSE OF SYMMETRY ONLY (N+1)//2 OF THE VALUES ARE DELIVERED); ZER: ; "ARRAY" ZER[1:N//2] ENTRY: THE NEGATIVE ZEROS OF THE N-TH DEGREE ORTHOGONAL POLYNOMIAL (ZER[I] < ZER[I+1], I=1,2,...,N//2-1); (IF N IS ODD THEN 0 IS ALSO A ZERO.) C: ; "ARRAY" C[1:N-1]; ENTRY: THE RECURRENCE COEFFICIENTS; W: ; "ARRAY" W[1:(N+1)//2]; EXIT : PART OF THE GAUSSIAN WEIGHTS DIVIDED BY THE INTEGRAL OF THE WEIGHT FUNCTION. (NOTE THAT W[N+1-K]=W[K] , K=1,2,...,(N+1)//2.) PROCEDURES USED: ALLORTPOLSYM = CP 31049. METHOD AND PERFORMANCE: SEE SUBSECTION GSSWTS; THIS PROCEDURE IS SUPPLIED FOR STORAGE ECONOMICAL REASONS. REFERENCES: SEE SUBSECTION GSSWTS. 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 7 EXAMPLE OF USE: THE FOLLOWING PROGRAM DELIVERS THE GAUSSIAN WEIGHTS FOR THE 5-POINTS GAUSS-CHEBYSHEV QUADRATURE RULE BY MEANS OF THE PROCEDURE GSSWTSSYM (C[1]=0.5; C[K]=0.25, K=2,3,...; ZER[I] = COS((2*(N-I) - 1) / (2*N) * PI), I=1,2,...,N//2. "BEGIN" "REAL" PI; "INTEGER" I; "ARRAY" ZER[1:2], W[1:3], C[1:4]; PI:=4*ARCTAN(1); C[1]:=.5; "FOR" I:=2 "STEP" 1 "UNTIL" 4 "DO" C[I]:=.25; ZER[1]:=COS(.9*PI); ZER[2]:=COS(.7*PI); GSSWTSSYM(5,ZER,C,W); OUTPUT(61,"("2/,5(3B,-ZD.3D)")",W[1]*PI,W[2]*PI,W[3]*PI, W[2]*PI,W[1]*PI); "END"; RESULTS: 0.628 0.628 0.628 0.628 0.628 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 8 SOURCE TEXT(S): 0"CODE" 31254; "PROCEDURE" RECCOF(N,M,X,WX,B,C,L,SYM); "VALUE" N,M,SYM;"INTEGER" N,M; "BOOLEAN" SYM; "REAL" X,WX;"ARRAY" B,C,L; "BEGIN" "INTEGER" I,J,UP;"REAL" R,S,PIM,H,HH,ARG,SA; PIM:=4*ARCTAN(1)/M; "IF" SYM "THEN" "BEGIN" "FOR" J:=0 "STEP" 1 "UNTIL" N"DO" "BEGIN" R:=B[J]:=0;UP:=M "DIV" 2; "FOR" I:=1 "STEP" 1 "UNTIL" UP"DO" "BEGIN" ARG:=(I-.5)*PIM;X:=COS(ARG); R:=R+SIN(ARG)*WX*ORTPOL(J,X,B,C)**2; "END";"IF" UP*2=M "THEN" L[J]:=2*R*PIM "ELSE" "BEGIN" X:=0;L[J]:=(2*R+WX*ORTPOL(J,0,B,C)**2)*PIM "END"; C[J]:="IF" J=0 "THEN" 0 "ELSE" L[J]/L[J-1]; "END" "END" "ELSE" "FOR" J:=0 "STEP" 1 "UNTIL" N "DO" "BEGIN" R:=S:=0;UP:=M"DIV" 2; "FOR" I:=1 "STEP" 1 "UNTIL" UP "DO" "BEGIN" ARG:=(I-.5)*PIM;SA:=SIN(ARG);X:=COS(ARG); H:=WX*ORTPOL(J,X,B,C)**2;X:=-X;HH:=WX*ORTPOL(J,X,B,C)**2; R:=R+(H+HH)*SA;S:=S+(HH-H)*X*SA; "END";B[J]:=S*PIM; "IF" UP*2=M "THEN" L[J]:=R*PIM"ELSE" "BEGIN" X:=0;L[J]:=(R+WX*ORTPOL(J,0,B,C)**2)*PIM "END"; C[J]:="IF" J=0 "THEN" 0 "ELSE" L[J]/L[J-1]; "END"; "END" RECCOF 1SECTION : 4.2.3.1 (NOVEMBER 1978) PAGE 9 ; "EOP" "CODE" 31253; "PROCEDURE" GSSWTS(N,ZER,B,C)RESULTS:(W); "VALUE" N; "INTEGER" N; "ARRAY" ZER,B,C,W; "BEGIN" "INTEGER" J,K; "REAL" S; "ARRAY" P[0:N-1]; "FOR" J:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" ALLORTPOL(N-1,ZER[J],B,C,P); S:=0.0; "FOR" K:=N-1 "STEP" -1 "UNTIL" 1 "DO" S:=(S+P[K]**2)/C[K]; W[J]:=1/(1+S); "END" "END" GSSWTS; "EOP" "CODE" 31252; "PROCEDURE" GSSWTSSYM(N,ZER,C)RESULTS:(W); "VALUE" N; "INTEGER" N; "ARRAY" ZER,C,W; "BEGIN" "INTEGER" LOW,UP,DUMMY; "ARRAY" P[0:N-1]; LOW:=1; UP:=N; "FOR" DUMMY:=1 "WHILE" LOW < UP "DO" "BEGIN" "INTEGER" I; "REAL" S; ALLORTPOLSYM( N-1,ZER[LOW],C,P ); S:=P[N-1]**2; "FOR" I:=N-1 "STEP" -1 "UNTIL" 1 "DO" S:=S/C[I] + (P[I-1])**2; W[LOW]:=1/S; LOW:=LOW+1; UP:=UP-1; "END"; "IF" LOW = UP "THEN" "BEGIN" "INTEGER" TWOI; "REAL" S; S:=1.0; "FOR" TWOI:=N-1 "STEP" -2 "UNTIL" 2 "DO" S:=S*C[TWOI-1]/C[TWOI]+1; W[LOW]:=1/S; "END"; "END" GSSWTSSYM; "EOP" 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 1 AUTHOR: M.BAKKER. INSTITUTE: MATHEMATICAL CENTRE, AMSTERDAM. RECEIVED: 760131. BRIEF DESCRIPTION: THIS SECTION CONTAINS THE FOLLOWING PROCEDURES: (1) GSS JAC WGHTS: GIVEN THE TWO PARAMETERS ALFA AND BETA,THIS PROCEDURE CALCULATES THE N ZEROS OF THE N-TH JACOBI POLYNOMIAL AND THE CORRESPONDING GAUSS-CHRISTOFFEL NUMBERS NEEDED FOR THE N-POINT GAUSS-JACOBI QUADRATURE OVER [-1,+1] WITH WEIGHT FUNCTION W(X) = (1-X)**ALFA*(1+X)**BETA; (2) GSS LAG WGHTS: GIVEN THE PARAMETER ALFA,THIS PROCEDURE CALCULATES THE N ZEROS OF THE N-TH LAGUERRE POLYNOMIAL AND THE GAUSS-CHRISTOFFEL NUMBERS NEEDED FOR THE N-POINT GAUSS-LAGUERRE QUADRATURE OF F(X) OVER (0, INFINITY) WITH RESPECT TO THE WEIGHT FUNCTION W(X) = X**ALFA*EXP(-X). THESE PROCEDURES CAN BE USED FOR GAUSSIAN QUADRATURE- RULES OF THE JACOBI AND LAGUERRE TYPE.LET THE WEIGHT FUNCTION W(X) AND THE INTERVAL(A,B) DETERMINE THE SYSTEM OF POLYNOMIALS ORTHOGONAL ON (A,B) WITH RESPECT TO W(X).THEN THE N-POINT GAUSSIAN QUADRATURE RULE APPROXIMATES THE INTEGRAL TO X=B INTEGRAL F(X) W(X) DX FROM X=A BY THE EXPRESSION J=N SUM W[J] F(X[J]) J=1 WHERE THE ABSCISSAS X[J] ARE THE ZEROS OF THE N-TH POLYNOMIAL AND W[J] ARE THE CORRESPONDING GAUSS-CHRISTOFFEL NUMBERS. 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 2 KEYWORDS: GAUSSIAN QUADRATURE, ZEROS OF ORTHOGONAL POLYNOMIALS, GAUSS-CHRISTOFFEL NUMBERS, GAUSSIAN WEIGHTS. LANGUAGE: ALGOL 60. REFERENCES: [1] M.ABRAMOWITZ AND I.A. STEGUN. HANDBOOK OF MATHEMATICAL FUNCTIONS, CH.22, [2] J.STOER, EINFUEHRUNG IN DIE NUMERISCHE MATHEMATIK 1, SPRINGER VERLAG, BERLIN, HEIDELBERG, GOETTINGEN. SUBSECTION: GSS JAC WGHTS. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" GSS JAC WGHTS(N, ALFA, BETA, X, W); "VALUE" N, ALFA, BETA; "INTEGER" N; "REAL" ALFA, BETA; "ARRAY" X, W; "CODE" 31425; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE UPPER BOUND OF THE ARRAYS X AND W; N>=1; ALFA, BETA: ; THE PARAMETERS OF THE WEIGHT FUNCTION FOR THE JACOBI POLYNOMIALS; ALFA, BETA > -1; X; ; "ARRAY" X[1:N]; EXIT: X[I] IS THE I-TH ZERO OF THE N-TH JACOBI POLYNOMIAL; W: ; "ARRAY" W[1:N]; EXIT: W[I] IS THE GAUSS-CHRISTOFFEL NUMBER ASSOCIATED WITH THE I-TH ZERO OF THE N-TH JACOBI POLYNOMIAL. 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 3 PROCEDURES USED: GAMMA = CP 35061; ALL JAC ZER = CP 31370. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF N REALS ARE USED. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. METHOD AND PERFORMANCE: AS IS WELL-KNOWN,THE GAUSSIAN QUADRATURE RULES ARE BASED ON THE ZEROS OF ORTHOGONAL POLYNOMIALS. PROCEDURES FOR THE COMPUTATION OF THESE ZEROS CAN BE FOUND IN SECTION 3.6.2. AFTER THE COMPUTATION OF THE ZEROS OF THE JACOBI POLYNOMIAL THE GAUSSIAN WEIGHTS ARE COMPUTED OF THE FORMULA J=N-1 W[I]=1/( SUM P(J, ALFA, BETA, X[I])**2) J=0 WHERE P(J, ALFA, BETA, X[I]) IS THE J-TH ORTHONORMAL JACOBI POLYNOMIAL;SEE FURTHER [2],CH.III. EXAMPLE OF USE: THE PROGRAM "BEGIN""COMMENT" EVALUATION OF THE INTEGRAL TO X=1 INTEGRAL (1+X)**2 * (1-X) * EXP(X) DX FROM X=-1 BY MEANS OF FIVE POINT GAUSS-JACOBI QUADRATURE. THE EXACT VALUE IS 2*EXP(1)-10/EXP(1); "REAL" ALFA, BETA, INT; "INTEGER" N; "ARRAY" X, W[1:5]; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; F:=EXP(X); ALFA:= 1; BETA:= 2; N:= 5; INT:= 0; GSS JAC WGHTS( N, ALFA, BETA, X, W); "FOR" N:= 1 "STEP" 1 "UNTIL" 5 "DO" INT:= INT + W[N] * F(X[N]); OUTPUT(61, "(" /, 4B+D. 4D"+ZD")", INT - 2 * EXP(1) + 10 / EXP(1)) "END" PRINTS THE FOLOWING RESULT: -1.5932"-10 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 4 SUBSECTION: GSS LAG WGHTS. CALLING SEQUENCE: THE DECLARATION OF THE PROCEDURE IN THE CALLING PROGRAM READS: "PROCEDURE" GSS LAG WGHTS (N, ALFA, X, W); "VALUE" N, ALFA; "INTEGER" N; "REAL" ALFA; "ARRAY" X, W; "CODE" 31427; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE UPPER BOUND OF THE ARRAYS X AND W; N>=1; ALFA: ; THE PARAMETER OF THE WEIGHT FUNCTION FOR THE LAGUERRE POLYNOMIALS; ALFA>-1; X: ; "ARRAY" X[1: N]; EXIT: X[I] IS THE I-TH ZERO OF THE N-TH LAGUERRE POLYNOMIAL; W: ; "ARRAY" W[1: N]; EXIT: W[I] IS THE GAUSSIAN WEIGHT CORRESPONDING WITH THE I-TH ZERO OF THE N-TH LAGUERRE POLYNOMIAL. PROCEDURES USED: GAMMA = CP 35061, ALL LAG ZER = CP 31371. REQUIRED CENTRAL MEMORY: TWO AUXILIARY ARRAYS OF N REALS ARE USED. RUNNING TIME: ROUGHLY PROPORTIONAL TO N CUBED. METHOD AND PERFORMANCE: THE ZEROS AND WEIGHTS ARE COMPUTED IN THE SAME WAY AS IN THE PROCEDURE GSS JAC WGHTS. 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 5 EXAMPLE OF USE: THE PROGRAM "BEGIN" "COMMENT" COMPUTATION OF THE INTEGRAL FROM 0 TO INFINITY OF SIN(X)*EXP(-X) BY MEANS OF A TEN POINT GAUSS-LAGUERRE QUADRATURE.THE EXACT VALUE IS 0.5; "REAL" INT;"INTEGER" N;"ARRAY" X, W[1:10]; "REAL""PROCEDURE" F(X); "VALUE"X; "REAL"X; F:=SIN(X); GSS LAG WGHTS(10, 0, X, W); INT:=0; "FOR" N:=10 "STEP" -1 "UNTIL" 1 "DO" INT:= INT + W[N] * F(X[N]); OUTPUT(61, "("-D.4D"-ZD")", INT-0.5) "END" PRINTS THE RESULT: 2.0497" -7 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 6 SOURCE TEXTS: 0"CODE" 31425; "PROCEDURE" GSS JAC WGHTS(N, ALFA, BETA, X, W); "VALUE" N, ALFA, BETA; "INTEGER" N; "REAL" ALFA, BETA; "ARRAY" X, W; "IF" ALFA = BETA "THEN" "BEGIN" "INTEGER" I, J, M; "ARRAY" B[1:N - 1]; "REAL" R0, R1, R2, S, H0, ALFA2, XI; ALL JAC ZER(N, ALFA, ALFA, X); ALFA2:= 2*ALFA; H0:= 2**(ALFA2 + 1)*GAMMA(1 + ALFA)**2/GAMMA(ALFA2 + 2); B[1]:= 1/SQRT(3 + ALFA2); M:= N - (N//2); "FOR" I:= 2 "STEP" 1 "UNTIL" N - 1 "DO" B[I]:= SQRT(I*(I + ALFA2)/(4*(I + ALFA)**2 - 1)); "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" XI:= ABS(X[I]); R0:= 1; R1:= XI/B[1]; S:= 1 + R1*R1; "FOR" J:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" R2:= (XI*R1 - B[J - 1]*R0)/B[J]; R0:= R1; R1:= R2; S:= S + R2*R2 "END"; W[I]:= W[N + 1 - I]:= H0/S "END" "END" "ELSE" "BEGIN" "INTEGER" I, J; "ARRAY" A, B[0:N]; "REAL" MIN, SUM, H0, R0, R1, R2, XI, ALFABETA; ALFABETA:= ALFA + BETA; MIN:= (BETA - ALFA)*ALFABETA; B[0]:= 0; SUM:= ALFABETA + 2; A[0]:= (BETA - ALFA)/SUM; A[1]:= MIN /SUM/(SUM + 2); B[1]:= 2*SQRT((1 + ALFA)*(1 + BETA)/(SUM + 1))/SUM; "FOR" I:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" SUM:= I + I + ALFABETA; A[I]:= MIN/SUM/(SUM + 2); B[I]:= (2/SUM)* SQRT(I*(SUM - I)*(I + ALFA)*(I + BETA)/(SUM*SUM - 1)) "END"; H0:= 2**(ALFABETA + 1)*GAMMA(1 + ALFA)*GAMMA(1 + BETA)/ GAMMA(2 + ALFABETA); ALL JAC ZER(N, ALFA, BETA, X); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" XI:= X[I]; R0:= 1; R1:= (XI - A[0])/B[1]; SUM:= 1 + R1*R1; "FOR" J:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" R2:= ((XI - A[J - 1])*R1 - B[J - 1]*R0)/B[J]; SUM:= SUM + R2*R2; R0:= R1; R1:= R2 "END"; W[I]:= H0/SUM "END" "END" GSS JAC WGHTS 1SECTION : 4.2.3.2 (NOVEMBER 1976) PAGE 7 ; "EOP" "CODE" 31427; "PROCEDURE" GSS LAG WGHTS(N, ALFA, X, W); "VALUE" N, ALFA; "INTEGER" N; "REAL" ALFA; "ARRAY" X, W; "BEGIN" "INTEGER" I, J; "REAL" H0, S, R0, R1, R2, XI; "ARRAY" A, B[0:N]; A[0]:= 1 + ALFA; A[1]:= 3 + ALFA; B[1]:= SQRT(A[0]); "FOR" I:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" A[I]:= I + I + ALFA + 1; B[I]:= SQRT(I*(I + ALFA)) "END"; ALL LAG ZER(N, ALFA, X); H0:= GAMMA(1 + ALFA); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" XI:= X[I]; R0:= 1; R1:= (XI - A[0])/B[1]; S:= 1 + R1*R1; "FOR" J:= 2 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" R2:= ((XI - A[J - 1])*R1 - B[J - 1]*R0)/B[J]; R0:= R1; R1:= R2; S:= S + R2*R2 "END"; W[I]:= H0/S "END" "END" GSS LAG WGHTS; "EOP" 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 1 AUTHOR: J.C.P.BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 740218. BRIEF DESCRIPTION: THIS SECTION CONTAINS PROCEDURES FOR CALCULATING THE DERIVATIVES OF FUNCTIONS OF MORE VARIABLES, USING DIFFERENCE FORMULAS; JACOBNNF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF N VARIABLES USING FORWARD DIFFERENCES; JACOBNMF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF M VARIABLES USING FORWARD DIFFERENCES; JACOBNBNDF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF N VARIABLES, IF THIS JACOBIAN IS KNOWN TO BE A BAND MATRIX AND HAVE TO BE STORED ROW-WISE IN A ONE-DIMENSIONAL ARRAY. KEYWORDS: NUMERICAL DIFFERENTIATION, FUNCTIONS OF MORE VARIABLES, DIFFERENCE FORMULAS. 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 2 SUBSECTION: JACOBNNF. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" JACOBNNF(N, X, F, JAC, I, DI, FUNCT); "VALUE" N; "INTEGER" I, N; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "CODE" 34437; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF INDEPENDENT VARIABLES AND THE DIMENSION OF THE FUNCTION; X: ; "ARRAY" X[1:N]; ENTRY: THE POINT AT WHICH THE JACOBIAN HAS TO BE CALCULATED F: ; "ARRAY" F[1:N]; ENTRY : THE VALUES OF THE FUNCTION-COMPONENTS AT THE POINT GIVEN IN ARRAY X; JAC: ; "ARRAY" JAC[1:N, 1:N]; EXIT : THE JACOBIAN MATRIX IN SUCH A WAY THAT THE PARTIAL DERIVATIVE OF F[I] TO X[J] IS GIVEN IN JAC[I, J], I, J = 1, ..., N; I: ; A JENSEN PARAMETER; DI MAY BE DEPENDENT OF I; DI: ; THE PARTIAL DERIVATIVES TO X[I] ARE APPROXIMATED WITH FORWARD DIFFERENCES, USING AN INCREMENT TO THE I-TH VARIABLE THAT EQUALS THE VALUE OF DI, I = 1, ..., N; FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD READ: "PROCEDURE" FUNCT(N, X, F); "VALUE" N; "INTEGER" N; "ARRAY" X, F; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF INDEPENDENT VARIABLES OF THE FUNCTION F; X: ; THE INDEPENDENT VARIABLES ARE GIVEN IN X[1:N]; F: ; AFTER A CALL OF FUNCT THE FUNCTION COMPONENTS SHOULD BE GIVEN IN F[1:N]. 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 3 PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY : EXECUTION FIELD LENGTH: JACOBNNF DECLARES ONE AUXILIARY ARRAY OF ORDER N. RUNNING TIME: PROPORTIONAL TO N ** 2. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: JACOBNNF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF N VARIABLES; THE ELEMENTS OF THIS MATRIX, WHICH ARE THE PARTIAL DERIVATIVES OF THE FUNCTION, ARE CALCULATED USING FORWARD DIFFERENCES WITH AN INCREMENT TO THE I-TH VARIABLE OF DI, (I = 1, ..., N). EXAMPLE OF USE: LET F BE DEFINED BY: F[1]= X[1] ** 3 + X[2], F[2]= 10 * X[2]; THE JACOBIAN MATRIX AT THE POINT (2, 1) MAY BE CALCULATED AND PRINTED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I; "ARRAY" JAC[1:2, 1:2], X, F[1:2]; "PROCEDURE" F1(N, X, F); "VALUE" N; "INTEGER" N; "ARRAY" X, F; "BEGIN" F[1]:= X[1] ** 3 + X[2]; F[2]:= X[2] * 10 "END" F1; X[1]:= 2; X[2]:= 1; F1(2, X, F); JACOBNNF(2, X, F, JAC, I, "IF" I = 1 "THEN" "-6 "ELSE" 1, F1); OUTPUT(71, "("/,4B,"("THE CALCULATED JACOBIAN IS:")",//, 2(4B,2(N),/)")", JAC[1, 1], JAC[1, 2], JAC[2, 1], JAC[2, 2]) "END" RESULTS: THE CALCULATED JACOBIAN IS: +1.2000005938262"+001 +1.0000000000000"+000 +0.0000000000000"+000 +1.0000000000000"+001 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 4 SUBSECTION: JACOBNMF. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" JACOBNMF(N, M, X, F, JAC, I, DI, FUNCT); "VALUE" N, M; "INTEGER" I, N, M; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "CODE" 34438; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF FUNCTION COMPONENTS; M: ; THE NUMBER OF INDEPENDENT VARIABLES; X: ; "ARRAY" X[1:M]; ENTRY: THE POINT AT WHICH THE JACOBIAN HAS TO BE CALCULATED F: ; "ARRAY" F[1:N]; ENTRY: THE VALUES OF THE FUNCTION-COMPONENTS AT THE POINT GIVEN IN ARRAY X; JAC: ; "ARRAY" JAC[1:N, 1:M]; EXIT : THE JACOBIAN MATRIX IN SUCH A WAY THAT THE PARTIAL DERIVATIVE OF F[I] TO X[J] IS GIVEN IN JAC[I, J], I = 1, ..., N, J = 1, ... M; I: ; A JENSEN PARAMETER; DI MAY BE DEPENDENT OF I; DI: ; THE PARTIAL DERIVATIVES TO X[I] ARE APPROXIMATED WITH FORWARD DIFFERENCES, USING AN INCREMENT TO THE I-TH VARIABLE THAT EQUALS THE VALUE OF DI, I = 1, ..., M; FUNCT: ; THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" FUNCT(N, M, X, F); "VALUE" N, M; "INTEGER" N, M; "ARRAY" X, F; THE MEANING OF THE FORMAL PARAMETERS IS : N: ; THE NUMBER OF FUNCTION COMPONENTS; M: ; THE NUMBER OF INDEPENDENT VARIABLES OF THE FUNCTION F; X: ; THE INDEPENDENT VARIABLES ARE GIVEN IN X[1:M]; F: ; AFTER A CALL OF FUNCT THE FUNCTION COMPONENTS SHOULD BE GIVEN IN F[1:N]. 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 5 PROCEDURES USED: NONE. EXECUTION FIELD LENGTH: JACOBNMF DECLARES ONE AUXILIARY ARRAY OF ORDER N. RUNNING TIME: PROPORTIONAL TO N * M. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: JACOBNMF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF M VARIABLES; THE ELEMENTS OF THIS MATRIX, WHICH ARE THE PARTIAL DERIVATIVES OF THE FUNCTION, ARE CALCULATED USING FORWARD DIFFERENCES WITH AN INCREMENT TO THE I-TH VARIABLE OF DI,(I = 1, ..., M). EXAMPLE OF USE: LET F BE DEFINED BY: F[1]= X[1] ** 3 + X[2], F[2]= 10 * X[2] + X[2] * X[1], F[3]= X[1] * X[2]; THE JACOBIAN MATRIX AT THE POINT (2, 1) MAY BE CALCULATED AND PRINTED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I; "ARRAY" JAC[1:3, 1:2], X[1:2], F[1:3]; "PROCEDURE" F1(N, M, X, F); "VALUE" N, M; "INTEGER" N, M; "ARRAY" X, F; "BEGIN" F[1]:= X[1] ** 3 + X[2]; F[2]:= X[2] * 10 + X[2] * X[1] ** 2; F[3]:= X[1] * X[2] "END" F1; X[1]:= 2; X[2]:= 1; F1(3, 2, X, F); JACOBNMF(3, 2, X, F, JAC, I, "IF" I=2 "THEN" 1 "ELSE" "-5, F1); OUTPUT(71, "("/,4B,"("THE CALCULATED JACOBIAN IS:")",//, 3(4B,2(N),/)")", JAC[1, 1], JAC[1, 2], JAC[2, 1], JAC[2, 2], JAC[3, 1], JAC[3, 2]) "END" RESULTS: THE CALCULATED JACOBIAN IS: +1.2000060002038"+001 +1.0000000000000"+000 +4.0000100000270"+000 +1.4000000000000"+001 +1.0000000003174"+000 +2.0000000000000"+000 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 6 SUBSECTION: JACOBNBNDF. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" JACOBNBNDF(N, LW, RW, X, F, JAC, I, DI, FUNCT); "VALUE" N, LW, RW; "INTEGER" N, I, LW, RW; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "CODE" 34439; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF INDEPENDENT VARIABLES AND THE DIMENSION OF THE FUNCTION; LW: ; THE NUMBER OF CODIAGONALS TO THE LEFT OF THE MAIN DIAGONAL OF THE JACOBIAN MATRIX, WHICH IS KNOWN TO BE A BAND MATRIX; RW: ; THE NUMBER OF CODIAGONALS TO THE RIGHT OF THE MAIN DIAGONAL OF THE JACOBIAN MATRIX; X: ; "ARRAY" X[1:N]; ENTRY: THE POINT AT WHICH THE JACOBIAN HAS TO BE CALCULATED F: ; "ARRAY" F[1:N]; ENTRY: THE VALUES OF THE FUNCTION-COMPONENTS AT THE POINT GIVEN IN ARRAY X; JAC: ; "ARRAY" JAC [1 : (LW + RW ) * (N - 1) + N]; EXIT: THE JACOBIAN MATRIX IN SUCH A WAY THAT THE (I, J)-TH ELEMENT OF THE JACOBIAN, I.E. THE PARTIAL DERIVATIVE OF F[I] TO X[J], IS GIVEN IN JAC[(LW + RW) * (I - 1) + J], FOR I = 1, ..., N J= MAX(1, I - LW), ..., MIN(N, I + RW); I: ; A JENSEN PARAMETER; DI MAY BE DEPENDENT OF I; DI: ; THE PARTIAL DERIVATIVES TO X[I] ARE APPROXIMATED WITH FORWARD DIFFERENCES, USING AN INCREMENT TO THE I-TH VARIABLE THAT EQUALS THE VALUE OF DI, I = 1, ..., N; 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 7 FUNCT: ; THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" FUNCT(N, L, U, X, F); "VALUE" N, L, U; "INTEGER" N, L, U; "ARRAY" X, F; THE MEANING OF THE FORMAL PARAMETERS IS : N: ; THE NUMBER OF FUNCTION COMPONENTS; L,U:; LOWER AND UPPER BOUND OF THE FUNCTION COMPONENT SUBSCRIPT; X: ; THE INDEPENDENT VARIABLES ARE GIVEN IN X[1:N]; F: ; AFTER A CALL OF FUNCT THE FUNCTION COMPONENTS F[I], I = L, ..., U, SHOULD BE GIVEN IN F[L:U]. PROCEDURES USED: NONE. EXECUTION FIELD LENGTH: JACOBNMF DECLARES ONE AUXILIARY ARRAY OF MAXIMUM ORDER LW + RW + 1; RUNNING TIME: PROPORTIONAL TO N * (LW + RW + 1). LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: JACOBNBNDF CALCULATES THE JACOBIAN MATRIX OF AN N-DIMENSIONAL FUNCTION OF N VARIABLES, IF THIS JACOBIAN IS KNOWN TO BE A BAND MATRIX AND HAVE TO BE STORED ROW-WISE IN A ONE-DIMENSIONAL ARRAY; THE ELEMENTS OF THIS JACOBIAN MATRIX ARE CALCULATED USING FORWARD DIFFERENCES, WITH AN INCREMENT TO THE I-TH VARIABLE OF DI, (I = 1, ..., N). 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 8 EXAMPLE OF USE: LET F BE DEFINED BY: F[1] = (3 - 2 * X[1]) * X[1] + 1 - 2 * X[2], F[I] = (3 - 2 * X[I]) * X[I] + 1 - X[I-1] - 2 * X[I+1], I= 2, 3, 4, F[5] = 4 - 2 * X[5] - X[4]; THE TRIDIAGONAL JACOBIAN MATRIX AT THE POINT X, GIVEN BY X[I] = -1, I = 1, ..., 5, MAY BE CALCULATED AND PRINTED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I; "ARRAY" JAC[1:13], X, F[1:5]; "PROCEDURE" F1(N, L, U, X, F); "VALUE" N, L, U; "INTEGER" N, L, U; "ARRAY" X, F; "BEGIN" "INTEGER" I; "FOR" I:= L "STEP" 1 "UNTIL" ("IF" U = 5 "THEN" 4 "ELSE" U) "DO" "BEGIN" F[I]:= (3 - 2 * X[I]) * X[I] + 1 - 2 * X[I + 1]; "IF" I ^= 1 "THEN" F[I]:= F[I] - X[I - 1] "END"; "IF" U = 5 "THEN" F[5]:= 4 - X[4] - X[5] * 2 "END" F1; "PROCEDURE" LIST(ITEM); "PROCEDURE" ITEM; "BEGIN" "INTEGER" I; ITEM("("THE CALCULATED TRIDIAGONAL JACOBIAN IS:")"); "FOR" I:= 1 "STEP" 1 "UNTIL" 13 "DO" ITEM(JAC[I]) "END" LIST; "PROCEDURE" LAYOUT; FORMAT("("//,4B,40S,//,4B,2(+.5D"+D2B),/,4B,3(+.5D"+D2B),/, 16B,3(+.5D"+D2B),/,28B,3(+.5D"+D2B),/,40B,2(+.5D"+D2B),/")"); "FOR" I := 1 "STEP" 1 "UNTIL" 5 "DO" X[I]:= -1; F1(5, 1, 5, X, F); JACOBNBNDF(5, 1, 1, X, F, JAC, I, "IF" I = 5 "THEN" 1 "ELSE" "-6, F1); OUTLIST(71, LAYOUT, LIST) "END" RESULTS: THE CALCULATED TRIDIAGONAL JACOBIAN IS: +.70000"+1 -.20000"+1 -.10000"+1 +.70000"+1 -.20000"+1 -.10000"+1 +.70000"+1 -.20000"+1 -.10000"+1 +.70000"+1 -.20000"+1 -.10000"+1 -.20000"+1 1SECTION : 4.3.2.1 (OCTOBER 1974) PAGE 9 SOURCE TEXT(S): "CODE" 34437; "PROCEDURE" JACOBNNF(N, X, F, JAC, I, DI, FUNCT); "VALUE" N; "INTEGER" N, I; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "BEGIN" "INTEGER" J; "REAL" STEP, AID; "ARRAY" F1[1:N]; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" STEP:= DI; AID:= X[I]; X[I]:= AID + STEP; STEP:= 1 / STEP; FUNCT(N, X, F1); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" JAC[J,I]:= (F1[J] - F[J]) * STEP; X[I]:= AID "END" "END" JACOBNNF; "EOP" "CODE" 34438; "PROCEDURE" JACOBNMF(N, M, X, F, JAC, I, DI, FUNCT); "VALUE" N, M; "INTEGER" N, M, I; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "BEGIN" "INTEGER" J; "REAL" STEP, AID; "ARRAY" F1[1:N]; "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" STEP:= DI; AID:= X[I]; X[I]:= AID + STEP; STEP:= 1 / STEP; FUNCT(N, M, X, F1); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" JAC[J,I]:= (F1[J] - F[J]) * STEP; X[I]:= AID "END" "END" JACOBNMF; "EOP" "CODE" 34439; "PROCEDURE" JACOBNBNDF(N, LW, RW, X, F, JAC, I, DI, FUNCT); "VALUE" N, LW, RW; "INTEGER" I, N, LW, RW; "REAL" DI; "ARRAY" X, F, JAC; "PROCEDURE" FUNCT; "BEGIN" "INTEGER" J, K, L, U, T, B; "REAL" AID, STEPI; L:= 1; U:= LW + 1; T:= RW + 1; B:= LW + RW; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "ARRAY" F1[L:U]; STEPI:= DI; AID:= X[I]; X[I]:= AID + STEPI; FUNCT(N, L, U, X, F1); X[I]:= AID; K:= I + ("IF" I <= T "THEN" 0 "ELSE" I - T) * B; "FOR" J:= L "STEP" 1 "UNTIL" U "DO" "BEGIN" JAC[K]:= (F1[J] - F[J]) / STEPI; K:=K + B "END"; "IF" I >= T "THEN" L:= L + 1; "IF" U < N "THEN" U:= U + 1 "END" "END" JACOBNBNDF; "EOP" 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 1 AUTHORS: J. C. P. BUS AND T. J. DEKKER. CONTRIBUTOR: J. C. P. BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 750615. BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO PROCEDURES FOR FINDING A ZERO OF A GIVEN FUNCTION IN A GIVEN INTERVAL; ZEROIN APPROXIMATES A ZERO MAINLY BY LINEAR INTERPOLATION AND EXTRAPOLATION; ZEROINRAT APPROXIMATES A ZERO BY INTERPOLATION WITH RATIONAL FUNCTIONS. ZEROIN IS PREFERABLE FOR SIMPLE (I.E. CHEAPLY TO CALCULATE) FUNCTIONS AND/OR WHEN NO HIGH PRECISION IS REQUIRED. ZEROINRAT IS PREFERABLE FOR COMPLICATED (I.E. EXPENSIVE) FUNCTIONS WHEN A ZERO IS REQUIRED IN RATHER HIGH PRECISION AND ALSO FOR FUNCTIONS HAVING A POLE NEAR THE ZERO. WHEN THE ANALYTIC DERIVATIVE OF THE FUNCTION IS EASILY OBTAINED, THEN ZEROINDER (SECTION 5.1.1.1.2) SHOULD BE TAKEN INTO CONSIDERATION. KEYWORDS: ZERO SEARCHING, ANALYTIC EQUATIONS, SINGLE NON-LINEAR EQUATION. 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 2 SUBSECTION: ZEROIN. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS : "BOOLEAN" "PROCEDURE" ZEROIN(X, Y, FX, TOLX); "REAL" X, Y, FX, TOLX; "CODE" 34150; ZEROIN SEARCHES FOR A ZERO OF A REAL FUNCTION F DEFINED ON A CERTAIN INTERVAL J; ZEROIN := "TRUE" WHEN A (SUFFICIENTLY SMALL) SUBINTERVAL OF J CONTAINING A ZERO OF F HAS BEEN FOUND; OTHERWISE, ZEROIN := "FALSE"; LET A REAL FUNCTION T DEFINED ON J, DENOTE A TOLERANCE FUNCTION DEFINING THE REQUIRED PRECISION OF THE ZERO; (FOR INSTANCE T(X) = ABS(X) * RE + AE, WHERE RE AND AE ARE THE REQUIRED RELATIVE AND ABSOLUTE PRECISION RESPECTIVELY); THE MEANING OF THE FORMAL PARAMETERS IS: X: ; A JENSEN VARIABLE; THE ACTUAL PARAMETERS FOR FX AND TOLX (MAY) DEPEND ON THE ACTUAL PARAMETER FOR X; ENTRY: ONE ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; EXIT: A VALUE APPROXIMATING THE ZERO WITHIN THE TOLERANCE 2 * T(X) WHEN ZEROIN HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE OTHERWISE; Y: ; ENTRY: THE OTHER ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; UPON ENTRY X < Y AS WELL AS Y < X IS ALLOWED; EXIT: THE OTHER STRADDLING APPROXIMATION OF THE ZERO, I.E. UPON EXIT THE VALUES OF Y AND X SATISFY 1. F(X) * F(Y) <= 0, 2. ABS(X - Y) <= 2 * T(X) AND 3. ABS(F(X)) <= ABS(F(Y)) WHEN ZEROIN HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE SATISFYING CONDITIONS 2 AND 3 BUT NOT 1 OTHERWISE; FX: ; DEFINES FUNCTION F AS A FUNCTION DEPENDING ON THE ACTUAL PARAMETER FOR X; TOLX: ; DEFINES THE TOLERANCE FUNCTION T WHICH MAY DEPEND ON THE ACTUAL PARAMETER FOR X; ONE SHOULD CHOOSE TOLX POSITIVE AND NEVER SMALLER THAN THE PRECISION OF THE MACHINE'S ARITHMETIC AT X, I.E. IN THIS ARITHMETIC X + TOLX AND X - TOLX SHOULD ALWAYS YIELD VALUES DISTINCT FROM X; OTHERWISE THE PROCEDURE MAY GET INTO A LOOP. 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 3 PROCEDURES USED: DWARF = CP30003; EXECUTION FIELD LENGTH: NO AUXILIARY ARRAYS ARE DECLARED IN ZEROIN. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE METHOD USED IS DESCRIBED IN DETAIL IN [1]. THE NUMBER OF EVALUATIONS OF FX AND TOLX IS AT MOST 4 * LOG(ABS(X - Y)) / TAU, WHERE X AND Y ARE THE ARGUMENT VALUES GIVEN UPON ENTRY, LOG DENOTES THE BASE 2 LOGARITHM AND TAU IS THE MINIMUM OF THE TOLERANCE FUNCTION TOLX ON THE INITIAL INTERVAL. IF UPON ENTRY X AND Y SATISFY F(X) * F(Y) <= 0, THEN CONVERGENCE IS GUARANTEED AND THE ASYMPTOTIC ORDER OF CONVERGENCE IS 1.618 FOR SIMPLE ZEROES. EXAMPLE OF USE: THE ZERO OF THE FUNCTION EXP(-X * 3) * (X - 1) + X ** 3, IN THE INTERVAL [0, 1], MAY BE CALCULATED BY THE FOLLOWING PROGRAM: "BEGIN" "REAL" X, Y; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; F:= EXP(-X * 3) * ( X - 1) + X ** 3; X:= 0; Y:= 1; "IF" ZEROIN(X, Y, F(X), ABS(X) * "-14 + "-14) "THEN" OUTPUT(71, "("/4B,"("CALCULATED ZERO:")"B+.15D"+3D")", X) "ELSE" OUTPUT(71, "("/4B,"("NO ZERO FOUND")"")") "END" RESULT: CALCULATED ZERO: +.489702748548240"+000 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 4 SUBSECTION: ZEROINRAT. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS: "BOOLEAN" "PROCEDURE" ZEROINRAT(X, Y, FX, TOLX); "REAL" X, Y, FX, TOLX; "CODE" 34436; ZEROINRAT SEARCHES FOR A ZERO OF A REAL FUNCTION F DEFINED ON A CERTAIN INTERVAL J; ZEROINRAT := "TRUE" WHEN A (SUFFICIENTLY SMALL) SUBINTERVAL OF J CONTAINING A ZERO OF F HAS BEEN FOUND; OTHERWISE, ZEROINRAT := "FALSE"; LET A REAL FUNCTION T DEFINED ON J, DENOTE A TOLERANCE FUNCTION DEFINING THE REQUIRED PRECISION OF THE ZERO; (FOR INSTANCE T(X) = ABS(X) * RE + AE, WHERE RE AND AE ARE THE REQUIRED RELATIVE AND ABSOLUTE PRECISION RESPECTIVELY); THE MEANING OF THE FORMAL PARAMETERS IS: X: ; A JENSEN VARIABLE; THE ACTUAL PARAMETERS FOR FX AND TOLX (MAY) DEPEND ON THE ACTUAL PARAMETER FOR X; ENTRY: ONE ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; EXIT: A VALUE APPROXIMATING THE ZERO WITHIN THE TOLERANCE 2 * T(X) WHEN ZEROINRAT HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE OTHERWISE; Y: ; ENTRY: THE OTHER ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; UPON ENTRY X < Y AS WELL AS Y < X IS ALLOWED; EXIT: THE OTHER STRADDLING APPROXIMATION OF THE ZERO, I.E. UPON EXIT THE VALUES OF Y AND X SATISFY 1. F(X) * F(Y) <= 0, 2. ABS(X - Y) <= 2 * T(X) AND 3. ABS(F(X)) <= ABS(F(Y)) WHEN ZEROINRAT HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE SATISFYING CONDITIONS 2 AND 3 BUT NOT 1 OTHERWISE; FX: ; DEFINES FUNCTION F AS A FUNCTION DEPENDING ON THE ACTUAL PARAMETER FOR X; TOLX: ; DEFINES THE TOLERANCE FUNCTION T WHICH MAY DEPEND ON THE ACTUAL PARAMETER FOR X; ONE SHOULD CHOOSE TOLX POSITIVE AND NEVER SMALLER THAN THE PRECISION OF THE MACHINE'S ARITHMETIC AT X, I.E. IN THIS ARITHMETIC X + TOLX AND X - TOLX SHOULD ALWAYS YIELD VALUES DISTINCT FROM X; OTHERWISE THE PROCEDURE MAY GET INTO A LOOP. 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 5 PROCEDURES USED: DWARF = CP30003; EXECUTION FIELD LENGTH: NO AUXILIARY ARRAYS ARE DECLARED IN ZEROINRAT. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE METHOD USED IS DESCRIBED IN DETAIL IN [1]. THE NUMBER OF EVALUATIONS OF FX AND TOLX IS AT MOST 5 * LOG(ABS(X - Y)) / TAU, WHERE X AND Y ARE THE ARGUMENT VALUES GIVEN UPON ENTRY, LOG DENOTES THE BASE 2 LOGARITHM AND TAU IS THE MINIMUM OF THE TOLERANCE FUNCTION TOLX ON THE INITIAL INTERVAL. IF UPON ENTRY X AND Y SATISFY F(X) * F(Y) <= 0, THEN CONVERGENCE IS GUARANTEED AND THE ASYMPTOTIC ORDER OF CONVERGENCE IS 1.839 FOR SIMPLE ZEROES. EXAMPLE OF USE: THE ZERO OF THE FUNCTION EXP(-X * 3) * (X - 1) + X ** 3, IN THE INTERVAL [0, 1], MAY BE CALCULATED BY THE FOLLOWING PROGRAM: "BEGIN" "REAL" X, Y; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; F:= EXP(-X * 3) * ( X - 1) + X ** 3; X:= 0; Y:= 1; "IF" ZEROINRAT(X, Y, F(X), ABS(X) * "-14 + "-14) "THEN" OUTPUT(71, "("/4B,"("CALCULATED ZERO:")"B+.15D"+3D")", X) "ELSE" OUTPUT(71, "("/4B,"("NO ZERO FOUND")"")") "END" RESULT: CALCULATED ZERO: +.489702748548240"+000 REFERENCES: [1]: BUS, J.C.P. AND DEKKER, T.J., TWO EFFICIENT ALGORITHMS WITH GUARANTEED CONVERGENCE FOR FINDING A ZERO OF A FUNCTION. MATHEMATICAL CENTRE, REPORT NW 13/74, AMSTERDAM (1974), (ALSO TO APPEAR IN TOMS 1975). 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 6 SOURCE TEXT(S): 0"CODE" 34150; "BOOLEAN" "PROCEDURE" ZEROIN(X, Y, FX, TOLX); "REAL" X, Y, FX, TOLX; "BEGIN" "INTEGER" EXT; "REAL" C, FC, B, FB, A, FA, D, FD, FDB, FDA, W, MB, TOL, M, P, Q, DW; DW:= DWARF; B:= X; FB:= FX; A:= X:= Y; FA:= FX; INTERPOLATE: C:= A; FC:= FA; EXT:= 0; EXTRAPOLATE: "IF" ABS(FC) < ABS(FB) "THEN" "BEGIN" "IF" C ^= A "THEN" "BEGIN" D:= A; FD:= FA "END"; A:= B; FA:= FB; B:= X:= C; FB:= FC; C:= A; FC:= FA "END" INTERCHANGE; TOL:= TOLX; M:= (C + B) * 0.5; MB:= M - B; "IF" ABS(MB) > TOL "THEN" "BEGIN" "IF" EXT > 2 "THEN" W:= MB "ELSE" "BEGIN" TOL:= TOL * SIGN(MB); P:= (B - A) * FB; "IF" EXT <= 1 "THEN" Q:= FA - FB "ELSE" "BEGIN" FDB:= (FD - FB) / (D - B); FDA:= (FD - FA) / (D - A); P:= FDA * P; Q:= FDB * FA - FDA * FB "END"; "IF" P < 0 "THEN" "BEGIN" P:= -P; Q:= -Q "END"; W:= "IF" P < DW "OR" P <= Q * TOL "THEN" TOL "ELSE" "IF" P < MB * Q "THEN" P / Q "ELSE" MB "END"; D:= A; FD:= FA; A:= B; FA:= FB; X:= B:= B + W; FB:= FX; "IF" ("IF" FC >= 0 "THEN" FB >= 0 "ELSE" FB <= 0) "THEN" "GOTO" INTERPOLATE "ELSE" "BEGIN" EXT:= "IF" W = MB "THEN" 0 "ELSE" EXT + 1; "GOTO" EXTRAPOLATE "END" "END"; Y:= C; ZEROIN:= "IF" FC >= 0 "THEN" FB <= 0 "ELSE" FB >= 0 "END" ZEROIN 1SECTION : 5.1.1.1.1 (OCTOBER 1975) PAGE 7 ; "EOP" "CODE" 34436; "BOOLEAN" "PROCEDURE" ZEROINRAT(X, Y, FX, TOLX); "REAL" X, Y, FX, TOLX; "BEGIN" "INTEGER" EXT; "BOOLEAN" FIRST; "REAL" B, FB, A, FA, D, FD, C, FC, FDB, FDA, W, MB, TOL, M, P, Q, DW; DW:= DWARF; B:= X; FB:= FX; A:= X:= Y; FA:= FX; FIRST:= "TRUE"; INTERPOLATE: C:= A; FC:= FA; EXT:= 0; EXTRAPOLATE: "IF" ABS(FC) < ABS(FB) "THEN" "BEGIN" "IF" C ^= A "THEN" "BEGIN" D:= A; FD:= FA "END"; A:= B; FA:= FB; B:= X:= C; FB:= FC; C:= A; FC:= FA "END" INTERCHANGE; TOL:= TOLX; M:= (C + B) * .5; MB:= M - B; "IF" ABS(MB) > TOL "THEN" "BEGIN" "IF" EXT > 3 "THEN" W:= MB "ELSE" "BEGIN" TOL:= TOL * SIGN(MB); P:= (B - A) * FB; "IF" FIRST "THEN" "BEGIN" Q:= FA - FB; FIRST:= "FALSE" "END" "ELSE" "BEGIN" FDB:= (FD - FB) / (D - B); FDA:= (FD - FA) / (D - A); P:= FDA * P; Q:= FDB * FA - FDA * FB "END"; "IF" P < 0 "THEN" "BEGIN" P:= -P; Q:= -Q "END"; "IF" EXT = 3 "THEN" P:= P * 2; W:= "IF" P < DW "OR" P <= Q * TOL "THEN" TOL "ELSE" "IF" P < MB * Q "THEN" P / Q "ELSE" MB "END"; D:= A; FD:= FA; A:= B; FA:= FB; X:= B:= B + W; FB:= FX; "IF" ("IF" FC >= 0 "THEN" FB >= 0 "ELSE" FB <= 0) "THEN" "GOTO" INTERPOLATE "ELSE" "BEGIN" EXT:= "IF" W = MB "THEN" 0 "ELSE" EXT + 1; "GOTO" EXTRAPOLATE "END" "END"; Y:= C; ZEROINRAT:= "IF" FC >= 0 "THEN" FB <= 0 "ELSE" FB >= 0 "END" ZEROINRAT; "EOP" 1SECTION : 5.1.1.1.2 (OCTOBER 1975) PAGE 1 AUTHOR: T.J. DEKKER. CONTRIBUTORS: T.J. DEKKER AND T.H.P. REYMER. INSTITUTE: UNIVERSITY OF AMSTERDAM. RECEIVED: 750615. BRIEF DESCRIPTION: THIS SECTION CONTAINS ONE PROCEDURE FOR FINDING A ZERO OF A GIVEN DIFFERENTIABLE FUNCTION IN A GIVEN INTERVAL; ZEROINDER APPROXIMATES A ZERO MAINLY BY MEANS OF CONFLUENT 3-POINT RATIONAL INTERPOLATION USING NOT ONLY VALUES OF THE GIVEN FUNCTION BUT ALSO OF ITS DERIVATIVE. ZEROINDER IS TO PREFER TO ZEROIN OR ZEROINRAT (SECTION 5.1.1.1.1), IF THE DERIVATIVE IS (MUCH) CHEAPER TO EVALUATE THAN THE FUNCTION. KEYWORDS: ZERO SEARCHING, ANALYTIC EQUATIONS, SINGLE NONLINEAR EQUATION, DERIVATIVE AVAILABLE. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS: "BOOLEAN" "PROCEDURE" ZEROINDER(X, Y, FX, DFX, TOLX); "REAL" X, Y, FX, DFX, TOLX; "CODE" 34453; ZEROINDER SEARCHES FOR A ZERO OF A DIFFERENTIABLE REAL FUNCTION F DEFINED ON A CERTAIN INTERVAL J; ZEROINDER := "TRUE" WHEN A (SUFFICIENTLY SMALL) SUBINTERVAL OF J CONTAINING A ZERO OF F HAS BEEN FOUND; OTHERWISE, ZEROINDER := "FALSE"; LET DF AND T DENOTE REAL FUNCTIONS DEFINED ON J, WHERE DF IS THE DERIVATIVE OF F AND T A TOLERANCE FUNCTION DEFINING THE REQUIRED PRECISION OF THE ZERO; ( FOR INSTANCE T(X) = ABS(X) * RE + AE, WHERE RE AND AE ARE THE REQUIRED RELATIVE AND ABSOLUTE PRECISION RESPECTIVELY); 1SECTION : 5.1.1.1.2 (OCTOBER 1975) PAGE 2 THE MEANING OF THE FORMAL PARAMETERS IS: X: ; A JENSEN VARIABLE; THE ACTUAL PARAMETERS FOR FX, DFX AND TOLX (MAY) DEPEND ON THE ACTUAL PARAMETER FOR X; ENTRY: ONE ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; EXIT: A VALUE APPROXIMATING THE ZERO WITHIN THE TOLERANCE 2 * T(X) WHEN ZEROINDER HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE OTHERWISE; Y: ; ENTRY: THE OTHER ENDPOINT OF INTERVAL J IN WHICH A ZERO IS SEARCHED FOR; UPON ENTRY X < Y AS WELL AS Y < X IS ALLOWED; EXIT: THE OTHER STRADDLING APPROXIMATION OF THE ZERO, I.E. UPON EXIT THE VALUES OF Y AND X SATISFY 1. F(X) * F(Y) <= 0, 2. ABS(X - Y) <= 2 * T(X) AND 3. ABS(F(X)) <= ABS(F(Y)) WHEN ZEROINDER HAS THE VALUE "TRUE", AND A PRESUMABLY WORTHLESS ARGUMENT VALUE SATISFYING CONDITIONS 2 AND 3 BUT NOT 1 OTHERWISE; FX: ; DEFINES FUNCTION F AS A FUNCTION DEPENDING ON THE ACTUAL PARAMETER FOR X; DFX: ; DEFINES THE DERIVATIVE DF OF F AS A FUNCTION DEPENDING ON THE ACTUAL PARAMETER FOR X; TOLX: ; DEFINES THE TOLERANCE FUNCTION T WHICH MAY DEPEND ON THE ACTUAL PARAMETER FOR X; ONE SHOULD CHOOSE TOLX POSITIVE AND NEVER SMALLER THAN THE PRECISION OF THE MACHINE'S ARITHMETIC AT X, I.E. IN THIS ARITHMETIC X + TOLX AND X - TOLX SHOULD ALWAYS YIELD VALUES DISTINCT FROM X; OTHERWISE THE PROCEDURE MAY GET INTO A LOOP. PROCEDURES USED: DWARF = CP30003; REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE DECLARED IN ZEROINDER. RUNNING TIME: SEE METHOD AND PERFORMANCE. LANGUAGE: ALGOL 60. 1SECTION : 5.1.1.1.2 (OCTOBER 1975) PAGE 3 METHOD AND PERFORMANCE: THE METHOD USED IS CONFLUENT 3-POINT RATIONAL INTERPOLATION, I.E. THE INTERPOLATION FUNCTION OF THE FORM (X - A) / (BX + C) IS FITTED EXACTLY AT 3 POINTS 2 OF WHICH ARE COINCIDING; MOREOVER, IN ORDER TO IMPROVE THE GLOBAL BEHAVIOUR OF THE PROCESS, LINEAR INTERPOLATION ON THE FUNCTION F / DF OR BISECTION ARE INCLUDED IN SOME STEPS; THE PERFORMANCE IS AS FOLLOWS: THE NUMBER OF EVALUATIONS OF FX, DFX AND TOLX IS AT MOST 4 LOG(ABS(X - Y)) / TAU, WHERE X AND Y ARE THE ARGUMENT VALUES GIVEN UPON ENTRY, LOG DENOTES THE BASE 2 LOGARITHM, AND TAU IS THE MINIMUM OF THE TOLERANCE FUNCTION ON J (I.E. ZEROINDER REQUIRES AT MOST 4 TIMES THE NUMBER OF STEPS REQUIRED FOR BISECTION); IF UPON ENTRY X AND Y SATISFY F(X) * F(Y) <= 0, THEN CONVERGENCE TO A ZERO IS GUARANTEED, SO THAT UPON EXIT X AND Y SATISFY CONDITION 1 TO 3 MENTIONED ABOVE (SEE PARAMETER Y) AND ZEROINDER HAS THE VALUE "TRUE"; FOR A SIMPLE ZERO OF F, THE ASYMPTOTIC ORDER OF CONVERGENCE APPROXIMATELY EQUALS 2.414. EXAMPLE OF USE: THE ZERO OF THE FUNCTION EXP(-X * 3) * (X - 1) + X ** 3, IN THE INTERVAL [0, 1], MAY BE CALCULATED BY THE FOLLOWING PROGRAM: "BEGIN" "REAL" X, Y; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; F:= EXP(-X * 3) * ( X - 1) + X ** 3; "REAL" "PROCEDURE" DF(X); "VALUE" X; "REAL" X; DF:= EXP(-X * 3) * (-3 * X + 4) + 3 * X ** 2; X:= 0; Y:= 1; "IF" ZEROINDER(X, Y, F(X), DF(X), ABS(X) * "-14 + "-14) "THEN" OUTPUT(71, "("/4B,"("CALCULATED ZERO AND FUNCTION VALUE:")", /8B, 2(B+.15D"+3D4B), /4B, "("OTHER STRADDLING APPROXIMATION AND FUNCTION VALUE:")", /8B, 2(B+.15D"+3D4B)")", X, F(X), Y, F(Y)) "ELSE" OUTPUT(71, "("/4B, "("NO ZERO FOUND")"")") "END" RESULT: CALCULATED ZERO AND FUNCTION VALUE: +.489702748548240"+000 -.444089209850060"-015 OTHER STRADDLING APPROXIMATION AND FUNCTION VALUE: +.489702748548250"+000 +.177635683940030"-013 1SECTION : 5.1.1.1.2 (OCTOBER 1975) PAGE 4 REFERENCES: [1]: BUS, J.C.P. AND DEKKER, T.J., TWO EFFICIENT ALGORITHMS WITH GUARANTEED CONVERGENCE FOR FINDING A ZERO OF A FUNCTION. MATHEMATICAL CENTRE, REPORT NW 13/74, AMSTERDAM (1974), (ALSO TO APPEAR IN TOMS 1975). [2]: OSTROWSKI, A.M., SOLUTION OF EQUATIONS AND SYSTEMS OF EQUATIONS. ACADEMIC PRESS 1966. CHAPTERS 3 AND 11. SOURCE TEXT(S): 0"CODE" 34453; "BOOLEAN" "PROCEDURE" ZEROINDER(X, Y, FX, DFX, TOLX); "REAL" X, Y, FX, DFX, TOLX; "BEGIN" "INTEGER" EXT; "REAL" B, FB, DFB, A, FA, DFA, C, FC, DFC, D, W, MB, TOL, M, P, Q, DW; DW:= DWARF; B:= X; FB:= FX; DFB:= DFX; A:= X:= Y; FA:= FX; DFA:= DFX; INTERPOLATE: C:= A; FC:= FA; DFC:= DFA; EXT:= 0; EXTRAPOLATE: "IF" ABS(FC) < ABS(FB) "THEN" "BEGIN" A:= B; FA:= FB; DFA:= DFB; B:= X:= C; FB:= FC; DFB:= DFC; C:= A; FC:= FA; DFC:= DFA "END" INTERCHANGE; TOL:= TOLX; M:= (C + B) * 0.5; MB:= M - B; "IF" ABS(MB) > TOL "THEN" "BEGIN" "IF" EXT > 2 "THEN" W:= MB "ELSE" "BEGIN" TOL:= TOL * SIGN(MB); D:= "IF" EXT = 2 "THEN" DFA "ELSE" (FB - FA) / (B - A); P:= FB * D * (B - A); Q:= FA * DFB - FB * D; "IF" P < 0 "THEN" "BEGIN" P:= -P; Q:= -Q "END"; W:= "IF" P < DW "OR" P <= Q * TOL "THEN" TOL "ELSE" "IF" P < MB * Q "THEN" P / Q "ELSE" MB; "END"; A:= B; FA:= FB; DFA:= DFB; X:= B:= B + W; FB:= FX; DFB:= DFX; "IF" ("IF" FC >= 0 "THEN" FB >= 0 "ELSE" FB <= 0) "THEN" "GOTO" INTERPOLATE "ELSE" "BEGIN" EXT:= "IF" W = MB "THEN" 0 "ELSE" EXT + 1; "GOTO" EXTRAPOLATE "END" "END"; Y:= C; ZEROINDER:= "IF" FC >= 0 "THEN" FB <= 0 "ELSE" FB >= 0 "END" ZEROINDER; "EOP" 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 1 AUTHOR: J.C.P.BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 740218. BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO PROCEDURES FOR SOLVING SYSTEMS OF NON-LINEAR EQUATIONS, OF WHICH THE JACOBIAN IS KNOWN TO BE A BAND MATRIX. QUANEWBND ASKS FOR AN APPROXIMATION OF THE JACOBIAN AT THE INITIAL GUESS. QUANEWBND1 CALCULATES AN APPROXIMATION OF THE JACOBIAN AT THE INITIAL GUESS, USING FORWARD DIFFERENCES. KEYWORDS: NON-LINEAR EQUATIONS, SYSTEMS OF EQUATIONS, NO EXPLICIT JACOBIAN. 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 2 SUBSECTION: QUANEWBND. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" QUANEWBND(N, LW, RW, X, F, JAC, FUNCT, IN, OUT); "VALUE" N, LW, RW; "INTEGER" N, LW, RW; "ARRAY" X, F, JAC, IN, OUT; "BOOLEAN" "PROCEDURE" FUNCT; "CODE" 34430; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF INDEPENDENT VARIABLES; THE NUMBER OF EQUATIONS SHOULD ALSO BE EQUAL TO N; LW: ; THE NUMBER OF CODIAGONALS TO THE LEFT OF THE MAIN DIAGONAL OF THE JACOBIAN; RW: ; THE NUMBER OF CODIAGONALS TO THE RIGHT OF THE MAIN DIAGONAL OF THE JACOBIAN; X: ; "ARRAY" X[1:N]; ENTRY: AN INITIAL ESTIMATE OF THE SOLUTION OF THE SYSTEM THAT HAS TO BE SOLVED; EXIT: THE CALCULATED SOLUTION OF THE SYSTEM; F: ; "ARRAY" F[1:N]; ENTRY: THE VALUES OF THE FUNCTION COMPONENTS AT THE INITIAL GUESS; EXIT: THE VALUES OF THE FUNCTION COMPONENTS AT THE CALCULATED SOLUTION; JAC: ; "ARRAY" JAC[1:(LW + RW) * (N - 1) + N]; ENTRY: AN APPROXIMATION OF THE JACOBIAN AT THE INITIAL ESTIMATE OF THE SOLUTION; EXIT: AN APPROXIMATION OF THE JACOBIAN AT THE CALCULATED SOLUTION; AN APPROXIMATION OF THE (I, J)-TH ELEMENT OF THE JACOBIAN IS GIVEN IN JAC[(LW + RW) * (I - 1) + J], FOR I = 1, ..., N AND J = MAX(1, I - LW), ..., MIN(N, I + RW); 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 3 FUNCT: ; THE HEADING OF THIS PROCEDURE READS : "BOOLEAN" "PROCEDURE" FUNCT(N, L, U, X, F); "VALUE" N, L, U; "INTEGER" N, L, U; "ARRAY" X, F; THE MEANING OF THE FORMAL PARAMETERS IS : N: ; THE NUMBER OF INDEPENDENT VARIABLES OF THE FUNCTION F; L, U: ; LOWER AND UPPER BOUND OF THE FUNCTION COMPONENT SUBSCRIPT; X: ; THE INDEPENDENT VARIABLES ARE GIVEN IN X[1:N]; F: ; AFTER A CALL OF FUNCT THE FUNCTION COMPONENTS F[I], I = L, ..., U, SHOULD BE GIVEN IN F[L:U]; IF THE VALUE OF THE PROCEDURE IDENTIFIER EQUALS FALSE, THEN THE EXECUTION OF QUANEWBND WILL BE TERMINATED, WHILE THE VALUE OF OUT[5] IS SET EQUAL TO 2; IN: ; "ARRAY" IN[0:4]; ENTRY : IN THIS AUXILIARY ARRAY ONE SHOULD GIVE THE FOLLOWING VALUES FOR CONTROLLING THE PROCESS: IN[0]: THE MACHINE PRECISION; IN[1]: THE RELATIVE PRECISION ASKED FOR; IN[2]: THE ABSOLUTE PRECISION ASKED FOR; IF THE VALUE, DELIVERED IN OUT[5] EQUALS ZERO, THEN THE LAST CORRECTION VECTOR D, SAY, WHICH IS A MEASURE FOR THE ERROR IN THE SOLUTION, SATIFIES THE INEQUALITY NORM(D) <= NORM(X) * IN[1] + IN[2], WHEREBY X DENOTES THE CALCULATED SOLUTION, GIVEN IN ARRAY X AND NORM(.) DENOTES THE EUCLIDIAN NORM; HOWEVER,WE CAN NOT GUARANTEE THAT THE TRUE ERROR IN THE SOLUTION SATISFIES THIS INEQUALITY, ESPECIALLY IF THE JACOBIAN IS (NEARLY) SINGULAR AT THE SOLUTION; IN[3]: THE MAXIMUM VALUE OF THE NORM OF THE RESIDUAL VECTOR ALLOWED; IF OUT[5] = 0, THEN THIS RESIDUAL VECTOR F, SAY, SATIFIES: NORM(F) <= IN[3]; IN[4]: THE MAXIMUM NUMBER OF FUNCTION COMPONENT EVALUATIONS ALLOWED; L - U + 1 FUNCTION COMPONENT EVALUATIONS ARE COUNTED EACH CALL OF FUNCT(N, L, U, X, F); IF OUT[5]=1, THEN THE PROCESS IS TERMINATED, BECAUSE THE NUMBER OF EVALUATIONS EXCEEDED THE VALUE GIVEN IN IN[4]; 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 4 OUT: ; "ARRAY" OUT[1:5]; EXIT : OUT[1]: THE EUCLIDIAN NORM OF THE LAST STEP ACCEPTED; OUT[2]: THE EUCLIDIAN NORM OF THE RESIDUAL VECTOR AT THE CALCULATED SOLUTION; OUT[3]: THE NUMBER OF FUNCTION COMPONENT EVALUATIONS PERFORMED; OUT[4]: THE NUMBER OF ITERATIONS CARRIED OUT; OUT[5]: THE INTEGER VALUE DELIVERED IN OUT[5] GIVES SOME INFORMATION ABOUT THE TERMINATION OF THE PROCESS; OUT[5] = 0: THE PROCESS IS TERMINATED IN A NORMAL WAY; THE LAST STEP AND THE NORM OF THE RESIDUAL VECTOR SATISFY THE CONDITIONS (SEE IN[2], IN[3]); IF OUT[5] ^= 0, THEN THE PROCESS IS TERMINATED PREMATURALY; OUT[5] = 1: THE NUMBER OF FUNCTION COMPONENT EVALUATIONS EXCEEDS THE VALUE GIVEN IN IN[4]; OUT[5] = 2: A CALL OF FUNCT DELIVERED THE VALUE FALSE; OUT[5] = 3: THE APPROXIMATION OF THE JACOBIAN MATRIX TURNS OUT TO BE SINGULAR. PROCEDURES USED: MULVEC = CP31020, DUPVEC = CP31030, VECVEC = CP34010, ELMVEC = CP34020, DECSOLBND = CP34322. EXECUTION FIELD LENGTH: THE MAXIMUM NUMBER OF WORDS, NECESSARY FOR THE ARRAYS DECLARED IN QUANEWBND EQUALS MAX(N * 3 + (N - 1) * (LW + RW), 4N). RUNNING TIME: PROPORTIONAL TO N * LW * ( LW + RW + 1). LANGUAGE: ALGOL 60. 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 5 METHOD AND PERFORMANCE: THE METHOD USED IN QUANEWBND IS THE SAME AS GIVEN IN [1]; THE SAME PROBLEMS HAVE BEEN TESTED AND THE RESULTS ARE THE SAME OR BETTER THAN THOSE REPORTED IN [1]; CITING [1], WE CAN SAY THAT "THIS METHOD OFFERS A USEFUL IF MODEST IMPROVEMENT UPON NEWTON'S METHOD, BUT THIS IMPROVEMENT TENDS TO VANISH AS THE NONLINEARITIES BECOME MORE PRONOUNCED". EXAMPLE OF USE: SEE QUANEWBND1 (THIS SECTION). REFERENCES: [1] BROYDEN C.G. THE CONVERGENCE OF AN ALGORITHM FOR SOLVING SPARSE NONLINEAR SYSTEMS. MATH. COMP., VOL.25 (1971). SUBSECTION: QUANEWBND1. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" QUANEWBND1(N, LW, RW, X, F, FUNCT, IN, OUT); "VALUE" N, LW, RW; "INTEGER" N, LW, RW; "ARRAY" X, F, IN, OUT; "BOOLEAN" "PROCEDURE" FUNCT; "CODE" 34431; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF INDEPENDENT VARIABLES; THE NUMBER OF EQUATIONS SHOULD ALSO BE EQUAL TO N; LW: ; THE NUMBER OF CODIAGONALS TO THE LEFT OF THE MAIN DIAGONAL OF THE JACOBIAN; RW: ; THE NUMBER OF CODIAGONALS TO THE RIGHT OF THE MAIN DIAGONAL OF THE JACOBIAN; X: ; "ARRAY" X[1:N]; ENTRY: AN INITIAL ESTIMATE OF THE SOLUTION OF THE SYSTEM THAT HAS TO BE SOLVED; EXIT: THE CALCULATED SOLUTION OF THE SYSTEM; F: ; "ARRAY" F[1:N]; EXIT: THE VALUES OF THE FUNCTION COMPONENTS AT THE CALCULATED SOLUTION; 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 6 FUNCT: ; THE HEADING OF THIS PROCEDURE READS : "BOOLEAN" "PROCEDURE" FUNCT(N, L, U, X, F); "VALUE" N, L, U; "INTEGER" N, L, U; "ARRAY" X, F; THE MEANING OF THE FORMAL PARAMETERS IS : N: ; THE NUMBER OF INDEPENDENT VARIABLES OF THE FUNCTION F; L, U: ; LOWER AND UPPER BOUND OF THE FUNCTION COMPONENT SUBSCRIPT; X: ; THE INDEPENDENT VARIABLES ARE GIVEN IN X[1:N]; F: ; AFTER A CALL OF FUNCT THE FUNCTION COMPONENTS F[I], I = L, ..., U, SHOULD BE GIVEN IN F[L:U]; IF THE VALUE OF THE PROCEDURE IDENTIFIER EQUALS FALSE, THEN THE EXECUTION OF QUANEWBND WILL BE TERMINATED, WHILE THE VALUE OF OUT[5] IS SET EQUAL TO 2; IN: ; "ARRAY" IN[0:4]; ENTRY : IN THIS AUXILIARY ARRAY ONE SHOULD GIVE THE FOLLOWING VALUES FOR CONTROLLING THE PROCESS: IN[0]: THE MACHINE PRECISION; IN[1]: THE RELATIVE PRECISION ASKED FOR; IN[2]: THE ABSOLUTE PRECISION ASKED FOR; IF THE VALUE, DELIVERED IN OUT[5] EQUALS ZERO, THEN THE LAST CORRECTION VECTOR D, SAY, WHICH IS A MEASURE FOR THE ERROR IN THE SOLUTION, SATIFIES THE INEQUALITY NORM(D) <= NORM(X) * IN[1] + IN[2], WHEREBY X DENOTES THE CALCULATED SOLUTION, GIVEN IN ARRAY X AND NORM(.) DENOTES THE EUCLIDIAN NORM; HOWEVER,WE CAN NOT GUARANTEE THAT THE TRUE ERROR IN THE SOLUTION SATISFIES THIS INEQUALITY, ESPECIALLY IF THE JACOBIAN IS (NEARLY) SINGULAR AT THE SOLUTION; IN[3]: THE MAXIMUM VALUE OF THE NORM OF THE RESIDUAL VECTOR ALLOWED; IF OUT[5] = 0, THEN THIS RESIDUAL VECTOR F, SAY, SATIFIES: NORM(F) <= IN[3]; IN[4]: THE MAXIMUM NUMBER OF FUNCTION COMPONENT EVALUATIONS ALLOWED; L - U + 1 FUNCTION COMPONENT EVALUATIONS ARE COUNTED EACH CALL OF FUNCT(N, L, U, X, F); IF OUT[5]=1, THEN THE PROCESS IS TERMINATED, BECAUSE THE NUMBER OF EVALUATIONS EXCEEDED THE VALUE GIVEN IN IN[4]; IN[5]: THE JACOBIAN MATRIX AT THE INITIAL GUESS IS APPROXIMATED USING FORWARD DIFFERENCES, WITH AN FIXED INCREMENT TO EACH VARIABLE THAT EQUALS THE VALUE GIVEN IN IN[5]; 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 7 OUT: ; "ARRAY" OUT[1:5]; EXIT : OUT[1]: THE EUCLIDIAN NORM OF THE LAST STEP ACCEPTED; OUT[2]: THE EUCLIDIAN NORM OF THE RESIDUAL VECTOR AT THE CALCULATED SOLUTION; OUT[3]: THE NUMBER OF FUNCTION COMPONENT EVALUATIONS PERFORMED; OUT[4]: THE NUMBER OF ITERATIONS CARRIED OUT; OUT[5]: THE INTEGER VALUE DELIVERED IN OUT[5] GIVES SOME INFORMATION ABOUT THE TERMINATION OF THE PROCESS; OUT[5] = 0: THE PROCESS IS TERMINATED IN A NORMAL WAY; THE LAST STEP AND THE NORM OF THE RESIDUAL VECTOR SATISFY THE CONDITIONS (SEE IN[2], IN[3]); IF OUT[5] ^= 0, THEN THE PROCESS IS TERMINATED PREMATURALY; OUT[5] = 1: THE NUMBER OF FUNCTION COMPONENT EVALUATIONS EXCEEDS THE VALUE GIVEN IN IN[4]; OUT[5] = 2: A CALL OF FUNCT DELIVERED THE VALUE FALSE; OUT[5] = 3: THE APPROXIMATION OF THE JACOBIAN MATRIX TURNS OUT TO BE SINGULAR. PROCEDURES USED: JACOBNBNDF = CP34439, QUANEWBND = CP34430. EXECUTION FIELD LENGTH: QUANEWBND1 DECLARES AN AUXILIARY ARRAY OF DIMENSION ONE AND ORDER N + (N - 1) * (LW + RW). RUNNING TIME: PROPORTIONAL TO N * LW * ( LW + RW + 1). LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: QUANEWBND1 USES JACOBNBNDF (SECTION 4.3.2.1) TO CALCULATE AN INITIAL APPROXIMATION OF THE JACOBIAN MATRIX AT THE INITIAL GUESS GIVEN IN X[1:N] AND SOLVES THE NONLINEAR SYSTEM BY CALLING QUANEWBND (THIS SECTION). 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 8 EXAMPLE OF USE: LET THE FUNCTION F BE DEFINED BY (SEE [1]): F[1] = (3 - 2 * X[1]) * X[1] + 1 - 2 * X[2], F[I] = (3 - 2 * X[I]) * X[I] + 1 - X[I - 1] - 2 * X[I + 1], I = 2, ..., N - 1, F[N] = (3 - 2 * X[N]) * X[N] + 1 - X[N - 1]; LET AN INITIAL ESTIMATE OF THE SOLUTION OF THE SYSTEM F(X) = 0 BE GIVEN BY X[I] = -1, I = 1, ..., N; THEN THE FOLLOWING PROGRAM MAY SOLVE THIS SYSTEM FOR N = 600 AND PRINTS SOME RESULTS. "BEGIN" "BOOLEAN" "PROCEDURE" FUN(N, L, U, X, F); "VALUE" N, L, U; "INTEGER" N, L, U; "ARRAY" X, F; "BEGIN" "INTEGER" I; "REAL" X1, X2, X3; X1:= "IF" L = 1 "THEN" 0 "ELSE" X[L - 1]; X2:= X[L]; X3:= "IF" L = N "THEN" 0 "ELSE" X[L + 1]; "FOR" I:= L "STEP" 1 "UNTIL" U "DO" "BEGIN" F[I]:= (3 - 2 * X2) * X2 + 1 - X1 - X3 * 2; X1:= X2; X2:= X3; X3:= "IF" I <= N - 2 "THEN" X[I + 2] "ELSE" 0 "END"; FUN:= "TRUE" "END" FUN; "INTEGER" I; "ARRAY" X, F[1:600], IN[0:5], OUT[1:5]; "FOR" I := 1 "STEP" 1 "UNTIL" 600 "DO" X[I]:= -1; IN[0]:= "-14; IN[1]:= IN[2]:= IN[3]:= "-6; IN[4]:= 20000; IN[5]:= 0.001; QUANEWBND1(600, 1, 1, X, F, FUN, IN, OUT); OUTPUT(71, "("//,"(" NORM RESIDUALVECTOR: ")"+.15D"+3D,/, "(" LENGTH OF LAST STEP: ")"+.15D"+3D,/, "(" NUMBER OF FUNCTION COMPONENT EVALUATIONS: ")"5ZD,/, "(" NUMBER OF ITERATIONS: ")"4ZD/,"("REPORT: ")"D/")", OUT[2], OUT[1], OUT[3], OUT[4], OUT[5]) "END" RESULTS: NORM RESIDUALVECTOR: +.221010684482660"-006 LENGTH OF LAST STEP: +.302712457332660"-006 NUMBER OF FUNCTION COMPONENT EVALUATIONS: 6598 NUMBER OF ITERATIONS: 7 REPORT: 0 REFERENCES: [1] BROYDEN C.G. THE CONVERGENCE OF AN ALGORITHM FOR SOLVING SPARSE NONLINEAR SYSTEMS. MATH. COMP., VOL.25 (1971). 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 9 SOURCE TEXT(S): 0"CODE" 34430; "PROCEDURE" QUANEWBND(N, LW, RW, X, F, JAC, FUNCT, IN, OUT); "VALUE" N, LW, RW; "INTEGER" N, LW, RW; "ARRAY" X, F, JAC, IN, OUT; "BOOLEAN" "PROCEDURE" FUNCT; "BEGIN" "INTEGER" L, IT, FCNT, FMAX, ERR, B; "REAL" MACHEPS, RELTOL, ABSTOL, TOLRES, ND, MZ, RES; "ARRAY" DELTA[1:N]; "REAL" "PROCEDURE" EVALUATE(N, X, F); "VALUE" N; "INTEGER" N; "ARRAY" X, F; "BEGIN" FCNT:= FCNT + N; "IF" ^ FUNCT(N, 1, N, X, F) "THEN" "BEGIN" ERR:= 2; "GOTO" EXIT "END"; "IF" FCNT > FMAX "THEN" ERR:= 1; EVALUATE:= SQRT(VECVEC(1, N, 0, F, F)) "END" EVAL; "BOOLEAN" "PROCEDURE" DIRECTION; "BEGIN" "ARRAY" LU[1:L], AUX[1:5]; AUX[2]:= MACHEPS; MULVEC(1, N, 0, DELTA, F, -1); DUPVEC(1, L, 0, LU, JAC); DECSOLBND(LU, N, LW, RW, AUX, DELTA); DIRECTION:= AUX[3] = N "END" SOLLINSYS; "BOOLEAN" "PROCEDURE" TEST(ND, TOLD, NRES, TOLRES, ERR); "VALUE" ND, TOLD; "INTEGER" ERR; "REAL" ND, TOLD, NRES, TOLRES; TEST:= ERR ^= 0 "OR" (NRES < TOLRES "AND" ND < TOLD); "PROCEDURE" UPDATE JAC; "BEGIN" "INTEGER" I, J, K, R, M; "REAL" MUL, CRIT; "ARRAY" PP, S[1:N]; CRIT:= ND * MZ; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" PP[I]:= DELTA[I] ** 2; R:= 1; K:= 1; M:= RW + 1; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" MUL:= 0; "FOR" J:= R "STEP" 1 "UNTIL" M "DO" MUL:= MUL + PP[J]; J:= R - K; "IF" ABS(MUL) > CRIT "THEN" ELMVEC(K, M - J, J, JAC, DELTA, F[I] / MUL); K:= K + B; "IF" I > LW "THEN" R:= R + 1 "ELSE" K:= K - 1; "IF" M < N "THEN" M:= M + 1 "END" "END" UPDATEJAC 1SECTION : 5.1.1.2.2 (OCTOBER 1974) PAGE 10 ; MACHEPS:= IN[0]; RELTOL:= IN[1]; ABSTOL:= IN[2]; TOLRES:= IN[3]; FMAX:= IN[4]; MZ:= MACHEPS ** 2; IT:= FCNT:= 0; B:= LW + RW; L:= (N - 1) * B + N; B:= B + 1; RES:= SQRT(VECVEC(1, N, 0, F, F)); ERR:= 0; ITERATE: "IF" ^ TEST(SQRT(ND), SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL, RES, TOLRES, ERR) "THEN" "BEGIN" IT:= IT + 1; "IF" IT ^= 1 "THEN" UPDATEJAC; "IF" ^ DIRECTION "THEN" ERR:= 3 "ELSE" "BEGIN" ELMVEC(1, N, 0, X, DELTA, 1); ND:= VECVEC(1, N, 0, DELTA, DELTA); RES:= EVALUATE(N, X, F); "GOTO" ITERATE "END" "END"; EXIT: OUT[1]:= SQRT(ND); OUT[2]:=RES; OUT[3]:= FCNT; OUT[4]:= IT; OUT[5]:= ERR "END" QUANEWBND; "EOP" "CODE" 34431; "PROCEDURE" QUANEWBND1(N, LW, RW, X, F, FUNCT, IN, OUT); "VALUE" N, LW, RW; "INTEGER" N, LW, RW; "ARRAY" X, F, IN, OUT; "BOOLEAN" "PROCEDURE" FUNCT; "BEGIN" "INTEGER" I, K; "REAL" S; "ARRAY" JAC[1:(LW + RW) * (N - 1) + N]; FUNCT(N, 1, N, X, F); S:= IN[5]; K:= (LW + RW)*(N - 1) + N*2 - ((LW - 1)*LW + (RW - 1)*RW) // 2; IN[4]:= IN[4] - K; JACOBNBNDF(N, LW, RW, X, F, JAC, I, S, FUNCT); QUANEWBND(N, LW, RW, X, F, JAC, FUNCT, IN, OUT); IN[4]:= IN[4] + K; OUT[3]:= OUT[3] + K "END" QUANEWBND1; "EOP" 1SECTION : 5.1.2.1.1 ( DECEMBER 1978 ) PAGE 1 AUTHOR : J. C. P. BUS INSTITUTE : MATHEMATICAL CENTRE. RECEIVED : 741101. BRIEF DESCRIPTION : THIS SECTION CONTAINS THE PROCEDURE MININ, FOR MINIMIZING A FUNCTION OF ONE VARIABLE IN A GIVEN INTERVAL; KEYWORDS : MINIMIZATION, FUNCTIONS OF ONE VARIABLE. CALLING SEQUENCE : THE HEADING OF THIS PROCEDURE IS : "REAL" "PROCEDURE" MININ(X, A, B, FX, TOLX); "REAL" X, A, B, FX, TOLX; "CODE" 34433; MININ DELIVERS THE CALCULATED MINIMUM VALUE OF THE FUNCTION, DEFINED BY FX, ON THE INTERVAL WITH ENDPOINTS A AND B. THE MEANING OF THE FORMAL PARAMETERS IS : X : ; A JENSEN VARIABLE; THE ACTUAL PARAMETERS FOR FX AND TOLX DEPEND ON X; EXIT : THE CALCULATED APPROXIMATION OF THE POSITION OF THE MINIMUM; A, B : ; ENTRY : THE ENDPOINTS OF THE INTERVAL ON WHICH A MINIMUM IS SEARCHED FOR; EXIT : THE ENDPOINTS OF THE INTERVAL WITH LENGTH LESS THAN 4 * TOL(X) SUCH THAT A < X < B; FX : ; THE FUNCTION ISGIVEN BY THE ACTUAL PARAMETER FX, WHICH DEPENDS ON X; TOLX : ; THE TOLERANCE IS GIVEN BY THE ACTUAL PARAMETER TOLX, WHICH MAY DEPEND ON X; A SUITABLE TOLERANCE FUNCTION IS : ABS(X)*RE + AE, WHERE RE IS THE RELATIVE PRECISION DESIRED AND AE IS AN ABSOLUTE PRECISION WHICH SHOULD NOT BE CHOSEN EQUAL TO ZERO. 1SECTION : 5.1.2.1.1 ( DECEMBER 1978 ) PAGE 2 DATA AND RESULTS : THE USER SHOULD BE AWARE OF THE FACT THAT THE CHOICE OF TOLX MAY HIGHLY AFFECT THE BEHAVIOUR OF THE ALGORITHM, ALTHOUGH CONVERGENCE TO A POINT FOR WHICH THE GIVEN FUNCTION IS MINIMAL ON THE INTERVAL IS ASSURED; THE ASYMPTOTIC BEHAVIOUR WILL USUALLY BE FINE AS LONG AS THE NUMERICAL FUNCTION IS STRICTLY DELTA-UNIMODAL ON THE GIVEN INTERVAL (SEE [1]) AND THE TOLERANCE FUNCTION SATISFIES TOL(X)>=DELTA, FOR ALL X IN THE GIVEN INTERVAL. PROCEDURES USED : NONE. REQUIRED CENTRAL MEMORY : NO AUXILIARY ARRAYS ARE DECLARED IN MININ. METHOD AND PERFORMANCE : MININ IS A SLIGHTLY MODIFIED VERSION OF THE ALGORITHM GIVEN IN [1]. EXAMPLE OF USE: THE FOLLOWING PROGRAM MAY BE USED TO CALCULATE THE MINIMUM OF THE FUNCTION F(X) = SUM(((I * 2 - 5)/(X - I ** 2)) ** 2; I = 1 (1) 20) ON THE INTERVAL [1 + TOL, 4 - TOL] (SEE [1]). "BEGIN" "REAL" M, X, A, B; "INTEGER" CNT; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; "BEGIN" "INTEGER" I; "REAL" S; S:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" 20 "DO" S:= S + ((I * 2 - 5) / (X - I ** 2)) ** 2; CNT:= CNT + 1; F:= S "END" F; "REAL" "PROCEDURE" TOL(X); "VALUE" X; "REAL" X; TOL:= ABS(X) * "-7 + "-7; A:= 1 + TOL(1); B:= 4 - TOL(4); CNT:= 0; M:= MININ(X, A, B, F(X), TOL(X)); OUTPUT(61, "("4B,"("MINIMUM IS ")",N,/4B, "("FOR X IS ")",N,/4B, "("IN THE INTERVAL WITH ENDPOINTS ")",/8B,2(N),/4B, "("THE NUMBER OF FUNCTION EVALUATIONS NEEDED IS ")",2ZD,/")", M, X, A, B, CNT) "END" 1SECTION : 5.1.2.1.1 ( DECEMBER 1978 ) PAGE 3 RESULTS : MINIMUM IS +3.6766990169019"+000 FOR X IS +3.0229153387991"+000 IN THE INTERVAL WITH ENDPOINTS +3.0229149365075"+000 +3.0229157410906"+000 THE NUMBER OF FUNCTION EVALUATIONS NEEDED IS 13 SOURCE TEXT: 0"CODE" 34433; "REAL" "PROCEDURE" MININ(X, A, B, FX, TOLX); "REAL" X, A, B, FX, TOLX; "BEGIN" "COMMENT" SEE BRENT, 1973, P79; "REAL" Z, C, D, E, M, P, Q, R, TOL, T, U, V, W, FU, FV, FW, FZ; C:= (3 - SQRT(5)) / 2; "IF" A > B "THEN" "BEGIN" Z:= A; A:= B; B:= Z "END"; W:= X:= A; FW:= FX; Z:= X:= B; FZ:= FX; "IF" FZ > FW "THEN" "BEGIN" Z:= W; W:= X; V:= FZ; FZ:= FW; FW:= V "END"; V:= W; FV:= FW; E:= 0; LOOP: M:= (A + B) * 0.5; TOL:= TOLX; T:= TOL * 2; "IF" ABS(Z - M) > T - (B - A) * 0.5 "THEN" "BEGIN" P:= Q:= R:= 0; "IF" ABS(E) > TOL "THEN" "BEGIN" R:= (Z - W) * (FZ - FV); Q:= (Z - V) * (FZ - FW); P:= (Z - V) * Q - (Z - W) * R; Q:= (Q - R) * 2; "IF" Q>0 "THEN" P:= -P "ELSE" Q:= -Q; R:= E; E:= D "END"; "IF" ABS(P) < ABS(Q * R * 0.5) "AND" P > (A - Z) * Q "AND" P < (B - Z) * Q "THEN" "BEGIN" D:= P / Q; U:= Z + D; "IF" U - A < T "OR" B - U < T "THEN" D:= "IF" Z < M "THEN" TOL "ELSE" -TOL "END" "ELSE" "BEGIN" E:= ("IF" Z < M "THEN" B "ELSE" A) - Z; D:= C * E "END"; U:= X:= Z + ("IF" ABS(D) >= TOL "THEN" D "ELSE" "IF" D > 0 "THEN" TOL "ELSE" -TOL); FU:= FX; "IF" FU <= FZ "THEN" "BEGIN" "IF" U < Z "THEN" B:= Z "ELSE" A:= Z; V:= W; FV:= FW; W:= Z; FW:= FZ; Z:= U; FZ:= FU "END" "ELSE" "BEGIN" "IF" U < Z "THEN" A:= U "ELSE" B:= U; "IF" FU <= FW "THEN" "BEGIN" V:= W; FV:= FW; W:= U; FW:= FU "END" "ELSE" "IF" FU <= FV "OR" V = W "THEN" "BEGIN" V:= U; FV:= FU "END" "END"; "GOTO" LOOP "END"; X:= Z; MININ:= FZ "END" MININ; "EOP" 1SECTION : 5.1.2.1.2 (OCTOBER 1975) PAGE 1 AUTHOR: J. C. P. BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 741101. BRIEF DESCRIPTION: THIS SECTION CONTAINS THE PROCEDURE MININDER, FOR MINIMIZING A FUNCTION OF ONE VARIABLE IN A GIVEN INTERVAL, WHEN THE ANALYTICAL DERIVATIVE OF THE FUNCTION IS AVAILABLE. KEYWORDS : MINIMIZATION, FUNCTIONS OF ONE VARIABLE. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "REAL" "PROCEDURE" MININDER(X, Y, FX, DFX, TOLX); "REAL" X, Y, FX, DFX, TOLX; "CODE" 34435; MININDER DELIVERS THE CALCULATED MINIMUM VALUE OF THE FUNCTION, DEFINED BY FX, ON THE INTERVAL WITH END POINTS A AND B. THE MEANING OF THE FORMAL PARAMETERS IS: X: ; A JENSEN VARIABLE; THE ACTUAL PARAMETERS FOR FX, DFX AND TOLX DEPEND ON X; ENTRY: ONE OF THE END POINTS OF THE INTERVAL ON WHICH THE FUNCTION HAS TO BE MINIMIZED; EXIT: THE CALCULATED APPROXIMATION OF THE POSITION OF THE MINIMUM; Y: ; ENTRY: THE OTHER END POINT OF THE INTERVAL ON WHICH THE FUNCTION HAS TO BE MINIMIZED; EXIT: A VALUE SUCH THAT ABS(X - Y) <= TOL(X) * 3; FX: ; THE FUNCTION IS GIVEN BY THE ACTUAL PARAMETER FX WHICH DEPENDS ON X; DFX: ; THE DERIVATIVE OF THE FUNCTION IS GIVEN BY THE ACTUAL PARAMETER DFX WHICH DEPENDS ON X; FX AND DFX ARE EVALUATED SUCCESSIVELY FOR A CERTAIN VALUE OF X; TOLX: ; THE TOLERANCE IS GIVEN BY THE ACTUAL PARAMETER TOLX, WHICH MAY DEPEND ON X; A SUITABLE TOLERANCE FUNCTION IS: ABS(X) * RE + AE, WHERE RE IS THE RELATIVE PRECISION DESIRED AND AE IS AN ABSOLUTE PRECISION WHICH SHOULD NOT BE CHOSEN EQUAL TO ZERO. 1SECTION : 5.1.2.1.2 (OCTOBER 1975) PAGE 2 DATA AND RESULTS: THE USER SHOULD BE AWARE OF THE FACT THAT THE CHOICE OF TOLX MAY HIGHLY AFFECT THE BEHAVIOUR OF THE ALGORITHM, ALTHOUGH CONVERGENCE TO A POINT FOR WHICH THE GIVEN FUNCTION IS MINIMAL ON THE GIVEN INTERVAL IS ASSURED; THE ASYMPTOTIC BEHAVIOUR WILL USUALLY BE FINE AS LONG AS THE NUMERICAL FUNCTION IS STRICTLY DELTA-UNIMODAL ON THE GIVEN INTERVAL (SEE [1]) AND THE TOLERANCE FUNCTION SATISFIES TOL(X) >= DELTA, FOR ALL X IN THE GIVEN INTERVAL; LET THE VALUE OF DFX AT THE BEGIN AND END POINT OF THE INITIAL INTERVAL BE DENOTED BY DFA AND DFB, RESPECTIVELY, THEN, FINDING A GLOBAL MINIMUM IS ONLY GUARANTEED IF THE FUNCTION IS CONVEX AND DFA <= 0 AND DFB >= 0; IF THESE CONDITIONS ARE NOT SATISFIED, THEN A LOCAL MINIMUM OR A MINIMUM AT ONE OF THE END POINTS MIGHT BE FOUND. PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE DECLARED IN MININDER. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: MININDER HAS ALMOST THE SAME STRUCTURE AS THE PROCEDURE GIVEN IN [1]; HOWEVER, CUBIC INTERPOLATION (SEE [2]) IS USED INSTEAD OF QUADRATIC INTERPOLATION TO APPROXIMATE THE MINIMIUM. REFERENCES: [1]: BRENT, R.P. ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES. CH.5. PRENTICE HALL, 1973. [2]: DAVIDON, W.C. VARIABLE METRIC METHODS FOR MINIMIZATION. REP. A.N.L. 5990, 1959. 1SECTION : 5.1.2.1.2 (OCTOBER 1975) PAGE 3 EXAMPLE OF USE: THE FOLLOWING PROGRAM MAY BE USED TO CALCULATE THE MINIMUM OF THE FUNCTION F(X) = SUM(((I * 2 - 5)/(X - I ** 2)) ** 2; I = 1 (1) 20) ON THE INTERVAL [1.01,3.99] (SEE [1]). "BEGIN" "REAL" M, X, Y; "INTEGER" CNT; "REAL" "PROCEDURE" F(X); "VALUE" X; "REAL" X; "BEGIN" "INTEGER" I; "REAL" S; S:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" 20 "DO" S:= S + ((I * 2 - 5) / (X - I ** 2)) ** 2; CNT:= CNT + 1; F:= S "END" F; "REAL" "PROCEDURE" DF(X); "VALUE" X; "REAL" X; "BEGIN" "INTEGER" I; "REAL" S; S:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" 20 "DO" S:= S + (I * 2 - 5) ** 2 / (X - I ** 2) ** 3; DF:= -S * 2 "END" DF; "REAL" "PROCEDURE" TOL(X); "VALUE" X; "REAL" X; TOL:= ABS(X) * "-7 + "-7; X:= 1.01; Y:= 3.99; CNT:= 0; M:= MININDER(X, Y, F(X), DF(X),TOL(X)); OUTPUT(61 ,"("4B,"("MINIMUM IS ")",N,/4B, "("FOR X IS ")",N,/4B, "("AND Y IS ")",N,/4B, "("THE NUMBER OF FUNCTION EVALUATIONS NEEDED IS ")",2ZD,/")", M, X, Y, CNT) "END" RESULTS: MINIMUM IS +3.6766990169021"+000 FOR X IS +3.0229155250302"+000 AND Y IS +3.0229151227386"+000 THE NUMBER OF FUNCTION EVALUATIONS NEEDED IS 9 1SECTION : 5.1.2.1.2 (NOVEMBER 1976) PAGE 4 SOURCE TEXT(S): "CODE"34435; "REAL" "PROCEDURE" MININDER(X, Y, FX, DFX, TOLX); "REAL" X, Y, FX, DFX, TOLX; "BEGIN" "COMMENT" THE FUNCTION IS APPROXIMATED BY A CUBIC AS GIVEN BY DAVIDON, 1958, THE STRUCTURE IS SIMILAR TO THE STRUCTURE OF THE PROGRAM GIVEN BY BRENT, 1973, THIS IS A REVISION OF 760407; "INTEGER" SGN; "REAL" A, B, C, FA, FB, FU, DFA, DFB, DFU, E, D, TOL, BA, Z, P, Q, S; "IF" X <= Y "THEN" "BEGIN" A:= X; FA:= FX; DFA:= DFX; B:= X:= Y; FB:= FX; DFB:= DFX "END" "ELSE" "BEGIN" B:= X; FB:= FX; DFB:= DFX; A:= X:= Y; FA:= FX; DFA:= DFX "END"; C:= (3 - SQRT(5)) / 2; D:= B - A; E:= D * 2; Z:= E * 2; LOOP: BA:= B - A; TOL:= TOLX; "IF" BA >= TOL * 3 "THEN" "BEGIN" "IF" ABS(DFA) <= ABS(DFB) "THEN" "BEGIN" X:=A; SGN:= 1 "END" "ELSE" "BEGIN" X:= B; SGN:= -1 "END"; "IF" DFA <= 0 "AND" DFB >= 0 "THEN" "BEGIN" Z:= (FA - FB) * 3 / BA + DFA + DFB; S:= SQRT(Z ** 2 - DFA * DFB); P:= "IF" SGN = 1 "THEN" DFA - S - Z "ELSE" DFB + S - Z; P:= P * BA; Q:= DFB - DFA + S * 2; Z:= E; E:= D; D:= "IF" ABS(P) <= ABS(Q) * TOL "THEN" TOL * SGN "ELSE" -P / Q "END" "ELSE" D:= BA; "IF" ABS(D) >= ABS(Z * 0.5) "OR" ABS(D) > BA * 0.5 "THEN" "BEGIN" E:= BA; D:= C * BA * SGN "END"; X:= X + D; FU:= FX; DFU:= DFX; "IF" DFU >= 0 "OR" (FU >= FA "AND" DFA <= 0) "THEN" "BEGIN" B:= X; FB:= FU; DFB:= DFU "END" "ELSE" "BEGIN" A:= X; FA:= FU; DFA:= DFU "END"; "GOTO" LOOP "END"; "IF" FA <= FB "THEN" "BEGIN" X:= A; Y:= B; MININDER:= FA "END" "ELSE" "BEGIN" X:= B; Y:= A; MININDER:= FB "END" "END" MININDER; "EOP" 1SECTION : 5.1.2.2.1 (DECEMBER 1975) PAGE 1 AUTHOR: J.C.P.BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730620. BRIEF DESCRIPTION: THIS SECTION CONTAINS FOUR PROCEDURES, LINEMIN, RNK1UPD, DAVUPD AND FLEUPD, THAT ARE AUXILIARY PROCEDURES FOR THE PROCEDURES RNK1MIN AND FLEMIN (SECTION 5.1.2.2.2). KEYWORDS: AUXILIARY PROCEDURE. SUBSECTION: LINEMIN. CALLING SEQUENCE: THE HEADING OF THIS AUXILIARY PROCEDURE IS: "PROCEDURE" LINEMIN(N, X, D, ND, ALFA, G, FUNCT, F0, F1, DF0, DF1, EVLMAX, STRONGSEARCH, IN); "VALUE" N, ND, F0, DF0, STRONGSEARCH; "INTEGER" N, EVLMAX; "BOOLEAN" STRONGSEARCH; "REAL" ND, ALFA, F0, F1, DF0, DF1; "ARRAY" X, D, G, IN; "REAL" "PROCEDURE" FUNCT;"CODE" 34210; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF VARIABLES OF THE GIVEN FUNCTION F; X: ; "ARRAY" X[1 : N]; ENTRY: A VECTOR X0, SUCH THAT F IS DECREASING IN X0, IN THE DIRECTION GIVEN BY D; EXIT: THE CALCULATED APPROXIMATION OF THE VECTOR FOR WHICH F IS MINIMAL ON THE LINE DEFINED BY: X0 + ALFA * D, (ALFA > 0); D: ; "ARRAY" D[1 : N]; ENTRY: THE DIRECTION OF THE LINE ON WHICH F HAS TO BE MINIMIZED; ND: ; ENTRY: THE EUCLIDEAN NORM OF THE VECTOR GIVEN IN D[1 : N]; ALFA: ; THE INDEPENDENT VARIABLE, THAT DEFINES THE POSITION ON THE LINE ON WHICH F HAS TO BE MINIMIZED; THIS LINE IS DEFINED BY X0 + ALFA * D, (ALFA > 0); ENTRY: AN ESTIMATE ALFA0 OF THE VALUE FOR WHICH H(ALFA) = F(X0 + ALFA * D), (ALFA > 0), IS MINIMAL; EXIT: THE CALCULATED APPROXIMATION ALFAM OF THE VALUE FOR WHICH H(ALFA) IS MINIMAL; 1SECTION : 5.1.2.2.1 (FEBRUARY 1979) PAGE 2 G: ; "ARRAY" G[1 : N]; EXIT: THE GRADIENT OF F AT THE CALCULATED APPROXIMATION OF THE MINIMUM; FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD BE: "REAL" "PROCEDURE" FUNCT(N, X, G); "VALUE" N; "INTEGER" N; "ARRAY" X, G; A CALL OF FUNCT SHOULD EFFECTUATE : 1: FUNCT:= F(X); 2: THE VALUE OF G[I], (I = 1, ..., N), BECOMES THE VALUE OF THE I - TH COMPONENT OF THE GRADIENT OF F AT X; F0: ; ENTRY: THE VALUE OF H(0), (SEE ALFA); F1: ; ENTRY: THE VALUE OF H(ALFA0); EXIT: THE VALUE OF H(ALFAM), (SEE ALFA); DF0: ; ENTRY: THE VALUE OF THE DERIVATIVE OF H AT ALFA = 0; DF1: ; ENTRY: THE VALUE OF THE DERIVATIVE OF H AT ALFA = ALFA0; EXIT: THE VALUE OF THE DERIVATIVE OF H AT ALFA = ALFAM; EVLMAX: ; ENTRY: THE MAXIMUM ALLOWED NUMBER OF CALLS OF FUNCT; EXIT: THE NUMBER OF TIMES FUNCT HAS BEEN CALLED; STRONGSEARCH: ; IF THE VALUE OF STRONGSEARCH IS TRUE, THEN THE PROCESS MAKES USE OF TWO STOPPING CRITERIA: A: THE NUMBER OF TIMES FUNCT HAS BEEN CALLED EXCEEDS THE GIVEN VALUE OF EVLMAX; B: AN INTERVAL IS FOUND WITH LENGTH LESS THAN TWO TIMES THE PRESCRIBED PRECISION,ON WICH A MINIMUM IS EXPECTED; IF THE VALUE OF STRONGSEARCH IS FALSE, THE PROCESS MAKES ALSO USE OF A THIRD STOPPING CRITERION : C: MU <= (H(ALFAK) - H(ALFA0)) / (ALFAK * DF0) <= 1 - MU, WHEREBY ALFAK IS THE CURRENT ITERATE AND MU A PRESCRIBED CONSTANT; IN: ; ENTRY: "ARRAY" IN[1:3]; IN[1]: THE RELATIVE PRECISION, EPSR, NECESSARY FOR THE STOPPING CRITERION B, (SEE STRONGSEARCH); IN[2]: THE ABSOLUTE PRECISION, EPSA, NECESSARY FOR THE STOPPING CRITERION B, (SEE STRONGSEARCH); THE PRESCRIBED PRECISION, EPS, AT ALFA = ALFAK IS GIVEN BY: EPS = NORM ( X0 + ALPHA * D ) * EPSR + EPSA, WHERE NORM ( . ) DENOTES THE EUCLIDEAN NORM. IN[3]: THE PARAMETER MU NECESSARY FOR STOPPING CRITERION C; THIS PARAMETER MUST SATISFY: 0 < MU < 0.5 ; IN PRACTICE,A CHOICE OF MU = 0.0001 IS ADVISED. 1SECTION : 5.1.2.2.1 (FEBRUARY 1979) PAGE 3 DATA AND RESULTS: LINEMIN CALCULATES AN APPROXIMATION OF A MINIMUM OF A HIGHER - DIMENSIONAL FUNCTION ON A GIVEN LINE; THE QUANTITY DF0 MUST SATIFY: DF0 < 0; IF MOREOVER DF1 > 0, THEN THE PROCEDURE WILL YIELD A RESULT THAT SATISFIES ONE OF THE CHOSEN STOPPING CRITERIA, (SEE STRONGSEARCH), OTHERWISE WE CAN NOT GUARANTEE SUCH A RESULT. PROCEDURES USED: VECVEC = CP34010, ELMVEC = CP34020, DUPVEC = CP31030. REQUIRED CENTRAL MEMORY: N WORDS. METHOD AND PERFORMANCE: AN APPROXIMATION TO THE MINIMUM ON THE GIVEN LINE IS CALCULATED WITH CUBIC INTERPOLATION ([2]);THE STOPPING CRITERION USED WHEN THE VALUE OF STRONGSEARCH IS FALSE IS DESCRIBED IN [3] AND [4]; A DETAILED DESCRIPTION OF THIS PROCEDURE IS GIVEN IN [1]. REFERENCES: [1] BUS, J. C. P. MINIMIZATION OF FUNCTIONS OF SEVERAL VARIABLES (DUTCH). MATHEMATICAL CENTRE, AMSTERDAM, NR 29/72 (1972). [2] DAVIDON, W. C. VARIABLE METRIC METHOD FOR MINIMIZATION. ARGONNE NAT. LAB. REPORT, ANL 5990 (1959). [3] FLETCHER, R. A NEW APPROACH TO VARIABLE METRIC ALGORITHMS. COMP. J. 6, (1963), P.163 - 168. [4] GOLDSTEIN, A. A. AND PRICE, J. F. AN EFFECTIVE ALGORITHM FOR MINIMIZATION. NUMER. MATH. 10, (1967), P.184 - 189. 1SECTION : 5.1.2.2.1 (FEBRUARY 1979) PAGE 4 SUBSECTION: RNK1UPD. CALLING SEQUENCE: THE HEADING OF THIS AUXILIARY PROCEDURE IS: "PROCEDURE" RNK1UPD(H, N, V, C); "VALUE" N, C; "INTEGER" N; "REAL" C; "ARRAY" H, V; "CODE" 34211; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE SYMMETRIC MATRIX, WHOSE UPPERTRIANGLE IS STORED COLUMNWISE IN THE ONE - DIMENSIONAL ARRAY H; C: ; SEE V; V: ; "ARRAY" V[1 : N]; THE GIVEN MATRIX IS UPDATED (ANOTHER MATRIX IS ADDED TO IT) WITH A SYMMETRIC MATRIX , U, OF RANK ONE, DEFINED BY: U[I,J] = C * V[I] * V[J]; H: ; "ARRAY" H[1 : N * (N + 1) // 2]; ENTRY: THE UPPERTRIANGLE (STORED COLUMNWISE, I.E. : A[I,J] = H[(J-1)*J//2+I], 1 <= I <= J <= N) OF THE SYMMETRIC MATRIX THAT HAS TO BE UPDATED; EXIT: THE UPPERTRIANGLE (STORED COLUMNWISE) OF THE UPDATED MATRIX. PROCEDURES USED: ELMVEC = CP34020. REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE DECLARED IN RNK1UPD. 1SECTION : 5.1.2.2.1 (FEBRUARY 1979) PAGE 5 SUBSECTION: DAVUPD. CALLING SEQUENCE: THE HEADING OF THIS AUXILIARY PROCEDURE IS: "PROCEDURE" DAVUPD(H, N, V, W, C1, C2); "VALUE" N, C1, C2; "INTEGER" N; "REAL" C1, C2; "ARRAY" H, V, W; "CODE" 34212; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE SYMMETRIC MATRIX WHOSE UPPERTRIANGLE IS STORED COLUMNWISE IN THE ONE - DIMENSIONAL ARRAY H; C1: ; SEE W; C2: ; SEE W; V: ; "ARRAY" V[1 : N]; SEE W; W: ; "ARRAY" W[1 : N]; THE GIVEN MATRIX IS UPDATED WITH A SYMMETRIC MATRIX U OF RANK TWO, DEFINED BY: U[I,J] = C1 * V[I] * V[J] - C2 * W[I] * W[J]; H: ; "ARRAY" H[1 : N * (N + 1) // 2]; ENTRY: THE UPPERTRIANGLE (STORED COLUMNWISE, I.E. : A[I,J] = H[(J - 1) * J // 2 + I], 1 <= I <= J <= N) OF THE MATRIX THAT HAS TO BE UPDATED; EXIT: THE UPPERTRIANGLE (STORED COLUMNWISE) OF THE UPDATED MATRIX. PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE DECLARED IN DAVUPD. 1SECTION : 5.1.2.2.1 (FEBRUARY 1979) PAGE 6 SUBSECTION: FLEUPD. CALLING SEQUENCE: THE HEADING OF THIS AUXILIARY PROCEDURE IS: "PROCEDURE" FLEUPD(H, N, V, W, C1, C2); "VALUE" N, C1, C2; "INTEGER" N; "REAL" C1, C2; "ARRAY" H, V, W; "CODE" 34213; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE ORDER OF THE SYMMETRIC MATRIX WHOSE UPPERTRIANGLE IS STORED COLUMNWISE IN THE ONE - DIMENSIONAL ARRAY H; C1: ; SEE W; C2: ; SEE W; V: ; "ARRAY" V[1 : N]; SEE W; W: ; "ARRAY" W[1 : N]; THE GIVEN MATRIX IS UPDATED WITH A SYMMETRIC MATRIX U OF RANK TWO, DEFINED BY: U[I,J]= C2 * V[I] * V[J] - C1 *(V[I] * W[J] + W[I] * V[J]); H: ; "ARRAY" H[1 : N * (N + 1) // 2]; ENTRY: THE UPPERTRIANGLE (STORED COLUMNWISE, I.E. : A[I,J] = H[(J - 1) * J // 2 + I], 1 <= I <= J <= N) OF THE MATRIX THAT HAS TO BE UPDATED; EXIT: THE UPPERTRIANGLE (STORED COLUMNWISE) OF THE UPDATED MATRIX. PROCEDURE USED: NONE. REQUIRED CENTRAL MEMORY: NO AUXILIARY ARRAYS ARE DECLARED IN FLEUPD. 1SECTION : 5.1.2.2.1 (DECEMBER 1975) PAGE 7 SOURCE TEXT(S): 0"CODE" 34210; "PROCEDURE" LINEMIN(N, X, D, ND, ALFA, G, FUNCT, F0, F1, DF0, DF1, EVLMAX, STRONGSEARCH, IN); "VALUE" N, ND, F0, DF0, STRONGSEARCH; "INTEGER" N, EVLMAX; "BOOLEAN" STRONGSEARCH; "REAL" ND, ALFA, F0, F1, DF0, DF1; "ARRAY" X, D, G, IN; "REAL" "PROCEDURE" FUNCT; "BEGIN" "INTEGER" I, EVL; "BOOLEAN" NOTININT; "REAL" F,OLDF,DF,OLDDF,MU,ALFA0,Q,W,Y,Z,RELTOL,ABSTOL ,EPS, AID; "ARRAY" X0[1:N]; RELTOL:= IN[1]; ABSTOL:= IN[2]; MU:= IN[3]; EVL:= 0; ALFA0:= 0; OLDF:= F0; OLDDF:= DF0; Y:= ALFA; NOTININT:= "TRUE"; DUPVEC(1, N, 0, X0, X); EPS:= (SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL) / ND; Q:= (F1 - F0) / (ALFA * DF0); INT: "IF" NOTININT "THEN" NOTININT:= DF1 < 0 "AND" Q > MU; AID:= ALFA; "IF" DF1 >= 0 "THEN" "BEGIN" Z:= 3 * (OLDF - F1) / ALFA + OLDDF + DF1; W:= SQRT(Z ** 2 - OLDDF * DF1); ALFA:= ALFA * (1 - (DF1 + W - Z) / (DF1 - OLDDF + W * 2)); "IF" ALFA < EPS "THEN" ALFA:= EPS "ELSE" "IF" AID - ALFA < EPS "THEN" ALFA:= AID - EPS "END" CUBIC INTERPOLATION "ELSE" "IF" NOTININT "THEN" "BEGIN" ALFA0:= ALFA:= Y; OLDDF:= DF1; OLDF:= F1 "END" "ELSE" ALFA:= 0.5 * ALFA; Y:= ALFA + ALFA0; DUPVEC(1, N, 0, X, X0); ELMVEC(1, N, 0, X, D, Y); EPS:= (SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL) / ND; F:= FUNCT(N, X, G); EVL:= EVL +1 ; DF:= VECVEC(1, N, 0, D, G); Q:= (F - F0) / (Y * DF0); "IF" ("IF" NOTININT "OR" STRONGSEARCH "THEN" "TRUE" "ELSE" Q < MU "OR" Q > 1 - MU) "AND" EVL < EVLMAX "THEN" "BEGIN" "IF" NOTININT "OR" DF > 0 "OR" Q < MU "THEN" "BEGIN" DF1:= DF; F1:= F "END" "ELSE" "BEGIN" ALFA0:= Y; ALFA:= AID - ALFA; OLDDF:= DF; OLDF:= F "END"; "IF" ALFA > EPS * 2 "THEN" "GOTO" INT "END"; ALFA:= Y; EVLMAX:= EVL; DF1:= DF; F1:= F "END" LINEMIN 1SECTION : 5.1.2.2.1 (DECEMBER 1975) PAGE 8 ; "EOP" 0"CODE" 34211; "PROCEDURE" RNK1UPD(H, N, V, C);"VALUE" N, C; "INTEGER" N; "REAL" C;"ARRAY" H, V; "BEGIN" "INTEGER" J, K; K:= 0; "FOR" J:= 1, J + K "WHILE" K < N "DO" "BEGIN" K:= K +1 ; ELMVEC(J, J + K - 1, 1 - J, H, V, V[K] * C) "END" "END" RNK1UPD; "EOP" 0"CODE" 34212; "PROCEDURE" DAVUPD(H, N, V, W, C1, C2); "VALUE" N, C1, C2; "INTEGER" N; "REAL" C1, C2; "ARRAY" H, V, W; "BEGIN" "INTEGER" I, J, K; "REAL" VK, WK; K:= 0; "FOR" J:= 1, J + K "WHILE" K < N "DO" "BEGIN" K:= K +1 ; VK:= V[K] * C1; WK:= W[K] * C2; "FOR" I:= 0 "STEP" 1 "UNTIL" K -1 "DO" H[I + J]:= H[I + J] + V[I + 1] * VK - W[I + 1] * WK "END" "END" DAVUPD; "EOP" 0"CODE" 34213; "PROCEDURE" FLEUPD(H, N, V, W, C1, C2); "VALUE" N, C1, C2; "INTEGER" N; "REAL" C1, C2; "ARRAY" H, V, W; "BEGIN" "INTEGER" I, J, K; "REAL" VK, WK; K:= 0; "FOR" J:= 1, J + K "WHILE" K < N "DO" "BEGIN" K:= K +1; VK:= - W[K] * C1 + V[K] * C2; WK:= V[K] * C1; "FOR" I:= 0 "STEP" 1 "UNTIL" K - 1 "DO" H[I + J]:= H[I + J] + V[I + 1] * VK -W[I + 1] * WK "END" "END" FLEUPD; "EOP" 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 1 AUTHOR: J.C.P. BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 741101. BRIEF DESCRIPTION: THIS SECTION CONTAINS THE PROCEDURE PRAXIS; PRAXIS MINIMIZES A FUNCTION OF SEVERAL VARIABLES. KEYWORDS: MINIMIZATION, FUNCTION OF SEVERAL VARIABLES. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "PROCEDURE" PRAXIS(N, X, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, IN, OUT; "REAL" "PROCEDURE" FUNCT; "CODE" 34432; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF VARIABLES OF THE FUNCTION TO BE MINIMIZED; X: ; "ARRAY" X[1 : N]; THE VARIABLES OF THE FUNCTION; ENTRY: AN APPROXIMATION OF THE POSITION OF THE MINIMUM; EXIT: THE CALCULATED POSITION OF THE MINIMUM; FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD BE: "REAL" "PROCEDURE" FUNCT(N, X); "VALUE" N; "INTEGER" N; "ARRAY" X; FUNCT SHOULD DELIVER THE VALUE OF THE FUNCTION TO BE MINIMIZED, AT THE POINT GIVEN BY X[1:N]; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF VARIABLES; X: ; "ARRAY" X[1:N]; THE VALUES OF THE VARIABLES FOR WHICH THE FUNCTION HAS TO BE EVALUATED; IN: ; "ARRAY" IN[0:9]; ENTRY: IN[0]: THE MACHINE PRECISION; FOR THE CYBER 73 A SUITABLE VALUE IS "-14; 1SECTION : 5.1.2.2.2 (OCTOBER 1975) PAGE 2 IN[1], IN[2]: RELATIVE AND ABSOLUTE TOLERANCE, RESPECTIVELY, FOR THE STEPVECTOR (RELATIVE TO THE CURRENT ESTIMATES OF THE VARIABLES); THE PROCESS IS TERMINATED WHEN IN IN[8] + 1 SUCCESSIVE ITERATION STEPS THE EUCLIDEAN NORM OF THE STEP VECTOR IS LESS THAN (IN[1] * NORM(X) + IN[2]) * 0.5; IN[1] SHOULD BE CHOSEN IN AGREEMENT WITH THE PRECISION IN WHICH THE FUNCTION IS CALCULATED; USUALLY IN[1] SHOULD BE CHOSEN SUCH THAT IN[1] >= SQRT(IN[0]); IN[0] SHOULD BE CHOSEN DIFFERENT FROM ZERO. IN[3], IN[4] ARE NEITHER USED NOR CHANGED; IN[5]: THE MAXIMUM NUMBER OF FUNCTION EVALUATIONS ALLOWED (I.E. CALLS OF FUNCT); IN[6]: THE MAXIMUM STEP SIZE; IN[6] SHOULD BE EQUAL TO THE MAXIMUM EXPECTED DISTANCE BETWEEN THE GUESS AND THE MINIMUM; IF IN[6] IS TOO SMALL OR TOO LARGE, THEN THE INITIAL RATE OF CONVERGENCE WILL BE SLOW; IN[7]: THE MAXIMUM SCALING FACTOR; THE VALUE OF IN[7] MAY BE USED TO OBTAIN AUTOMATIC SCALING OF THE VARIABLES; HOWEVER, THIS SCALING IS WORTHWHILE BUT MAY BE UNRELIABLE; THEREFORE, THE USER SHOULD TRY TO SCALE HIS PROBLEM HIMSELF AS WELL AS POSSIBLE AND SET IN[7]:= 1; IN EITHER CASE, IN[7] SHOULD NOT BE CHOSEN GREATER THAN 10; IN[8]: THE PROCESS TERMINATES IF NO SUBSTANTIAL IMPROVEMENT OF THE VALUES OF THE VARIABLES IS OBTAINED IN IN[8] + 1 SUCCESSIVE ITERATION STEPS (SEE IN[1], IN[2]); IN[8] = 4 IS VERY CAUTIOUS; USUALLY, IN[8] = 1 IS SATISFACTORY; IN[9]: IF THE PROBLEM IS KNOWN TO BE ILL-CONDITIONED (SEE [1]), THEN THE VALUE OF IN[9] SHOULD BE NEGATIVE, OTHERWISE IN[9] >= 0; OUT: ; "ARRAY" OUT[1:6]; EXIT: OUT[1]: THIS VALUE GIVES INFORMATION ABOUT THE TERMINATION OF THE PROCESS; OUT[1] = 0: NORMAL TERMINATION; OUT[1] = 1: THE PROCESS IS BROKEN OFF, BECAUSE, AT THE END OF AN ITERATION STEP, THE NUMBER OF CALLS OF FUNCT EXCEEDED THE VALUE GIVEN IN IN[5]; OUT[1] = 2: THE PROCESS IS BROKEN OFF, BECAUSE THE CONDITION OF THE PROBLEM IS TOO BAD; OUT[2]: THE CALCULATED MINIMUM OF THE FUNCTION; OUT[3]: THE VALUE OF THE FUNCTION AT THE INITIAL GUESS; OUT[4]: THE NUMBER OF FUNCTION EVALUATIONS NEEDED TO OBTAIN THIS RESULT; OUT[5]: THE NUMBER OF LINE SEARCHES (SEE [1]); OUT[6]: THE STEP SIZE IN THE LAST ITERATION STEP. 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 3 PROCEDURES USED: INIVEC = CP31010, INIMAT = CP31011, DUPVEC = CP31030, DUPMAT = CP31035, DUPCOLVEC = CP31034, MULROW = CP31021, MULCOL = CP31022, VECVEC = CP34010, TAMMAT = CP34014, MATTAM = CP34015, ICHROWCOL = CP34033, ELMVECCOL = CP34021, QRISNGVALDEC = CP34273, SETRANDOM = CP11014, RANDOM = CP11015, DWARF = CP30003. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: ONE ARRAY OF LENGTH N SQUARED AND FIVE ARRAYS OF LENGTH N ARE DECLARED; RUNNING TIME: THE NUMBER OF ITERATION STEPS DEPENDS STRONGLY ON THE PROBLEM TO BE SOLVED. METHOD AND PERFORMANCE: THIS PROCEDURE IS ADOPTED FROM [1]. REFERENCES: [1] R. P. BRENT, ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES, CH. 7. PRENTICE HALL, 1973. 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 4 EXAMPLE OF USE: THE FOLLOWING PROGRAM MAY BE USED TO CALCULATE THE MINIMUM OF THE FUNCTION F(X) = 100 * (X[2] - X[1] ** 2) ** 2 + (1 - X[1]) ** 2, USING (-1.2, 1) AS AN INITIAL ESTIMATE. "BEGIN" "ARRAY" X[1:2], IN[0:9], OUT[1:6]; "REAL" "PROCEDURE" F(N, X); "VALUE" N; "INTEGER" N; "ARRAY" X; F:= (X[2] - X[1] ** 2) ** 2 * 100 + (1 - X[1]) ** 2; IN[0]:= "-14; IN[1]:= IN[2]:= "-6; IN[5]:= 250; IN[6]:= 1; IN[7]:= 1; IN[8]:= 1; IN[9]:= 1; X[1]:= -1.2; X[2]:= 1; PRAXIS(2, X, F, IN, OUT); "IF" OUT[1] = 0 "THEN" OUTPUT(61,"(""(" NORMAL TERMINATION")" ,//")"); OUTPUT(61, "("4B,"("MINIMUM IS ")",N,/,4B, "("FOR X IS ")", 2(N),/,4B, "("THE INITIAL FUNCTION VALUE WAS ")" ,N,/,4B, "("THE NUMBER OF FUNCTION EVALUATIONS NEEDED WAS ")",3ZD,/,4B, "("THE NUMBER OF LINE SEARCHES WAS ")",3ZD,/,4B, "("THE STEP SIZE IN THE LAST ITERATION STEP WAS ")", N,/")", OUT[2], X[1], X[2], OUT[3], OUT[4], OUT[5], OUT[6]) "END" RESULTS: NORMAL TERMINATION MINIMUM IS +1.5694986738789"-021 FOR X IS +1.0000000000389"+000 +1.0000000000785"+000 THE INITIAL FUNCTION VALUE WAS +2.4200000000001"+001 THE NUMBER OF FUNCTION EVALUATIONS NEEDED WAS 189 THE NUMBER OF LINE SEARCHES WAS 72 THE STEP SIZE IN THE LAST ITERATION STEP WAS +5.3830998470105"-009 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 5 SOURCE TEXT(S): 0"CODE" 34432; "PROCEDURE" PRAXIS(N, X, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, IN, OUT; "REAL" "PROCEDURE" FUNCT; "BEGIN" "COMMENT"THIS PROCEDURE MINIMIZES FUNCT(N,X),WITH THE PRINCIPAL AXIS METHOD (SEE BRENT,R.P, 1973, ALGORITHMS FOR MINIMIZATION WITHOUT DERIVATIVES,CH.7); "PROCEDURE" SORT; "BEGIN" "INTEGER" I, J, K; "REAL" S; "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" K:= I; S:= D[I]; "FOR" J:= I+1 "STEP" 1 "UNTIL" N "DO" "IF" D[J]>S "THEN" "BEGIN" K:= J; S:= D[J] "END"; "IF" K>I "THEN" "BEGIN" D[K]:= D[I]; D[I]:= S; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" S:=V[J,I]; V[J,I]:= V[J,K]; V[J,K]:= S "END" "END" "END" "END" SORT 1SECTION : 5.1.2.2.2 (OCTOBER 1975) PAGE 6 ; "PROCEDURE" MIN(J, NITS, D2, X1, F1, FK); "VALUE" J, NITS, FK; "INTEGER" J, NITS; "REAL" D2, X1, F1; "BOOLEAN" FK; "BEGIN" "REAL" "PROCEDURE" FLIN(L); "VALUE" L; "REAL" L; "BEGIN" "INTEGER" I; "ARRAY" T[1:N]; "IF" J > 0 "THEN" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" T[I]:= X[I] + L * V[I,J] "END" "ELSE" "BEGIN" "COMMENT" SEARCH ALONG PARABOLIC SPACE CURVE; QA:= L * (L - QD1) / (QD0 * (QD0 + QD1)); QB:= (L + QD0) * (QD1 - L) /(QD0 * QD1); QC:= L * (L + QD0) / (QD1 * (QD0 + QD1)); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" T[I]:= QA * Q0[I] +QB * X[I] + QC * Q1[I] "END"; NF:= NF + 1; FLIN:= FUNCT(N, T) "END" FLIN; "INTEGER" K; "BOOLEAN" DZ; "REAL" X2, XM, F0, F2, FM, D1, T2, S, SF1, SX1; SF1:= F1; SX1:= X1; K:= 0; XM:= 0; F0:= FM:= FX; DZ:= D2 < RELTOL; S:= SQRT(VECVEC(1,N,0,X,X)); T2:= M4 * SQRT(ABS(FX) / ("IF" DZ "THEN" DMIN "ELSE" D2) + S * LDT) + M2 * LDT; S:= S * M4 + ABSTOL; "IF" DZ "AND" T2 > S "THEN" T2:= S; "IF"T20.01*H "THEN"T2:= 0.01*H; "IF"FK"AND"F1<=FM "THEN" "BEGIN"XM:=X1; FM:= F1 "END"; "IF" ^ FK"OR"ABS(X1)0 "THEN"T2"ELSE"-T2; F1:= FLIN(X1) "END"; "IF"F1<= FM"THEN" "BEGIN"XM:= X1; FM:= F1 "END"; L0: "IF" DZ "THEN" "BEGIN" "COMMENT"EVALUATE FLIN AT ANOTHER POINT AND ESTIMATE THE SECOND DERIVATIVE; X2:= "IF" F0 < F1 "THEN" -X1 "ELSE" X1 * 2; F2:= FLIN(X2); "IF"F2 <= FM "THEN" "BEGIN" XM:= X2; FM:= F2 "END"; D2:=(X2*(F1-F0)-X1*(F2-F0))/(X1*X2*(X1-X2)) "END"; "COMMENT"ESTIMATE FIRST DERIVATIVE AT 0; D1:=(F1-F0)/X1-X1*D2; DZ:="TRUE"; X2:= "IF"D2<=SMALL"THEN" ("IF"D1<0"THEN"H"ELSE"-H) "ELSE"-0.5*D1/D2; "IF"ABS(X2)>H"THEN"X2:="IF"X2>0"THEN"H"ELSE"-H; "COMMENT" 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 7 ; L1: F2:=FLIN(X2); "IF"KF0"THEN" "BEGIN"K:=K+1; "IF"F00"THEN" "GOTO"L0; X2:= 0.5*X2; "GOTO"L1 "END"; NL:= NL+1; "IF"F2>FM"THEN"X2:=XM"ELSE"FM:=F2; D2:="IF"ABS(X2*(X2-X1))>SMALL"THEN" (X2*(F1-F0)-X1*(FM-F0))/(X1*X2*(X1-X2)) "ELSE" "IF"K>0"THEN"0"ELSE"D2; "IF"D2<=SMALL"THEN"D2:=SMALL; X1:=X2; FX:=FM; "IF"SF10"THEN"ELMVECCOL(1,N,J,X,V,X1) "END" MIN; "PROCEDURE"QUAD; "BEGIN" "INTEGER" I; "REAL" L, S; S:= FX; FX:= QF1; QF1:= S; QD1:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN"S:=X[I]; X[I]:= L:= Q1[I]; Q1[I]:= S; QD1:= QD1 + (S - L) ** 2 "END"; L:=QD1:=SQRT(QD1); S:= 0; "IF"(QD0*QD1>DWARF)"AND"NL>=3*N*N"THEN" "BEGIN"MIN(0,2,S,L,QF1,"TRUE"); QA:= L*(L-QD1)/(QD0*(QD0+QD1)); QB:=(L+QD0)*(QD1-L)/(QD0*QD1); QC:= L*(L+QD0)/(QD1*(QD0+QD1)) "END" "ELSE" "BEGIN" FX:= QF1; QA:= QB:= 0; QC:= 1 "END"; QD0:= QD1;"FOR"I:= 1"STEP"1"UNTIL"N"DO" "BEGIN"S:=Q0[I]; Q0[I]:=X[I]; X[I]:= QA*S + QB*X[I]+QC*Q1[I] "END" "END" QUAD; "BOOLEAN" ILLC; "INTEGER" I, J, K, K2, NL, MAXF, NF, KL, KT, KTM; "REAL" S, SL, DN, DMIN, FX, F1, LDS, LDT, SF, DF, QF1, QD0, QD1, QA, QB, QC, M2, M4, SMALL, VSMALL, LARGE, VLARGE, SCBD, LDFAC,T2, MACHEPS, RELTOL, ABSTOL, H; "ARRAY" V[1:N,1:N], D, Y, Z, Q0, Q1[1:N]; MACHEPS:= IN[0]; RELTOL:= IN[1]; ABSTOL:= IN[2]; MAXF:= IN[5]; H:= IN[6]; SCBD:= IN[7]; KTM:= IN[8]; ILLC:= IN[9] < 0; SMALL:= MACHEPS ** 2; VSMALL:= SMALL ** 2; LARGE:= 1/SMALL; VLARGE:= 1/VSMALL; M2:= RELTOL; M4:= SQRT(M2); SETRANDOM(0.5); LDFAC:= "IF" ILLC "THEN" 0.1 "ELSE" 0.01; KT:=NL:=0; NF:=1; OUT[3]:= QF1:=FX:=FUNCT(N,X); "COMMENT" 1SECTION : 5.1.2.2.2 (DECEMBER 1979) PAGE 8 ; ABSTOL:=T2:= SMALL+ABS(ABSTOL); DMIN:= SMALL; "IF" H= D[1] "THEN" INIVEC(2,N,D,0); "FOR" K:= 2"STEP"1"UNTIL"N"DO" "BEGIN" DUPVEC(1,N,0,Y,X); SF:=FX; ILLC:= ILLC "OR" KT>0; L1: KL:=K; DF:= 0; "IF" ILLC "THEN" "BEGIN" "COMMENT"RANDOM STOP TO GET OFF RESULTION VALLEY; "FOR"I:= 1 "STEP"1"UNTIL"N"DO" "BEGIN"S:=Z[I]:=(0.1*LDT+T2*10**KT) *(RANDOM-0.5); ELMVECCOL(1,N,I,X,V,S) "END"; FX:= FUNCT(N,X); NF:= NF+1 "END"; "FOR"K2:= K "STEP" 1 "UNTIL" N "DO" "BEGIN" SL:=FX; S:= 0; MIN (K2, 2, D[K2], S, FX, "FALSE"); S:="IF" ILLC "THEN" D[K2] * (S + Z[K2]) ** 2 "ELSE"SL-FX;"IF"DF SMALL "THEN" "BEGIN" "FOR" I:= KL - 1 "STEP" -1 "UNTIL" K "DO" "BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" V[J, I + 1]:= V[J,I]; D[I + 1]:= D[I] "END"; D[K]:= 0; DUPCOLVEC(1, N, K, V, Y); MULCOL(1, N, K, K, V, V, 1 / LDS); MIN(K, 4, D[K], LDS, F1, "TRUE"); "IF" LDS <= 0 "THEN" "BEGIN" LDS:= LDS; MULCOL(1, N, K, K, V, V, -1) "END" "END"; LDT:= LDFAC * LDT; "IF" LDT < LDS "THEN" LDT:= LDS; T2:= M2 * SQRT(VECVEC(1, N, 0, X, X)) + ABSTOL; "COMMENT" 1SECTION : 5.1.2.2.2 (OCTOBER 1975) PAGE 9 ; KT:= "IF" LDT > 0.5 * T2 "THEN" 0 "ELSE" KT + 1; "IF" KT > KTM "THEN" "BEGIN" OUT[1]:= 0; "GOTO" L2 "END" "END"; QUAD; DN:= 0;"FOR"I:= 1"STEP"1"UNTIL"N"DO" "BEGIN"D[I]:= 1/SQRT(D[I]); "IF"DN1"THEN" "BEGIN"S:=VLARGE; "FOR"I:=1 "STEP"1"UNTIL"N"DO" "BEGIN" SL:= Z[I]:= SQRT(MATTAM(1, N, I, I, V, V)); "IF"SLSL "THEN" S:= SL "END"; "FOR"I:=1"STEP"1"UNTIL"N"DO" "BEGIN"SL:=S/Z[I];Z[I]:= 1/SL; "IF"Z[I]>SCBD"THEN" "BEGIN"SL:=1/SCBD; Z[I]:= SCBD"END"; MULROW(1, N, I, I, V, V, SL) "END" "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" ICHROWCOL(I + 1, N, I, I, V); "BEGIN" "ARRAY" A[1:N,1:N], EM[0:7]; EM[0]:= EM[2]:= MACHEPS; EM[4]:= 10 * N; EM[6]:= VSMALL; DUPMAT(1, N, 1, N, A, V); "IF" QRISNGVALDEC(A, N, N, D, V, EM) ^= 0 "THEN" "BEGIN" OUT[1]:= 2; "GOTO" L2 "END"; "END"; "IF"SCBD>1"THEN" "BEGIN" "FOR"I:=1"STEP"1"UNTIL"N"DO" MULROW(1,N,I,I,V,V,Z[I]); "FOR"I:= 1"STEP"1"UNTIL"N"DO" "BEGIN"S:= SQRT(TAMMAT(1,N,I,I,V,V)); D[I]:= S*D[I]; S:= 1/S; MULCOL(1,N,I,I,V,V,S) "END" "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" S:= DN * D[I]; D[I]:= "IF" S > LARGE "THEN" VSMALL "ELSE" "IF" S < SMALL "THEN" VLARGE "ELSE" S ** (-2) "END"; SORT; DMIN:= D[N]; "IF" DMIN < SMALL "THEN" DMIN:= SMALL; ILLC:= (M2 * D[1]) > DMIN; "IF" NF < MAXF "THEN" "GOTO" L0 "ELSE" OUT[1]:= 1; L2: OUT[2]:= FX; OUT[4]:= NF; OUT[5]:= NL; OUT[6]:= LDT "END"PRAXIS; "EOP" 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 1 AUTHOR: J.C.P.BUS. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730620. BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO PROCEDURES, RNK1MIN AND FLEMIN, FOR MINIMIZING A GIVEN DIFFERENTIABLE FUNCTION OF SEVERAL VARIABLES; BOTH PROCEDURES USE A VARIABLE METRIC METHOD; THE USER HAS TO PROGRAM THE EVALUATION OF THE FUNCTION AND ITS GRADIENT; THE CHOICE OF RNK1MIN AND FLEMIN IS DEPENDENT ON THE PROBLEM INVOLVED; IF THE NUMBER OF VARIABLES OF THE FUNCTION TO BE MINIMIZED IS VERY LARGE AND THE CALCULATION OF THE FUNCTION AND ITS GRADIENT IS RELATIVELY CHEAP (THE NUMBER OF ARITHMETIC OPERATIONS IS OF ORDER AT MOST N ** 2),THEN THE USER IS ADVISED TO USE FLEMIN; IF THE HESSIAN OF THE FUNCTION IS EXPECTED TO BE (ALMOST) SINGULAR AT THE MINIMUM, THEN RNK1MIN IS PREFERRED; KEYWORDS: OPTIMIZATION, HIGHER - DIMENSIONAL, UNCONSTRAINED, VARIABLE METRIC METHOD. 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 2 SUBSECTION: RNK1MIN. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "REAL" "PROCEDURE" RNK1MIN(N, X, G, H, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, G, H, IN, OUT; "REAL" "PROCEDURE" FUNCT; "CODE" 34214; RNK1MIN: DELIVERS THE CALCULATED LEAST VALUE OF THE GIVEN FUNCTION; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF VARIABLES OF THE FUNCTION TO BE MINIMIZED; X: ; "ARRAY" X[1 : N]; THE INDEPENDENT VARIABLES; ENTRY: AN APPROXIMATION OF A MINIMUM OF THE FUNCTION; EXIT: THE CALCULATED MINIMUM OF THE FUNCTION; G: ; "ARRAY" G[1 : N]; EXIT: THE GRADIENT OF THE FUNCTION AT THE CALCULATED MINIMUM; H: ; "ARRAY" H[1 : N * (N + 1) // 2]; THE UPPERTRIANGLE OF AN APPROXIMATION OF THE INVERSE HESSIAN IS STORED COLUMNWISE IN H (I.E. THE I,J-TH ELEMENT= H[ (J - 1) * J // 2 + I], 1 <= I<= J <= N ); IF IN[6] > 0 INITIALIZING OF H WILL BE DONE AUTOMATICALLY AND THE INITIAL APPROXIMATION OF THE INVERSE HESSIAN WILL EQUAL THE UNITMATRIX MULTIPLIED WITH THE VALUE OF IN[6]; IF IN[6] < 0 NO INITIALIZING OF H WILL BE DONE AND THE USER SHOULD GIVE IN H AN APPROXIMATION OF THE INVERSE HESSIAN, AT THE STARTING POINT;THE UPPERTRIANGLE OF AN APPROXIMATION OF THE INVERSE HESSIAN AT THE CALCULATED MINIMUM IS DELIVERED IN H; FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD BE: "REAL" "PROCEDURE" FUNCT(N, X, G); "VALUE" N; "INTEGER" N; "ARRAY" X, G; A CALL OF FUNCT MUST EFFECTUATE IN: 1: FUNCT BECOMES THE VALUE OF THE FUNCTION TO BE MINIMIZED AT THE POINT X; 2: THE VALUE OF G[I], (I = 1, ..., N), BECOMES THE VALUE OF THE I - TH COMPONENT OF THE GRADIENT OF THE FUNCTION AT X; 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 3 IN: ; "ARRAY" IN[0 : 8]; ENTRY: IN[0]: THE MACHINE PRECISION; FOR THE CYBER 73-26 A SUITABLE VALUE IS "-14; IN[1]: THE RELATIVE TOLERANCE FOR THE SOLUTION; THIS TOLERANCE SHOULD NOT BE CHOSEN SMALLER THAN IN[0]; IN[2]: THE ABSOLUTE TOLERANCE FOR THE SOLUTION; IN[3]; A PARAMETER USED FOR CONTROLLING LINEMINIMIZATION, ([3], [4]); USUALLY A SUITABLE VALUE IS: 0.0001; IN[4]: THE ABSOLUTE TOLERANCE FOR THE EUCLIDEAN NORM OF THE GRADIENT AT THE SOLUTION; IN[5]: A LOWERBOUND FOR THE FUNCTIONVALUE; IN[6]: THIS PARAMETER CONTROLS THE INITIALIZATION OF THE APPROXIMATION OF THE INVERSE HESSIAN (METRIC), SEE H; USUALLY THE CHOICE IN[6] = 1 WILL GIVE GOOD RESULTS; IN[7]: THE MAXIMUM ALLOWED NUMBER OF CALLS OF FUNCT; IN[8]: A PARAMETER USED FOR CONTROLLING THE UPDATING OF THE METRIC; IT IS USED TO AVOID UNBOUNDEDNESS OF THE METRIC (SEE: [6], FORMULA (19)); THE VALUE OF IN[8] SHOULD SATISFY: SQRT(IN[0] / IN[1]) / N < IN[8] < 1; USUALLY A SUITABLE VALUE WILL BE 0.01; OUT: ; "ARRAY" OUT[0:4]; EXIT: OUT[0]: THE EUCLIDEAN NORM OF THE PRODUCT OF THE METRIC AND THE GRADIENT AT THE CALCULATED MINIMUM; OUT[1]: THE EUCLIDEAN NORM OF THE GRADIENT AT THE CALCULATED MINIMUM; OUT[2]: THE NUMBER OF CALLS OF FUNCT, NECESSARY TO ATTAIN THIS RESULT; OUT[3]: THE NUMBER OF ITERATIONS IN WHICH A LINESEARCH WAS NECESSARY; OUT[4]: THE NUMBER OF ITERATIONS IN WHICH A DIRECTION HAD TO BE CALCULATED WITH THE METHOD GIVEN IN [5]; IN SUCH AN ITERATION A CALCULATION OF THE EIGENVALUES AND EIGENVECTORS OF THE METRIC IS NECESSARY. DATA AND RESULTS: USUALLY THE CALCULATED SOLUTION WILL SATISFY: NORM ( XMIN - XCAL ) < NORM ( XCAL ) * IN[1] + IN[2]. WHERE AT XMIN THE GIVEN FUNCTION IS MINIMAL, XCAL THE CALCULATED APPROXIMATION OF XMIN AND NORM( . ) DENOTES THE EUCLIDEAN NORM OF X; HOWEVER, WE CANNOT GUARANTEE SUCH A RESULT; THE CALCULATED SOLUTION POSSIBLY WILL NOT SATISFY THE ABOVE INEQUALITY IF THE PROBLEM IS VERY ILL - CONDITIONED; THE USER CAN DISCOVER SUCH A SITUATION BY LOOKING AT THE EUCLIDEAN NORM OF THE METRIC, DELIVERED IN H; THE PROBLEM IS ILL - CONDITIONED IF THIS NORM IS LARGE RELATIVE TO 1. 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 4 PROCEDURES USED: VECVEC = CP34010, MATVEC = CP34011, TAMVEC = CP34012, SYMMATVEC = CP34018, INIVEC = CP31010, INISYMD = CP31013, MULVEC = CP31020, DUPVEC = CP31030, EIGSYM1 = CP34156, LINEMIN = CP34210, RNK1UPD = CP34211, DAVUPD = CP34212, FLEUPD = CP34213. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: VARIES FROM 5 * N + 26 TO N ** 2 + N * (N + 1) // 2 + 5 * N + 35 WORDS. RUNNING TIME: DEPENDS STRONGLY ON THE PROBLEM TO BE SOLVED; LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: RNK1MIN CALCULATES AN APPROXIMATION OF A MINIMUM OF A GIVEN FUNCTION BY MEANS OF A VARIABLE METRIC METHOD; THE RANK - ONE UPDATING FORMULA, USED IN THIS ALGORITHM IS GIVEN IN [6], (FORMULA (4)); TO AVOID UNBOUNDEDNESS OF THE METRIC (SEE [8]), SOMETIMES A RANK - TWO UPDATING FORMULA IS USED ([3], FORMULAS (1) AND (5)); TO AVOID LINESEARCHES AS MUCH AS POSSIBLE A STRATEGY GIVEN IN [4] IS USED; IF IN AN ITERATION THE FUNCTION IS INCREASING IN THE DIRECTION GIVEN BY THE VARIABLE METRIC ALGORITHM, BECAUSE THE METRIC IS NOT POSITIVE DEFINITE, THEN A METHOD GIVEN IN [5] IS USED TO CALCULATE A NEW DIRECTION; THIS METHOD REQUIRES THE CALCULATION OF THE EIGENVECTORS AND EIGENVALUES OF THE METRIC; USUALLY,THE NUMBER OF TIMES SUCH A CALCULATION IS NECESSARY IS VERY SMALL RELATIVE TO THE NUMBER OF ITERATIONS (AND OFTEN EQUALS ZERO); IF THE NUMBER OF VARIABLES OF THE FUNCTION IS VERY LARGE AND THE CALCULATION OF THE FUNCTION AND ITS GRADIENT IS RELATIVELY CHEAP (THE NUMBER OF ARITHMETICAL OPERATIONS IS OF ORDER AT MOST N ** 2), THEN THE USER IS ADVISED TO USE FLEMIN (CP32105); A DETAILED DESCRIPTION OF THE ALGORITHM AND SOME RESULTS ABOUT ITS CONVERGENCE IS GIVEN IN [1]. 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 5 SUBSECTION: FLEMIN. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "REAL" "PROCEDURE" FLEMIN(N, X, G, H, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, G, H, IN, OUT; "REAL" "PROCEDURE" FUNCT; "CODE" 34215; FLEMIN: DELIVERS THE CALCULATED LEAST VALUE OF THE GIVEN FUNCTION; THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF VARIABLES OF THE FUNCTION TO BE MINIMIZED; X: ; "ARRAY" X[1 : N]; THE INDEPENDENT VARIABLES; ENTRY: AN APPROXIMATION OF A MINIMUM OF THE FUNCTION; EXIT: THE CALCULATED MINIMUM OF THE FUNCTION; G: ; "ARRAY" G[1 : N]; EXIT: THE GRADIENT OF THE FUNCTION AT THE CALCULATED MINIMUM; H: ; "ARRAY" H[1 : N * (N + 1) // 2]; THE UPPERTRIANGLE OF AN APPROXIMATION OF THE INVERSE HESSIAN IS STORED COLUMNWISE IN H (I.E. THE I,J-TH ELEMENT= H[ (J - 1) * J // 2 + I], 1 <= I <= J <= N); IF IN[6] > 0 INITIALIZING OF H WILL BE DONE AUTOMATICALLY AND THE INITIAL APPROXIMATION OF THE INVERSE HESSIAN WILL EQUAL THE UNITMATRIX MULTIPLIED WITH THE VALUE OF IN[6]; IF IN[6] < 0 NO INITIALIZING OF H WILL BE DONE AND THE USER SHOULD GIVE IN H AN APPROXIMATION OF THE INVERSE HESSIAN AT THE STARTING POINT; THE UPPERTRIANGLE OF AN APPROXIMATION OF THE INVERSE HESSIAN AT THE CALCULATED MINIMUM IS DELIVERED IN H; FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD BE : "REAL" "PROCEDURE" FUNCT(N, X, G); "VALUE" N; "INTEGER" N; "ARRAY" X, G; A CALL OF FUNCT SHOULD EFFECTUATE IN: 1: FUNCT BECOMES THE VALUE OF THE FUNCTION TO BE MINIMIZED AT THE POINT X; 2: THE VALUE OF G[I], (I = 1, ..., N), BECOMES THE VALUE OF THE I - TH COMPONENT OF THE GRADIENT OF THE FUNCTION AT X; 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 6 IN: ; "ARRAY" IN[1 : 7]; ENTRY: IN[1]: THE RELATIVE TOLERANCE FOR THE SOLUTION; IN[2]: THE ABSOLUTE TOLERANCE FOR THE SOLUTION; IN[3]: A PARAMETER USED FOR CONTROLLING LINEMINIMIZATION ([3], [4]); USUALLY A SUITABLE VALUE IS 0.0001; IN[4]: THE ABSOLUTE TOLERANCE FOR THE EUCLIDEAN NORM OF THE GRADIENT AT THE SOLUTION; IN[5]: A LOWERBOUND FOR THE FUNCTION VALUE; IN[6]: THIS PARAMETER CONTROLS THE INITIALIZATION OF THE APPROXIMATION OF THE INVERSE HESSIAN (METRIC) (SEE H); USUALLY IN[6] = 1 WILL GIVE GOOD RESULTS; IN[7]: THE MAXIMUM ALLOWED NUMBER OF CALLS OF FUNCT; OUT: ; "ARRAY" OUT[0:4]; EXIT: OUT[0]: THE EUCLIDEAN NORM OF THE PRODUCT OF THE METRIC AND THE GRADIENT AT THE CALCULATED MINIMUM; OUT[1]: THE EUCLIDEAN NORM OF THE GRADIENT AT THE CALCULATED MINIMUM; OUT[2]: THE NUMBER OF CALLS OF FUNCT, NECESSARY TO ATTAIN THESE RESULTS; OUT[3]: THE NUMBER OF ITERATIONS IN WHICH A LINESEARCH WAS NECESSARY; OUT[4]: IF OUT[4] = - 1, THEN THE PROCESS IS BROKEN OFF BECAUSE NO DOWNHILL DIRECTION COULD BE CALCULATED; THE PRECISION ASKED FOR MAY NOT BE ATTAINED AND IS POSSIBLY CHOSEN TOO HIGH; NORMALLY OUT[4] = 0; DATA AND RESULTS: USUALLY THE CALCULATED SOLUTION WILL SATISFY: NORM ( XMIN - XCAL ) < NORM ( XCAL ) * IN[1] + IN[2]. WHERE AT XMIN THE GIVEN FUNCTION IS MINIMAL, XCAL THE CALCULATED APPROXIMATION OF XMIN AND NORM ( . ) DENOTES THE EUCLIDEAN NORM OF X; HOWEVER, WE CAN NOT GUARANTEE SUCH A RESULT; THE CALCULATED SOLUTION POSSIBLY WILL NOT SATISFY THE ABOVE INEQUALITY IF THE PROBLEM IS VERY ILL - CONDITIONED; THE USER CAN DISCOVER SUCH A SITUATION BY LOOKING AT THE EUCLIDEAN NORM OF THE METRIC, DELIVERED IN H; THE PROBLEM IS ILL - CONDITIONED IF THIS NORM IS LARGE RELATIVE TO 1. 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 7 PROCEDURES USED: VECVEC = CP34010, ELMVEC = CP34020, SYMMATVEC = CP34018, INIVEC = CP31010, INISYMD = CP31013, MULVEC = CP31020, DUPVEC = CP31030, LINEMIN = CP34210, DAVUPD = CP34212, FLEUPD = CP34213. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: 3 * N + 19 WORDS. RUNNING TIME: DEPENDS STRONGLY ON THE PROBLEM TO BE SOLVED; LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: FLEMIN CALCULATES AN APPROXIMATION OF A MINIMUM OF A GIVEN FUNCTION BY MEANS OF THE VARIABLE METRIC ALGORITHM GIVEN IN [3], EXCEPT FOR SOME DETAILS (SEE [1]). REFERENCES: [1] BUS, J. C. P. MINIMIZATION OF FUNCTIONS OF SEVERAL VARIABLES (DUTCH). MATHEMATICAL CENTRE, AMSTERDAM, NR 29/72 (1972). [2] DAVIDON, W. C. VARIABLE METRIC METHOD FOR MINIMIZATION. ARGONNE NAT. LAB. REPORT, ANL 5990 (1959). [3] FLETCHER, R. A NEW APPROACH TO VARIABLE METRIC ALGORITHMS. COMP. J. 6, (1963), P.163 - 168. [4] GOLDSTEIN, A. A. AND PRICE, J. F. AN EFFECTIVE ALGORITHM FOR MINIMIZATION. NUMER. MATH. 10, (1967), P.184 - 189. [5] GREENSTADT, J. L. ON THE RELATIVE EFFICIENCIES OF GRADIENT METHODS. MATH. COMP. 21, (1967), P.360 - 367. [6] POWELL, M. J. D. RANK ONE METHODS FOR UNCONSTRAINED OPTIMIZATION. IN: ABADIE, J. (ED.) INTEGER AND NONLINEAR PROGRAMMING. NORTH - HOLLAND, (1970). 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 8 EXAMPLE OF USE: THE MINIMUM OF THE FUNCTION: F(X) = (X[2] - X[1] ** 2) ** 2 * 100 + (1 - X[1]) ** 2, CALCULATED WITH BOTH RNK1MIN AND FLEMIN MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "REAL" "PROCEDURE" ROSENBROCK(N, X, G); "VALUE" N; "INTEGER" N; "ARRAY" X, G; "BEGIN" ROSENBROCK:= (X[2] - X[1] ** 2) ** 2 * 100 + (1 - X[1]) ** 2; G[1]:= ((X[1] ** 2 - X[2]) * 400 + 2) * X[1] - 2; G[2]:=(X[2] - X[1] ** 2) * 200 "END" ROSENBROCK; "INTEGER" I; "BOOLEAN" AGAIN; "REAL" F; "ARRAY" X, G[1:2], H[1:3], IN[0:8], OUT[0:4]; IN[0]:= "-14; IN[1]:= "-5; IN[2]:= "-5; IN[3]:= "-4; IN[4]:= "-5; IN[5]:= -10; IN[6]:= 1; IN[7]:= 100; IN[8]:= 0.01; X[1]:= -1.2; X[2]:= 1; AGAIN:= "TRUE"; F:= RNK1MIN(2, X, G, H, ROSENBROCK, IN, OUT); "GOTO" PRINT; NEXT: X[1]:= -1.2; X[2]:= 1; AGAIN:= "FALSE"; F:= FLEMIN(2, X, G, H, ROSENBROCK, IN, OUT); PRINT: OUTPUT(61, "(""("LEAST VALUE:")"B+.15D"+3D,//, "("X:")", 2(B+.15D"+3DB),//,"("GRADIENT:")", 2(B+.15D"+3DB),//, "("METRIC:")" ,2(B+.15D"+3DB),/,32B+.15D"+3D,//,"("OUT:")", 5(B+.15D"+3DB,/),/// ")",F, X[1], X[2], G[1], G[2], H[1], H[2], H[3], OUT[0], OUT[1], OUT[2], OUT[3], OUT[4]); "IF" AGAIN "THEN" "GOTO" NEXT "END" DELIVERS: 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 9 LEAST VALUE: +.200699798801180"-018 X: +.999999999944840"+000 +.999999999845220"+000 GRADIENT: +.176731305145950"-007 -.889173179530190"-008 METRIC: +.499982414863250"+000 +.999957383810230"+000 +.200489757679290"+001 OUT: +.164157123774660"-009 +.197838933606480"-007 +.550000000000000"+002 +.800000000000000"+001 +.400000000000000"+001 LEAST VALUE: +.811973499921290"-016 X: +.999999999758770"+000 +.999999998616780"+000 GRADIENT: +.359826657359010"-006 -.180154557938290"-006 METRIC: +.501085356975550"+000 +.100198139199600"+001 +.200861655543510"+001 OUT: +.133802289387830"-008 +.402406371833370"-006 +.440000000000000"+002 +.700000000000000"+001 +.000000000000000"+000 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 10 SOURCE TEXT(S): 0"CODE" 34214; "REAL" "PROCEDURE" RNK1MIN(N, X, G, H, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, G, H, IN, OUT; "REAL" "PROCEDURE" FUNCT; "BEGIN" "INTEGER" I, IT, N2, CNTL, CNTE, EVL, EVLMAX; "BOOLEAN" OK; "REAL" F, F0, FMIN, MU, DG, DG0, GHG, GS, NRMDELTA, ALFA, MACHEPS, RELTOL, ABSTOL, EPS, TOLG, ORTH, AID; "ARRAY" V, DELTA, GAMMA, S, P[1:N]; MACHEPS:= IN[0]; RELTOL:= IN[1]; ABSTOL:= IN[2]; MU:= IN[3]; TOLG:= IN[4]; FMIN:= IN[5]; IT := 0; ALFA:= IN[6]; EVLMAX:= IN[7]; ORTH:= IN[8]; N2:= N * (N + 1) // 2; CNTL:= CNTE:= 0; "IF" ALFA > 0 "THEN" "BEGIN" INIVEC(1, N2, H, 0); INISYMD(1, N, 0, H, ALFA) "END"; F:= FUNCT(N, X, G); EVL:= 1; DG:= SQRT(VECVEC(1, N, 0, G, G)); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DELTA[I]:= - SYMMATVEC(1, N, I, H, G); NRMDELTA:= SQRT(VECVEC(1, N, 0, DELTA, DELTA)); DG0:= VECVEC(1, N, 0, DELTA, G); OK:= DG0 < 0; EPS:= SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL; "FOR" IT:= IT + 1 "WHILE" (NRMDELTA > EPS "OR" DG > TOLG "OR" ^ OK) "AND" EVL < EVLMAX "DO" "BEGIN" "IF" ^OK "THEN" "BEGIN" "ARRAY" VEC[1:N,1:N], TH[1:N2], EM[0:9]; EM[0]:= MACHEPS; EM[2]:= AID:= SQRT(MACHEPS * RELTOL); EM[4]:= ORTH; EM[6]:= AID * N; EM[8]:= 5; CNTE:= CNTE + 1; DUPVEC(1, N2, 0, TH, H); EIGSYM1(TH,N,N,V,VEC,EM); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" AID:= - TAMVEC(1, N, I, VEC, G); S[I]:= AID * ABS(V[I]); V[I]:= AID * SIGN(V[I]) "END" 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 11 ; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" DELTA[I]:= MATVEC(1, N, I, VEC, S); P[I]:= MATVEC(1, N, I, VEC, V) "END"; DG0:= VECVEC(1, N, 0, DELTA, G); NRMDELTA:= SQRT(VECVEC(1, N, 0, DELTA, DELTA)) "END" CALCULATING GREENSTADTS DIRECTION; DUPVEC(1, N, 0, S, X); DUPVEC(1, N, 0, V, G); "IF" IT > N "THEN" ALFA:= 1 "ELSE" "BEGIN" "IF" IT ^= 1 "THEN" ALFA:= ALFA / NRMDELTA "ELSE" "BEGIN" ALFA:= 2 * (FMIN - F) / DG0; "IF" ALFA > 1 "THEN" ALFA:= 1 "END" "END"; ELMVEC(1, N, 0, X, DELTA, ALFA); F0:= F; F:= FUNCT(N, X, G); EVL:= EVL +1 ; DG:= VECVEC(1, N, 0, DELTA, G); "IF" IT = 1 "OR" F0 - F < -MU * DG0 * ALFA "THEN" "BEGIN" I:= EVLMAX - EVL; CNTL:= CNTL +1 ; LINEMIN(N, S, DELTA, NRMDELTA, ALFA, G, FUNCT, F0, F, DG0, DG, I, "FALSE", IN); EVL:= EVL + I; DUPVEC(1, N, 0, X, S); "END" LINEMINIMIZATION; DUPVEC(1, N, 0, GAMMA, G); ELMVEC(1, N, 0, GAMMA, V, -1); "IF" ^ OK "THEN" MULVEC(1, N, 0, V, P, -1); DG:= DG - DG0; "IF" ALFA ^= 1 "THEN" "BEGIN" MULVEC(1, N, 0, DELTA, DELTA, ALFA); MULVEC(1, N, 0, V, V, ALFA); NRMDELTA:= NRMDELTA * ALFA; DG:= DG * ALFA "END"; DUPVEC(1, N, 0, P, GAMMA); ELMVEC(1, N, 0, P, V, 1); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" V[I]:= SYMMATVEC(1, N, I, H, GAMMA); DUPVEC(1, N, 0, S, DELTA); ELMVEC(1, N, 0, S, V, -1); GS:= VECVEC(1, N, 0, GAMMA, S); GHG:= VECVEC(1, N, 0, V, GAMMA); AID:= DG / GS; "IF" VECVEC(1, N, 0, DELTA, P) ** 2 > VECVEC(1, N, 0, P, P) * (ORTH * NRMDELTA) ** 2 "THEN" RNK1UPD(H, N, S, 1 / GS) "ELSE" "IF" AID >= 0 "THEN" FLEUPD(H, N, DELTA, V, 1 / DG, (1 + GHG / DG) / DG) "ELSE" DAVUPD(H, N, DELTA, V, 1 / DG, 1 / GHG); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DELTA[I]:= -SYMMATVEC(1, N, I, H, G); ALFA:= NRMDELTA; NRMDELTA:= SQRT(VECVEC(1, N, 0, DELTA, DELTA)); EPS:= SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL; DG:= SQRT(VECVEC(1, N, 0, G, G)); DG0:= VECVEC(1, N, 0, DELTA, G); OK:= DG0 <= 0 "END" ITERATION; OUT[0]:= NRMDELTA; OUT[1]:= DG; OUT[2]:= EVL; OUT[3]:= CNTL; OUT[4]:= CNTE; RNK1MIN:= F "END" RNK1MIN 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 12 ; "EOP" 0"CODE" 34215; "REAL" "PROCEDURE" FLEMIN(N, X, G, H, FUNCT, IN, OUT); "VALUE" N; "INTEGER" N; "ARRAY" X, G, H, IN, OUT; "REAL" "PROCEDURE" FUNCT; "BEGIN" "INTEGER" I, IT, CNTL, EVL, EVLMAX; "REAL" F,F0,FMIN,MU,DG,DG0,NRMDELTA,ALFA,RELTOL,ABSTOL, EPS, TOLG, AID; "ARRAY" V, DELTA, S[1:N]; RELTOL:= IN[1]; ABSTOL:= IN[2]; MU:= IN[3]; TOLG:= IN[4]; FMIN:= IN[5]; ALFA:= IN[6]; EVLMAX:= IN[7]; OUT[4]:= 0; IT := 0; F:= FUNCT(N, X, G); EVL:= 1; CNTL:= 0;"IF" ALFA > 0 "THEN" "BEGIN" INIVEC(1, N * (N + 1) // 2, H, 0); INISYMD(1, N, 0, H, ALFA) "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DELTA[I]:= - SYMMATVEC(1, N, I, H, G); DG:= SQRT(VECVEC(1, N, 0, G, G)); NRMDELTA:= SQRT(VECVEC(1, N, 0, DELTA, DELTA)); EPS:= SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL; DG0:= VECVEC(1, N, 0, DELTA, G);"COMMENT" 1SECTION : 5.1.2.2.3 (DECEMBER 1975) PAGE 13 ; "FOR" IT := IT +1 "WHILE" (NRMDELTA > EPS "OR" DG > TOLG ) "AND" EVL < EVLMAX "DO" "BEGIN" DUPVEC(1, N, 0, S, X); DUPVEC(1, N, 0, V, G); "IF" IT >= N "THEN" ALFA:= 1 "ELSE" "BEGIN" "IF" IT ^= 1 "THEN" ALFA:= ALFA / NRMDELTA "ELSE" "BEGIN" ALFA:= 2 * (FMIN - F) / DG0; "IF" ALFA > 1 "THEN" ALFA:= 1 "END" "END"; ELMVEC(1, N, 0, X, DELTA, ALFA); F0:= F; F:= FUNCT(N, X, G); EVL:= EVL +1 ; DG:= VECVEC(1, N, 0, DELTA, G); "IF" IT = 1 "OR" F0 - F < - MU * DG0 * ALFA "THEN" "BEGIN" I:= EVLMAX - EVL; CNTL:= CNTL +1 ; LINEMIN(N, S, DELTA, NRMDELTA, ALFA, G, FUNCT, F0, F, DG0, DG, I, "FALSE", IN); EVL:= EVL + I; DUPVEC(1, N, 0, X, S); "END" LINEMINIMIZATION; "IF" ALFA ^= 1 "THEN" MULVEC(1, N, 0, DELTA, DELTA, ALFA); MULVEC(1, N, 0, V, V, -1); ELMVEC(1, N, 0, V, G, 1); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" S[I]:= SYMMATVEC(1, N, I, H, V); AID:= VECVEC(1, N, 0, V, S); DG:= (DG - DG0) * ALFA; "IF" DG > 0 "THEN" "BEGIN" "IF" DG >= AID "THEN" FLEUPD(H, N, DELTA, S, 1 / DG, (1 + AID / DG) / DG) "ELSE" DAVUPD(H, N, DELTA, S, 1 / DG, 1 / AID) "END" UPDATING; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DELTA[I]:= -SYMMATVEC(1, N, I, H, G); ALFA:= NRMDELTA * ALFA; NRMDELTA:= SQRT(VECVEC(1, N, 0, DELTA, DELTA)); EPS:= SQRT(VECVEC(1, N, 0, X, X)) * RELTOL + ABSTOL; DG:= SQRT(VECVEC(1, N, 0, G, G)); DG0:= VECVEC(1, N, 0, DELTA, G); "IF" DG0 > 0 "THEN" "BEGIN" OUT[4]:= -1 ; "GOTO" EXIT "END" "END" ITERATION; EXIT: OUT[0]:= NRMDELTA; OUT[1]:= DG; OUT[2]:= EVL; OUT[3]:= CNTL; FLEMIN:= F "END" FLEMIN; "EOP" 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 1 AUTHORS: J.C.P. BUS, B. VAN DOMSELAAR, J. KOK. CONTRIBUTORS: B. VAN DOMSELAAR, J. KOK. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 741101. BRIEF DESCRIPTION: THIS SECTION CONTAINS TWO PROCEDURES; MARQUARDT CALCULATES THE LEAST SQUARES SOLUTION OF AN OVERDETERMINED SYSTEM OF NONLINEAR EQUATIONS WITH MARQUARDT'S METHOD; GSSNEWTON CALCULATES THE LEAST SQUARES SOLUTION OF AN OVERDETERMINED SYSTEM OF NONLINEAR EQUATIONS WITH THE GAUSS-NEWTON METHOD; THE USER HAS TO PROGRAM THE EVALUATION OF THE RESIDUAL VECTOR OF THE SYSTEM AND THE JACOBIAN MATRIX OF ITS PARTIAL DERIVATIVES. KEYWORDS: NONLINEAR EQUATIONS, LEAST SQUARES SOLUTION, OVERDETERMINED SYSTEM, MARQUARDT'S METHOD, GAUSS-NEWTON METHOD, CURVE FITTING. 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 2 SUBSECTION: MARQUARDT. CALLING SEQUENCE: THE HEADING OF THIS PROCEDURE IS: "PROCEDURE" MARQUARDT(M, N, PAR, RV, JJINV, FUNCT, JACOBIAN, IN, OUT); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV, JJINV, IN, OUT; "BOOLEAN" "PROCEDURE" FUNCT; "PROCEDURE" JACOBIAN; "CODE" 34440; THE MEANING OF THE FORMAL PARAMETERS IS: M: ; THE NUMBER OF EQUATIONS; N: ; THE NUMBER OF UNKNOWN VARIABLES; N SHOULD SATISFY N<=M; PAR: ; "ARRAY" PAR[1 : N]; THE UNKNOWN VARIABLES OF THE SYSTEM; ENTRY: AN APPROXIMATION TO A LEAST SQUARES SOLUTION OF THE SYSTEM; EXIT: THE CALCULATED LEAST SQUARES SOLUTION; RV: ; "ARRAY" RV[1 : M]; EXIT: THE RESIDUAL VECTOR AT THE CALCULATED SOLUTION; JJINV: ; "ARRAY" JJINV[1 : N, 1 : N]; EXIT: THE INVERSE OF THE MATRIX J' * J WHERE J DENOTES THE MATRIX OF PARTIAL DERIVATIVES DRV[I] / DPAR[J] (I=1,...,M; J=1,...,N) AND J' DENOTES THE TRANSPOSE OF J. FUNCT: ; THE HEADING OF THIS PROCEDURE SHOULD BE: "BOOLEAN" "PROCEDURE" FUNCT(M, N, PAR, RV); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV; ENTRY: M, N, PAR; M, N HAVE THE SAME MEANING AS IN THE PROCEDURE MARQUARDT; "ARRAY" PAR[1:N] CONTAINS THE CURRENT VALUES OF THE UNKNOWNS AND SHOULD NOT BE ALTERED; EXIT: "ARRAY" RV[1 : M]; UPON COMPLETION OF A CALL OF FUNCT, THIS ARRAY RV SHOULD CONTAIN THE RESIDUAL VECTOR, OBTAINED WITH THE CURRENT VALUES OF THE UNKNOWNS; E.G. IN CURVE FITTING PROBLEMS: RV[I] := THEORETICAL VALUE F(X[I], PAR) - OBSERVED VALUE Y[I]; AFTER A SUCCESSFUL CALL OF FUNCT, THE BOOLEAN PROCEDURE SHOULD DELIVER THE VALUE TRUE; HOWEVER, IF FUNCT DELIVERS THE VALUE FALSE, THEN IT IS ASSUMED THAT THE CURRENT ESTIMATES OF THE UNKNOWNS LIE OUTSIDE A FEASIBLE REGION AND THE PROCESS IS TERMINATED (SEE OUT[1]); 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 3 HENCE, PROPER PROGRAMMING OF FUNCT MAKES IT POSSIBLE TO AVOID CALCULATION OF A RESIDUAL VECTOR WITH VALUES OF THE UNKNOWN VARIABLES WHICH MAKE NO SENSE OR WHICH EVEN MAY CAUSE OVERFLOW IN THE COMPUTATION; JACOBIAN: ; THE HEADING OF THIS PROCEDURE SHOULD BE: "PROCEDURE" JACOBIAN(M, N, PAR, RV, JAC, LOCFUNCT); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV, JAC; "PROCEDURE" LOCFUNCT; ENTRY: M, N, PAR, RV, LOCFUNCT; FOR M,N,PAR SEE: FUNCT; RV CONTAINS THE RESIDUAL VECTOR OBTAINED WITH THE CURRENT VALUES OF THE UNKNOWNS AND SHOULD NOT BE ALTERED; A CALL OF LOCFUNCT(M,N,PAR,RV) IS EQUIVALENT WITH A CALL OF THE USER-DEFINED PROCEDURE FUNCT(M,N,PAR,RV), BUT, IN ADDITION, THIS CALL IS COUNTED TO THE TOTAL NUMBER OF CALLS OF FUNCT (SEE OUT[4]) AND, MOREOVER, IF FUNCT DELIVERS THE VALUE FALSE THEN THE PROCESS IS TERMINATED; EXIT: "ARRAY" JAC[1 : M, 1 : N]; UPON COMPLETION OF A CALL OF JACOBIAN, JAC SHOULD CONTAIN THE PARTIAL DERIVATIVES DRV[I] / DPAR[J], OBTAINED WITH THE CURRENT VALUES OF THE UNKNOWN VARIABLES GIVEN IN PAR[1:N]; IT IS A PREREQUISITE FOR THE PROPER OPERATION OF THE PROCEDURE MARQUARDT THAT THE PRECISION OF THE ELEMENTS OF THE MATRIX JAC IS AT LEAST THE PRECISION DEFINED BY IN[3] AND IN[4]; IN: ; "ARRAY" IN[0 : 6]; ENTRY: IN THIS ARRAY THE USER SHOULD GIVE SOME DATA TO CONTROL THE PROCESS; IN[0]: THE MACHINE PRECISION; FOR THE CYBER 73 A SUITABLE VALUE IS "-14; IN[1], IN[2] ARE NOT USED BY THE PROCEDURE MARQUARDT; IN[3], IN[4]: THE RELATIVE AND ABSOLUTE TOLERANCE FOR THE DIFFERENCE BETWEEN THE EUCLIDEAN NORM OF THE ULTIMATE AND PENULTIMATE RESIDUAL VECTOR; THE PROCESS IS TERMINATED IF THE IMPROVEMENT OF THE SUM OF SQUARES IS LESS THAN IN[3] * (SUM OF SQUARES) + IN[4] * IN[4]; THESE TOLERANCES SHOULD BE CHOSEN GREATER THAN THE CORRESPONDING ERRORS OF THE CALCULATED RESIDUAL VECTOR; NOTE THAT THE EUCLIDEAN NORM OF THE RESIDUAL VECTOR IS DEFINED AS THE SQUARE ROOT OF THE SUM OF SQUARES; IN[5]: THE MAXIMUM NUMBER OF CALLS OF FUNCT ALLOWED; 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 4 IN[6]: A STARTING VALUE USED FOR THE RELATION BETWEEN THE GRADIENT AND THE GAUSS-NEWTON DIRECTION (SEE [2]); IF THE PROBLEM IS WELL CONDITIONED THEN A SUITABLE VALUE FOR IN[6] WILL BE 0.01; IF THE PROBLEM IS ILL CONDITIONED THEN IN[6] SHOULD BE GREATER, BUT THE VALUE OF IN[6] SHOULD SATISFY: IN[0] < IN[6] <= 1/IN[0]; OUT: ; "ARRAY" OUT[1 : 7]; EXIT : IN ARRAY OUT SOME BY-PRODUCTS ARE DELIVERED; OUT[1]: THIS VALUE GIVES INFORMATION ABOUT THE TERMINATION OF THE PROCESS; OUT[1]=0: NORMAL TERMINATION; OUT[1]=1: THE PROCESS HAS BEEN BROKEN OFF, BECAUSE THE NUMBER OF CALLS OF FUNCT EXCEEDED THE NUMBER GIVEN IN IN[5]; OUT[1]=2: THE PROCESS HAS BEEN BROKEN OFF, BECAUSE A CALL OF FUNCT DELIVERED THE VALUE FALSE; OUT[1]=3: FUNCT BECAME FALSE WHEN CALLED WITH THE INITIAL ESTIMATES OF PAR[1:N]; THE ITERATION PROCESS WAS NOT STARTED AND SO JJINV[1:N,1:N] CAN NOT BE USED; OUT[1]=4: THE PROCESS HAS BEEN BROKEN OFF, BECAUSE THE PRECISION ASKED FOR CAN NOT BE ATTAINED; THIS PRECISION IS POSSIBLY CHOSEN TOO HIGH, RELATIVE TO THE PRECISION IN WHICH THE RESIDUAL VECTOR IS CALCULATED (SEE IN[3]); OUT[2]: THE EUCLIDEAN NORM OF THE RESIDUAL VECTOR CALCULATED WITH VALUES OF THE UNKNOWNS DELIVERED; OUT[3]: THE EUCLIDEAN NORM OF THE RESIDUAL VECTOR CALCULATED WITH THE INITIAL VALUES OF THE UNKNOWN VARIABLES; OUT[4]: THE NUMBER OF CALLS OF FUNCT NECESSARY TO OBTAIN THE CALCULATED RESULT; OUT[5]: THE TOTAL NUMBER OF ITERATIONS PERFORMED; NOTE THAT IN EACH ITERATION ONE EVALUATION OF THE JACOBIAN MATRIX HAD TO BE MADE; OUT[6]: THE IMPROVEMENT OF THE EUCLIDEAN NORM OF THE RESIDUAL VECTOR IN THE LAST ITERATION STEP; OUT[7]: THE CONDITION NUMBER OF J' * J , I.E. THE RATIO OF ITS LARGEST TO SMALLEST EIGENVALUES; DATA AND RESULTS: IF THIS PROCEDURE IS USED FOR CURVE FITTING THEN THE RELATIVE ACCURACY IN THE CALCULATION OF THE RESIDUAL VECTOR DEPENDS STRONGLY ON THE ERRORS IN THE EXPERIMENTAL DATA AND THIS SHOULD BE REFLECTED IN THE PARAMETERS IN[3] AND IN[4]; THE MATRIX JJINV CAN BE USED IF SOME STATISTICAL INFORMATION ABOUT THE FITTED PARAMETERS IS REQUIRED; THE STANDARD DEVIATION, COVARIANCE MATRIX AND CORRELATION MATRIX MAY BE CALCULATED EASILY FROM JJINV ; 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 5 PROCEDURES USED: MULCOL = CP31022, DUPVEC = CP31030, VECVEC = CP34010, MATVEC = CP34011, TAMVEC = CP34012, MATTAM = CP34015, QRISNGVALDEC = CP34273. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: ONE ARRAY OF LENGTH M * N AND FOUR ARRAYS OF LENGTH N ARE DECLARED; RUNNING TIME: THE NUMBER OF ITERATION STEPS DEPENDS STRONGLY ON THE PROBLEM TO BE SOLVED; HOWEVER, THE WORK DONE EACH ITERATION STEP IS PROPORTIONAL TO N CUBED; LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: MARQUARDT USES MARQUARDT'S METHOD; FOR DETAILS SEE [2]; REFERENCES: [1] D. W. MARQUARDT, AN ALGORITHM FOR LEAST-SQUARES ESTIMATION OF NONLINEAR PARAMETERS, J. SIAM 11 (1963), PP.431-441. [2] J. C. P. BUS, B. VAN DOMSELAAR, J. KOK, NONLINEAR LEAST SQUARES ESTIMATION, MATHEMATICAL CENTRE, AMSTERDAM. (TO APPEAR) 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 6 EXAMPLE OF USE: THE PARAMETERS PAR[1 : 3] IN THE CURVE FITTING PROBLEM: RV[I] = PAR[1] + PAR[2] * EXP(PAR[3] * X[I]) - Y[I] WITH (X[I], Y[I]), I=1,...,6 AS THE EXPERIMENTAL DATA, MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I; "ARRAY" IN[0:6],OUT[0:7],X,Y,RV[1:6],JJINV[1:3,1:3],PAR[1:3]; "BOOLEAN" "PROCEDURE" EXPFUNCT(M,N,PAR,RV); "VALUE" M,N; "INTEGER" M,N; "ARRAY" PAR,RV; "BEGIN" "INTEGER" I; "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" "IF" PAR[3]*X[I]>680 "THEN" "BEGIN" EXPFUNCT:="FALSE"; "GOTO" STOP "END"; RV[I]:=PAR[1]+PAR[2]*EXP(PAR[3]*X[I])-Y[I] "END"; EXPFUNCT:="TRUE"; STOP: "END" EXPFUNCT; "PROCEDURE" JACOBIAN(M,N,PAR,RV,JAC,LOCFUNCT); "VALUE" M,N; "INTEGER" M,N; "ARRAY" PAR,RV,JAC; "PROCEDURE" LOCFUNCT; "BEGIN" "INTEGER" I; "REAL" EX; "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" JAC[I,1]:=1; JAC[I,2]:=EX:=EXP(PAR[3]*X[I]); JAC[I,3]:=X[I]*PAR[2]*EX "END" "END" JACOBIAN; IN[0]:="-14; IN[3]:="-4; IN[4]:="-1; IN[5]:=75; IN[6]:="-2; "FOR" I:=1 "STEP" 1 "UNTIL" 6 "DO" "BEGIN" INREAL(70,X[I]); INREAL(70,Y[I]) "END"; PAR[1]:=580; PAR[2]:=-180; PAR[3]:=-0.160; MARQUARDT(6,3,PAR,RV,JJINV,EXPFUNCT,JACOBIAN,IN,OUT); OUTPUT(61,"("3/,"("X[I] Y[I]")",/,6(B,+D,5B,3D.D,/),2/, "("PARAMETERS")",/,3(+D.3D"+ZD,/),2/,"("OUT:")",/,7(5B+D.6D"+ZD, /),2/,"("LAST RESIDUAL VECTOR")",/,6(6B+3Z.D,/)")",X[1],Y[1], X[2],Y[2],X[3],Y[3],X[4],Y[4],X[5],Y[5],X[6],Y[6],PAR[1],PAR[2], PAR[3],OUT[7],OUT[2],OUT[6],OUT[3],OUT[4],OUT[5],OUT[1],RV[1], RV[2],RV[3],RV[4],RV[5],RV[6]) "END" 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 7 WITH THE DATA GIVEN IN THE X - Y - TABLE THIS PROGRAM DELIVERS: X[I] Y[I] -5 127.0 -3 151.0 -1 379.0 +1 421.0 +3 460.0 +5 426.0 PARAMETERS +5.232" +2 -1.568" +2 -1.998" -1 OUT: +7.220828" +7 +1.157156" +2 +1.728008" -3 +1.654588" +2 +2.300000" +1 +2.200000" +1 +0.000000" +0 LAST RESIDUAL VECTOR -29.6 +86.6 -47.3 -26.2 -22.9 +39.5 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 8 SUBSECTION : GSSNEWTON. CALLING SEQUENCE : THE HEADING OF THE PROCEDURE READS : "PROCEDURE" GSSNEWTON(M, N, PAR, RV, JJINV, FUNCT, JACOBIAN, IN, OUT); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV, JJINV, IN, OUT; "BOOLEAN""PROCEDURE" FUNCT; "PROCEDURE" JACOBIAN; "CODE" 34441; THE MEANING OF THE FORMAL PARAMETERS IS : M : ; THE NUMBER OF EQUATIONS; N : ; THE NUMBER OF UNKNOWNS IN THE M EQUATIONS (N <= M); PAR : ; "ARRAY" PAR[1 : N]; THE UNKNOWNS OF THE EQUATIONS. ENTRY : AN APPROXIMATION TO A LEAST SQUARES SOLUTION OF THE SYSTEM. EXIT : THE CALULATED LEAST SQUARES SOLUTION; RV : ; "ARRAY" RV[1 : M]; EXIT : THE RESIDUAL VECTOR OF THE SYSTEM AT THE CALCULATED SOLUTION; JJINV : ; "ARRAY" JJINV[1 : N,1 : N]; EXIT : THE INVERSE OF THE MATRIX J' * J, WHERE J IS THE JACOBIAN MATRIX AT THE SOLUTION AND J' IS J TRANSPOSED; FUNCT : ; THE HEADING OF THIS PROCEDURE SHOULD BE : "BOOLEAN""PROCEDURE" FUNCT(M, N, PAR, RV); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV; ENTRY: M, N, PAR; M, N HAVE THE SAME MEANING AS IN THE PROCEDURE GSSNEWTON; "ARRAY" PAR[1:N] CONTAINS THE CURRENT VALUES OF THE UNKNOWNS AND SHOULD NOT BE ALTERED. EXIT: "ARRAY" RV[1 : M]; UPON COMPLETION OF A CALL OF FUNCT, THIS ARRAY RV SHOULD CONTAIN THE RESIDUAL VECTOR, OBTAINED WITH THE CURRENT VALUES OF THE UNKNOWNS. THE PROGRAMMER OF FUNCT MAY DECIDE THAT SOME CURRENT ESTIMATES OF THE UNKNOWNS LIE OUTSIDE A FEASIBLE REGION; IN THIS CASE FUNCT SHOULD DELIVER THE VALUE FALSE AND THE PROCESS IS TERMINATED (SEE OUT[1]). OTHERWISE FUNCT SHOULD DELIVER THE VALUE TRUE; 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 9 JACOBIAN : ; THE HEADING OF THIS PROCEDURE SHOULD BE : "PROCEDURE" JACOBIAN(M, N, PAR, RV, JAC, LOCFUNCT); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, RV, JAC; "PROCEDURE" LOCFUNCT; THE MEANING OF THE PARAMETERS OF JACOBIAN IS : M, N : SEE GSSNEWTON. PAR : ; "ARRAY" PAR[1 : N]; ENTRY : CURRENT ESTIMATES OF THE UNKNOWNS. THESE VALUES SHOULD NOT BE CHANGED. RV : ; "ARRAY" RV[1 : M]; ENTRY : THE RESIDUAL VECTOR OF THE SYSTEM OF EQUATIONS CORRESPONDING TO THE VECTOR OF UNKNOWNS AS GIVEN IN PAR. EXIT : THE ENTRY VALUES. JAC : ; "ARRAY" JAC[1 : M, 1 : N]; EXIT : THE JACOBIAN MATRIX AT THE CURRENT ESTIMATES GIVEN IN PAR, I.E. THE MATRIX OF PARTIAL DERIVATIVES D(RV)[I] / DPAR[J], I = 1(1)M, J = 1(1)N. LOCFUNCT : ; THE HEADING OF THIS PROCEDURE IS THE SAME AS THE HEADING OF FUNCT. A CALL OF THE PROCEDURE JACOBIAN SHOULD DELIVER THE JACOBIAN MATRIX EVALUATED WITH THE CURRENT ESTIMATES OF THE UNKNOWN VARIABLES GIVEN IN PAR IN SUCH A WAY, THAT THE PARTIAL DERIVATIVE D(RV)[I] / DPAR[J] IS DELIVERED IN JAC[I,J], I = 1(1)M, J = 1(1)N. FOR THE CALCULATION OF THE DERIVATIVES ONE CAN USE THE VALUES OF THE CURRENT ESTIMATES OF THE UNKNOWNS AS GIVEN IN PAR AND THE RESIDUAL VECTOR AS GIVEN IN RV. ONE CAN ALSO USE THE PROCEDURE FUNCT (PARAMETER OF GSSNEWTON) THROUGH CALLS OF THE PROCEDURE LOCFUNCT (PARAMETER OF JACOBIAN). THIS PARAMETER OF JACOBIAN MAY BE USED WHEN THE JACOBIAN MATRIX IS APPROXIMATED USING (FORWARD) DIFFERENCES. AN APPROPRIATE PROCEDURE TO THIS PURPOSE IS JACOBNMF (SECTION 4.3.2.1). SUCH A PROCEDURE MAY BE USED ONLY IF THE MATRIX ELEMENTS ARE COMPUTED SUFFICIENTLY ACCURATE; IN : ; "ARRAY" IN[0 : 7]; IN THIS ARRAY TOLERANCES AND CONTROL PARAMETERS SHOULD BE GIVEN. ENTRY : IN[0] : THE MACHINE PRECISION. FOR CALCULATION ON THE CYBER 73 A SUITABLE VALUE IS "-14. 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 10 IN[1], IN[2] : RELATIVE AND ABSOLUTE TOLERANCE FOR THE STEP VECTOR (RELATIVE TO THE VECTOR OF CURRENT ESTIMATES IN PAR). THE PROCESS IS TERMINATED IF IN SOME ITERATION (BUT NOT THE FIRST) THE EUCLIDEAN NORM OF THE CALCULATED NEWTON STEP IS LESS THAN IN[1] * NORM(PAR) + IN[2]. IN[1] SHOULD NOT BE CHOSEN SMALLER THAN IN[0]. IN[3] IS NOT USED BY THE PROCEDURE GSSNEWTON; IN[4] : ABSOLUTE TOLERANCE FOR THE EUCLIDEAN NORM OF THE RESIDUAL VECTOR. THE PROCESS IS TERMINATED WHEN THIS NORM IS LESS THAN IN[4]. IN[5] : THE MAXIMUM ALLOWED NUMBER OF FUNCTION EVALUATIONS (I.E. CALLS OF FUNCT). IN[6] : THE MAXIMUM ALLOWED NUMBER OF HALVINGS OF A CALCULATED NEWTON STEP VECTOR (SEE METHOD AND PERFORMANCE). A SUITABLE VALUE IS 15. IN[7] : THE MAXIMUM ALLOWED NUMBER OF SUCCESSIVE IN[6] TIMES HALVED STEP VECTORS. SUITABLE VALUES ARE 1 AND 2; OUT : ; "ARRAY" OUT[1 : 9]; IN ARRAY OUT INFORMATION ABOUT THE TERMINATION OF THE PROCESS IS DELIVERED. EXIT : OUT[1] : THE PROCESS WAS TERMINATED BECAUSE (OUT[1] = ) 1.THE NORM OF THE RESIDUAL VECTOR IS SMALL WITH RESPECT TO IN[4], 2.THE CALCULATED NEWTON STEP IS SUFFICIENTLY SMALL (SEE IN[1], IN[2]), 3.THE CALCULATED STEP WAS COMPLETELY DAMPED (HALVED) IN IN[7] SUCCESSIVE ITERATIONS, 4.OUT[4] EXCEEDS IN[5], THE MAXIMUM ALLOWED NUMBER OF CALLS OF FUNCT, 5.THE JACOBIAN WAS NOT FULL-RANK (SEE OUT[8]), 6.FUNCT DELIVERED FALSE AT A NEW VECTOR OF ESTIMATES OF THE UNKNOWNS, 7.FUNCT DELIVERED FALSE IN A CALL FROM JACOBIAN. OUT[2] : THE EUCLIDEAN NORM OF THE LAST RESIDUAL VECTOR. OUT[3] : THE EUCLIDEAN NORM OF THE INITIAL RESIDUAL VECTOR. OUT[4] : THE TOTAL NUMBER OF CALLS OF FUNCT. OUT[4] WILL BE LESS THAN IN[5] + IN[6]. OUT[5] : THE TOTAL NUMBER OF ITERATIONS. OUT[6] : THE EUCLIDEAN NORM OF THE LAST STEP VECTOR. OUT[7] : ITERATION NUMBER OF THE LAST ITERATION IN WHICH THE NEWTON STEP WAS HALVED. OUT[8], OUT[9] : RANK AND MAXIMUM COLUMN NORM OF THE JACOBIAN MATRIX IN THE LAST ITERATION, AS DELIVERED BY LSQORTDEC (SECTION 3.1.1.2.1.1) IN AUX[3] AND AUX[5]. 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 11 DATA AND RESULTS : THE PROCEDURE GSSNEWTON CAN BE USED FOR APPROXIMATING AN EXACT OR A LEAST SQUARES SOLUTION OF A SYSTEM OF NONLINEAR EQUATIONS. WHEN AN EXACT SOLUTION IS REQUIRED, THE PROCEDURE MAY TERMINATE ONLY WITH OUT[1] = 1, AND VERY SMALL VALUES SHOULD BE ASSIGNED TO IN[1] AND IN[2]. WHEN A LEAST SQUARES SOLUTION IS REQUIRED, POSITIVE RESULTS OF THE PROCEDURE ARE SIGNALED BY OUT[1] = 1 OR 2. WHENEVER THE PROCEDURE TERMINATES WITH OUT[1] < 5, THEN THE INVERSE OF J' * J (SEE MEANING OF THE PARAMETER JJINV) IS DELIVERED IN JJINV. IN THAT CASE THE COVARIANCE MATRIX AND THE STANDARD DEVIATIONS OF THE SOLUTION CAN BE CALCULATED. FOR A CURVE FITTING PROBLEM, SAY : "ESTIMATE PARAMETERS PAR[1], ... , PAR[N] OF A FUNCTION "Y = F(X; PAR[1], ... , PAR[N]), WHEN A SET OF DATA (X[I],Y[I]), "I = 1(1)M, HAS TO BE FITTED," THE FOLLOWING SYSTEM OF M EQUATIONS IN THE N UNKNOWN PARAMETERS PAR[1], ... , PAR[N] CAN BE DERIVED : F(X[I]; PAR[1], ... , PAR[N]) - Y[I] = 0, I = 1(1)M. PROCEDURES USED : VECVEC = CP34010, DUPVEC = CP31030, ELMVEC = CP34020, LSQORTDEC = CP34134, LSQSOL = CP34131, LSQINV = CP34136. REQUIRED CENTRAL MEMORY : EXECUTION FIELD LENGTH : AN ARRAY OF (M + 1) * N ELEMENTS, FOUR ARRAYS OF N ELEMENTS AND ONE ARRAY OF M ELEMENTS ARE DECLARED. RUNNING TIME : THE RUNNING TIME IS PROPORTIONAL TO THE TOTAL NUMBER OF CALLS OF FUNCT. BESIDES, IN EACH ITERATION A LINEAR LEAST SQUARES SYSTEM IS SOLVED (NUMBER OF OPERATIONS PROPORTIONAL TO M * (N SQUARED)). LANGUAGE : ALGOL 60. METHOD AND PERFORMANCE : THE PROCEDURE GSSNEWTON IS BASED UPON THE GAUSS-NEWTON METHOD FOR CALCULATING A LEAST SQUARES SOLUTION OF A SYSTEM OF NONLINEAR EQUATIONS (SEE E.G. [1], [2]). IN SEVERAL ITERATIONS A STEP VECTOR (FOR THE VECTOR OF UNKNOWNS) IS OBTAINED BY SOLVING A LINEARIZED SYSTEM OF EQUATIONS. THIS STEP IS PERFORMED ONLY IF THE NORM OF THE RESIDUAL VECTOR DECREASES. OTHERWISE THE NEWTON STEP VECTOR IS HALVED A NUMBER OF TIMES UNTIL THE NORM OF THE RESIDUAL VECTOR IS SMALLER THAN THE NORMS OF THE LAST AND SUBSEQUENT RESIDUAL VECTORS (THIS PROCESS IS CALLED STEP SIZE CONTROL). FOR FURTHER DETAILS SEE [3] (SEE ALSO [2]). 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 12 REFERENCES : [1] H.O. HARTLEY : THE MODIFIED GAUSS-NEWTON METHOD. TECHNOMETRICS, V.3 (1961), PP. 269 - 280. [2] H. SPAETH : THE DAMPED TAYLOR'S SERIES METHOD FOR MINIMIZING A SUM OF SQUARES AND FOR SOLVING SYSTEMS OF NONLINEAR EQUATIONS. ALGORITHM 315, COLLECTED ALGORITHMS FROM CACM, COMMUNICATIONS OF THE ACM, VOL. 10 (NOV. 1967), PP. 726 - 728. [3] J.C.P. BUS, B. VAN DOMSELAAR, J. KOK : NONLINEAR LEAST SQUARES ESTIMATION. MATHEMATICAL CENTRE (TO APPEAR). EXAMPLE OF USE : THE PARAMETERS PAR[1 : 3] IN THE CURVE FITTING PROBLEM: G[I] = PAR[1] + PAR[2] * EXP(PAR[3] * X[I]) - Y[I] WITH (X[I], Y[I]), I=1,...,6 AS THE EXPERIMENTAL DATA, MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" I; "ARRAY" IN[0:7], OUT[1:9], X, Y, G[1:6], V[1:3, 1:3], PAR[1:3]; "BOOLEAN""PROCEDURE" EXPFUNCT(M, N, PAR, G); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, G; "BEGIN""INTEGER" I; "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN""IF" PAR[3] * X[I] > 680 "THEN" "BEGIN" EXPFUNCT:= "FALSE"; "GO TO" STOP "END"; G[I]:= PAR[1] + PAR[2] * EXP(PAR[3] * X[I]) - Y[I] "END"; EXPFUNCT:="TRUE"; STOP: "END" EXPFUNCT; "PROCEDURE" JACOBIAN(M, N, PAR, G, JAC, LOCFUNCT); "VALUE" M, N; "INTEGER" M, N; "ARRAY" PAR, G, JAC; "PROCEDURE" LOCFUNCT; "BEGIN" "INTEGER" I; "REAL" EX; "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" JAC[I,1]:=1; JAC[I,2]:= EX:= EXP(PAR[3] * X[I]); JAC[I,3]:= X[I] * PAR[2] * EX "END" "END" JACOBIAN; 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 13 IN[0]:= "-14; IN[1]:= IN[2]:= "-6; IN[5]:= 75; IN[4]:="-10; IN[6]:= 14; IN[7]:= 1; "FOR" I:= 1 "STEP" 1 "UNTIL" 6 "DO" "BEGIN" INREAL(70,X[I]); INREAL(70,Y[I]) "END"; PAR[1]:= 580; PAR[2]:= - 180; PAR[3]:= - 0.160; GSSNEWTON(6, 3, PAR, G, V, EXPFUNCT, JACOBIAN, IN, OUT); OUTPUT(61,"("3/4B,"("X[I] Y[I]")",/, 6(5B+D, 5B3D.D, /), 2/, 4B"("PARAMETERS")",/,3(4B+D.3D"+ZD,/),2/,4B"("OUT:")",/, 3(9B+D.6D"+ZD,/), 5(14B3ZD,/), 9B+D.6D"+ZD,2/4B, "("LAST RESIDUAL VECTOR")",/,6(10B+3Z.D,/)")", X[1], Y[1], X[2],Y[2],X[3],Y[3],X[4],Y[4],X[5],Y[5],X[6],Y[6],PAR[1],PAR[2], PAR[3],OUT[6],OUT[2],OUT[3],OUT[4],OUT[5],OUT[1],OUT[7],OUT[8], OUT[9], G[1],G[2],G[3],G[4],G[5],G[6]) "END" WITH THE DATA GIVEN IN THE X - Y - TABLE THIS PROGRAM DELIVERS: X[I] Y[I] -5 127.0 -3 151.0 -1 379.0 +1 421.0 +3 460.0 +5 426.0 PARAMETERS +5.233" +2 -1.569" +2 -1.997" -1 OUT: +5.260478" -4 +1.157156" +2 +1.654588" +2 16 16 2 0 3 +2.339529" +3 LAST RESIDUAL VECTOR -29.6 +86.6 -47.3 -26.2 -22.9 +39.5 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 14 SOURCE TEXTS : 0"CODE" 34440; "PROCEDURE" MARQUARDT(M,N,PAR,G,V,FUNCT,JACOBIAN,IN,OUT); "VALUE" M,N; "INTEGER" M,N; "ARRAY" PAR,G,V,IN,OUT; "BOOLEAN" "PROCEDURE" FUNCT; "PROCEDURE" JACOBIAN; "BEGIN" "INTEGER" MAXFE,FE,IT,I,J,ERR; "REAL" VV,WW,W,MU,RES,FPAR,FPARPRES,LAMBDA,LAMBDAMIN, P,PW,RELTOLRES,ABSTOLRES; "ARRAY" EM[0:7],VAL,B,BB,PARPRES[1:N],JAC[1:M,1:N]; "PROCEDURE" LOCFUNCT(M,N,PAR,G); "VALUE" M, N; "INTEGER" M,N; "ARRAY" PAR,G; "BEGIN" FE:= FE+1; "IF" FE >= MAXFE "THEN" ERR:= 1 "ELSE" "IF" "NOT" FUNCT(M,N,PAR,G) "THEN" ERR:= 2; "IF" ERR^=0 "THEN" "GOTO" EXIT "END" LOCFUNCT; VV:=10; W:=0.5; MU:= 0.01; WW:=("IF" IN[6]<"-7 "THEN" "-8 "ELSE" "-1*IN[6]); EM[0]:=EM[2]:=EM[6]:=IN[0]; EM[4]:=10*N; RELTOLRES:=IN[3]; ABSTOLRES:=IN[4]**2; MAXFE:=IN[5]; ERR:= 0; FE:= IT:= 1; P:=FPAR:= RES:= 0; PW:=-LN(WW*IN[0])/2.30; "IF" "NOT" FUNCT(M,N,PAR,G) "THEN" "BEGIN" ERR:= 3; "GOTO" ESCAPE "END"; FPAR:= VECVEC(1,M,0,G,G); OUT[3]:=SQRT(FPAR); "FOR" IT:= 1, IT+1 "WHILE" FPAR > ABSTOLRES "AND" RES > RELTOLRES*FPAR+ABSTOLRES "DO" "BEGIN" JACOBIAN(M,N,PAR,G,JAC,LOCFUNCT); I:=QRISNGVALDEC(JAC,M,N,VAL,V,EM); "IF" IT=1 "THEN" LAMBDA:= IN[6] * VECVEC(1,N,0,VAL,VAL) "ELSE" "IF" P =0 "THEN" LAMBDA:= LAMBDA*W "ELSE" P:= 0; "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" B[I]:=VAL[I]*TAMVEC(1,M,I,JAC,G); "COMMENT" 1SECTION : 5.1.3.1.3 (DECEMBER 1975) PAGE 15 ; L: "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" BB[I]:=B[I]/(VAL[I]*VAL[I]+LAMBDA); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" PARPRES[I]:= PAR[I] - MATVEC(1,N,I,V,BB); LOCFUNCT(M,N,PARPRES,G); FPARPRES:= VECVEC(1,M,0,G,G); RES:=FPAR-FPARPRES; "IF" RES < MU * VECVEC(1,N,0,B,BB) "THEN" "BEGIN" P:= P+1; LAMBDA:= VV * LAMBDA; "IF" P=1 "THEN" "BEGIN" LAMBDAMIN:= WW * VECVEC(1,N,0,VAL,VAL); "IF" LAMBDALAMBDA "THEN" LAMBDA := VAL[I] "ELSE" "IF" VAL[I] RELTOLPAR * NORMX + ABSTOLPAR "OR" IT = 1 "AND" STAP > 0 "THEN" "BEGIN""FOR" INR:= 0, INR + 1 "WHILE""IF" INR = 1 "THEN" DAMPING ON "OR" RES2 >= RN "ELSE""NOT" CONV "AND" (RN <= RES1 "OR" RES2 < RES1) "DO" "BEGIN""COMMENT" DAMPING STOPS WHEN R0 > R1 "AND" R1 <= R2 (BEST RESULT IS X1, R1) WITH X1 = X0 + I * DX, I:= 1, .5, .25, .125, ETC. ; RHO:= RHO / 2; "IF" INR > 0 "THEN" "BEGIN" RES1:= RES2; DUPVEC(1, M, 0, RV, FU2); DAMPING ON:= INR > 1 "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" PR[I]:= PAR[I] - SOL[I] * RHO; FEVAL:= FEVAL + 1; "IF" "NOT" FUNCT(M, N, PR, FU2) "THEN" "BEGIN" TEXT:= 6; "GO TO" FAIL "END"; RES2:= VECVEC(1, M, 0, FU2, FU2); CONV:= INR >= INRMAX "END" DAMPING OF STEP VECTOR; "IF" CONV "THEN" "BEGIN""COMMENT" RESIDUE CONSTANT; MIT:= MIT + 1; "IF" MIT < TIM "THEN" CONV:= "FALSE" "END" "ELSE" MIT:= 0; "IF" INR > 1 "THEN" "BEGIN" RHO:= RHO * 2; ELMVEC(1, N, 0, PAR, SOL, - RHO); RN:= RES1; "IF" INR > 2 "THEN" OUT[7]:= IT "END""ELSE" "BEGIN" DUPVEC(1, N, 0, PAR, PR); RN:= RES2; DUPVEC(1, M, 0, RV, FU2) "END"; "IF" RN <= ABSTOLRES "THEN" "BEGIN" TEXT:= 1; ITMAX:= IT "END""ELSE" "IF" CONV "AND" INRMAX > 0 "THEN" "BEGIN" TEXT:= 3; ITMAX:= IT "END" "ELSE" DUPVEC(1, M, 0, FU2, RV) "END" ITERATION WITH DAMPING AND TESTS "ELSE" "BEGIN" TEXT:= 2; RHO:= 1; ITMAX:= IT "END" "END" OF ITERATIONS; LSQINV(JAC, N, AID, CI); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" JJINV[I,I]:= JAC[I,I]; "FOR" J:= I + 1 "STEP" 1 "UNTIL" N "DO" JJINV[I,J]:= JJINV[J,I]:= JAC[I,J] "END" CALCULATION OF INVERSE MATRIX OF NORMAL EQUATIONS; FAIL : OUT[6]:= SQRT(STAP) * RHO; OUT[2]:= SQRT(RN); OUT[4]:= FEVAL; OUT[1]:= TEXT; OUT[8]:= AUX[3]; OUT[9]:= AUX[5] "END" GSSNEWTON; "EOP" 1SECTION : 5.2.1.1.1.1 (DECEMBER 1979) PAGE 1 SECTION 5.2.1.1.1.1 CONTAINS NINE PROCEDURES FOR INTIAL-VALUE PROBLEMS FOR FIRST ORDER ORDINARY DIFERENTIAL EQUATIONS. A. RK1 SOLVES AN IVP FOR A SINGLE ODE BY MEANS OF A 5-TH ORDER RUNGE-KUTTA METHOD. B. RKE SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A 5-TH ORDER RUNGE-KUTTA METHOD. C. RK4A SOLVES AN IVP FOR A SINGLE ODE BY MEANS OF A 5-TH ORDER RUNGE-KUTTA METHOD. RK4A INTERCHANGES THE DEPENDENT AND INDEPENDENT VARIABLE. D. RK4NA SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A 5-TH ORDER RUNGE-KUTTA METHOD. RK4NA INTERCHANGES THE INDEPENDENT AND ONE DEPENDENT VARIABLE. E. RK5NA SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A 5-TH ORDER RUNGE-KUTTA METHOD. RK5NA USES THE ARC LENGTH AS INTEGRATION VARIABLE. F. MULTISTEP SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A LINEAR MULTISTEP METHOD. IT USES EITHER THE BACKWARD DIFFERENTIATION METHODS, OR THE ADAMS-BASHFORTH-MOULTON-METHOD. G. DIFFSYS SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF A HIGH ORDER EXTRAPOLATION METHOD BASED ON THE MODIFIED MIDPOINT RULE. H. ARK SOLVES AN IVP FOR A LARGE SYSTEM OF ODE'S WHICH IS OBTAINED FROM SEMI-DISCRETIZATION OF AN INITIAL BOUNDARY VALUE PROBLEM FOR A PARABOLIC OR HYPERBOLIC EQUATION. ARK IS BASED ON STABILIZED, EXPLICIT RUNGE-KUTTA METHODS OF LOW ORDER. I. EFRK SOLVES AN IVP FOR A SYSTEM OF ODE'S BY MEANS OF AN EXPONENTIALLY FITTED EXPLICIT RUNGE-KUTTA METHOD OF FIRST, SECOND OR THIRD ORDER. RK1 AND RKE ARE INTENDED FOR NON-STIFF EQUATIONS,WHILE RK4A, RK4NA AND RK5NA ARE INTENDED FOR NON-STIFF EQUATIONS WHERE DERIVATIVES BECOME VERY LARGE, SUCH AS IN THE NEIGHBOURHOOD OF SINGULARITIES. MULTISTEP CAN BE USED FOR NON-STIFF, AS WELL AS STIFF EQUATIONS. DIFFSYS SHOULD BE USED FOR PROBLEMS FOR WHICH A VERY HIGH ACCURACY IS DESIRED. ARK IS RECOMMENDED FOR THE INTEGRATION OF SEMI-DISCRETE PARABOLIC OR HYPERBOLIC PROBLEMS. EFRK IS A SPECIAL PURPOSE PROCEDURE FOR STIFF EQUATIONS WITH A KNOWN, CLUSTERED EIGENVALUE SPECTRUM. EXCEPT EFRK, ALL PROCEDURES PERFORM STEPSIZE CONTROL. 1SECTION : 5.2.1.1.1.1.A (FEBRUARY 1979) PAGE 1 PROCEDURE : RK1 AUTHOR:J.A.ZONNEVELD. CONTRIBUTORS: M.BAKKER AND I.BRINK. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730715. BRIEF DESCRIPTION: RK1 SOLVES AN INITIAL VALUE PROBLEM FOR A SINGLE FIRST ORDER ORDINARY DIFFERENTIAL EQUATION DY / DX = F(X,Y). THE EQUATION IS SUPPOSED TO BE NON-STIFF. KEYWORDS: INITIAL VALUE PROBLEM. SINGLE FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" RK1(X,A,B,Y,YA,FXY,E,D,FI); "VALUE" B,FI; "REAL" X,A,B,Y,YA,FXY; "BOOLEAN" FI; "ARRAY" E,D; "CODE" 33010; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE. UPON COMPLETION OF A CALL X IS EQUAL TO B; A: ; ENTRY: THE INITIAL VALUE OF X; B: ; ENTRY: A VALUE PARAMETER,GIVING THE END VALUE OF X; Y: ; THE DEPENDENT VARIABLE; 1SECTION : 5.2.1.1.1.1.A (FEBRUARY 1979) PAGE 2 YA: ; ENTRY : THE VALUE OF Y AT X=A; FXY: ; AN EXPRESSION,DEPENDING ON X AND Y,GIVING THE VALUE OF DY/DX; E: ; "ARRAY" E[1:2]; ENTRY: E[1] : A RELATIVE TOLERANCE, E[2] : AN ABSOLUTE TOLERANCE ; D: ; "ARRAY" D[1:4]; EXIT: ENTIER(D[1]+.5) GIVES THE NUMBER OF STEPS SKIPPED; D[2] : EQUALS THE STEP LENGTH; D[3] : IS EQUAL TO B; D[4] : IS EQUAL TO Y(B); FI: ; IF FI="TRUE" RK1 INTEGRATES FROM X=A TO X=B WITH INITIAL VALUE VALUE Y(A)=YA AND TRIAL STEP B-A. IF FI="FALSE" RK1 INTEGRATES FROM X=D[3] TO X=B WITH INITIAL VALUE Y(D[3])=D[4] AND FIRST STEP H=D[2]*SIGN(B-D[3]), WHILE A AND YA ARE IGNORED. PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY : NEGLIGIBLE SMALL. METHOD AND PERFORMANCE: RK1 IS BASED ON AN EXPLICIT, 5-TH ORDER RUNGE-KUTTA METHOD AND IS PROVIDED WITH STEPLENGTH AND ERRORCONTROL. THE ERRORCONTROL IS BASED ON THE LAST TERM OF THE TAYLOR SERIES WICH IS TAKEN INTO ACCOUNT. A STEP IS REJECTED IF THE ABSOLUTE VALUE OF THIS LAST TERM GREATER THAN (ABS(FXY)*E[1]+E[2])*ABS(H)/INT, WHERE INT = ABS( B-( "IF" FI "THEN" A "ELSE" D[3])) DENOTES THE LENGTH OF THE INTEGRATION INTERVAL, OTHERWISE, A STEP IS ACCEPTED. RK1 USES AS ITS MINIMAL ABBSOLUTE STEPLENGTH HMIN = E[1]*INT+E[2]. IF A STEP OF LENGTH ABS(H) = HMIN IS REJECTED, THE STEP IS SKIPPED. FOR FURTHER DETAILS SEE [1]. 1SECTION : 5.2.1.1.1.1.A (FEBRUARY 1979) PAGE 3 REFERENCES: [1]J.A.ZONNEVELD. AUTOMATIC NUMERICAL INTEGRATION. MATHEMATICAL CENTRE TRACT 8(1970). EXAMPLE OF USE: THE SOLUTION OF THE DIFFERENTIAL EQUATION DY/DX=-Y WITH INITIAL CONDITION Y(0)=1 AT X=1 IS COMPUTED BY MEANS OF THE FOLLOWING PROGRAM: "BEGIN" "REAL" X,Y; "BOOLEAN" FI,FIRST; "REAL" "ARRAY" E[1:2],D[1:4]; E[1]:=+"-4;E[2]:=+"-4;FIRST:="TRUE"; RK1(X,0.0,1,Y,1.0,-Y,E,D,FIRST); OUTPUT(61,"("//10B"("X=")".12D"2D,//10B"("Y=")".12D"2D, 10B"("YEXACT=")".12D"2D")",X,Y,EXP(-X)); "END" IT DELIVERS WITH E[1]=E[2]="-4: X=.100000000000"01 Y=.367876846355"00 YEXACT=.367879441171"00 SOURCE TEXT(S): 0"CODE" 33010; "PROCEDURE" RK1(X, A, B, Y, YA, FXY, E, D, FI); "VALUE" B, FI; "REAL" X, A, B, Y, YA, FXY; "BOOLEAN" FI; "ARRAY" E, D; "BEGIN" "REAL" E1, E2, XL, YL, H, INT, HMIN, ABSH, K0, K1, K2, K3, K4, K5, DISCR, TOL, MU, MU1, FH, HL; "BOOLEAN" LAST, FIRST, REJECT; "COMMENT" 1SECTION : 5.2.1.1.1.1.A (AUGUST 1974) PAGE 4 ; "IF" FI "THEN" "BEGIN" D[3]:= A; D[4]:= YA "END"; D[1]:= 0; XL:= D[3]; YL:= D[4]; "IF" FI "THEN" D[2]:= B - D[3]; ABSH:= H:= ABS(D[2]); "IF" B - XL < 0 "THEN" H:= - H; INT:= ABS(B - XL); HMIN:= INT * E[1] + E[2]; E1:= E[1] / INT; E2:= E[2] / INT; FIRST:= "TRUE"; "IF" FI "THEN" "BEGIN" LAST:= "TRUE"; "GOTO" STEP "END"; TEST: ABSH:= ABS(H); "IF" ABSH < HMIN "THEN" "BEGIN" H:= "IF" H > 0 "THEN" HMIN "ELSE" - HMIN; ABSH:= HMIN "END"; "IF" H >= B - XL "EQV" H >= 0 "THEN" "BEGIN" D[2]:= H; LAST:= "TRUE"; H:= B - XL; ABSH:= ABS(H) "END" "ELSE" LAST:= "FALSE"; STEP: X:= XL; Y:= YL; K0:= FXY * H; X:= XL + H / 4.5; Y:= YL + K0 / 4.5; K1:= FXY * H; X:= XL + H / 3; Y:= YL + (K0 + K1 * 3) / 12; K2:= FXY * H; X:= XL + H * .5; Y:= YL + (K0 + K2 * 3) / 8; K3:= FXY * H; X:= XL + H * .8; Y:= YL + (K0 * 53 - K1 * 135 + K2 * 126 + K3 * 56) / 125; K4:= FXY * H; X:= "IF" LAST "THEN" B "ELSE" XL + H; Y:= YL + (K0 * 133 - K1 * 378 + K2 * 276 + K3 * 112 + K4 * 25) / 168; K5:= FXY * H; DISCR:= ABS(K0 * 21 - K2 * 162 + K3 * 224 - K4 * 125 + K5 * 42) / 14; TOL:= ABS(K0) * E1 + ABSH * E2; REJECT:= DISCR > TOL; MU:= TOL / (TOL + DISCR) + .45; "IF" REJECT "THEN" "BEGIN" "IF" ABSH <= HMIN "THEN" "BEGIN" D[1]:= D[1] + 1; Y:= YL; FIRST:= "TRUE"; "GOTO" NEXT "END"; H:= MU * H; "GOTO" TEST "END"; "IF" FIRST "THEN" "BEGIN" FIRST:= "FALSE"; HL:= H; H:= MU * H; "GOTO" ACC "END"; FH:= MU * H / HL + MU - MU1; HL:= H; H:= FH * H; ACC: MU1:= MU; Y:= YL + ( - K0 * 63 + K1 * 189 - K2 * 36 - K3 * 112 + K4 * 50) / 28; K5:= FXY * HL; Y:= YL + (K0 * 35 + K2 * 162 + K4 * 125 + K5 * 14) / 336; NEXT: "IF" B ^= X "THEN" "BEGIN" XL:= X; YL:= Y; "GOTO" TEST "END"; "IF" "NOT"LAST "THEN" D[2]:= H; D[3]:= X; D[4]:= Y "END" RK1; "EOP" 1SECTION : 5.2.1.1.1.1.B (FEBRUARY 1979) PAGE 1 PROCEDURE : RKE. AUTHOR: P.A. BEENTJES. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 740520. BRIEF DESCRIPTION: RKE SOLVES AN INITIAL VALUE PROBLEM FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DY / DX = F(X,Y). THE SYSTEM IS SUPPOSED TO BE NON-STIFF. KEYWORDS: INITIAL VALUE PROBLEM. SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. 1SECTION : 5.2.1.1.1.1.B (FEBRUARY 1979) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" RKE (X, XE, N, Y, DER, DATA, FI, OUT); "VALUE" N, FI; "INTEGER" N; "REAL" X, XE; "BOOLEAN" FI; "ARRAY" Y, DATA; "PROCEDURE" DER, OUT; "CODE" 33033; RKE INTEGRATES THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS DY / DX = F(X, Y), FROM X = X0 TO X = XE WHERE Y(X0) = Y0. THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE; ENTRY: THE INITIAL VALUE X0; XE: ; THE FINAL VALUE OF X; N: ; THE NUMBER OF EQUATIONS OF THE SYSTEM; Y: ; "ARRAY" Y[1 : N]; THE DEPENDENT VARIABLES; ENTRY: THE INITIAL VALUES AT X = X0; EXIT : THE VALUES OF THE SOLUTION AT X = XE; DER: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DER (T, V); "VALUE" T; "REAL" T; "ARRAY" V; THIS PROCEDURE PERFORMS AN EVALUATION OF THE RIGHT HAND SIDE OF THE SYSTEM WITH DEPENDENT VARIABLES V[1 : N] AND INDEPENDENT VARIABLE T; UPON COMPLETION OF DER THE RIGHT HAND SIDE SHOULD BE OVERWRITTEN ON V[1 : N]; DATA: ; "ARRAY" DATA[1 : 6]; IN ARRAY DATA ONE SHOULD GIVE: DATA[1]: THE RELATIVE TOLERANCE; DATA[2]: THE ABSOLUTE TOLERANCE; AFTER EACH STEP DATA[3:6] CONTAINS : DATA[3]: THE STEPLENGTH USED FOR THE LAST STEP; DATA[4]: THE NUMBER OF INTEGRATION STEPS PERFORMED; DATA[5]: THE NUMBER OF INTEGRATION STEPS REJECTED; DATA[6]: THE NUMBER OF INTEGRATION STEPS SKIPPED; IF UPON COMPLETION OF RKE DATA[6] > 0 , RESULTS SHOULD BE CONSIDERED MOST CRITICALLY; FI: ; IF FI = "TRUE" THE INTEGRATION STARTS AT X0 WITH A TRIAL STEP XE - X0; IF FI = "FALSE" THE INTEGRATION IS CONTINUED WITH A STEPLENGTH DATA[3] * SIGN(XE - X0); OUT: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" OUT; AFTER EACH INTEGRATION STEP PERFORMED OUT CAN BE USED TO OBTAIN INFORMATION FROM THE SOLUTION PROCESS, E.G. THE VALUES OF X, Y[1 : N] AND DATA[3 : 6]. POUT CAN ALSO BE USED TO UPDATE DATA. 1SECTION : 5.2.1.1.1.1.B (FEBRUARY 1979) PAGE 3 PROCEDURES USED : NONE. REQUIRED CENTRAL MEMORY: CIRCA 5 * N MEMORY PLACES. METHOD AND PERFORMANCE: THE METHOD UPON WHICH THE PROCEDUDRE IS BASED, IS A MEMBER OF A CLASS OF FIFTH ORDER RUNGE KUTTA FORMULAS PRESENTED IN REFERENCE [1]. AUTOMATIC STEPSIZE CONTROL IS IMPLEMENTED IN A WAY AS PROPOSED IN REFERENCE [2]. FOR TESTRESULTS AND FURTHER INFORMATION SEE REFERENCE [3]. REFERENCES: [1]. R. ENGLAND. ERROR ESTIMATES FOR RUNGE KUTTA TYPE SOLUTIONS TO SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. THE COMPUTER JOURNAL , VOLUME 12, P 166 - 169, 1969. [2]. J.A. ZONNEVELD. AUTOMATIC NUMERICAL INTEGRATION. MATH. CENTRE TRACT 8(1970). [3]. P.A. BEENTJES. SOME SPECIAL FORMULAS OF THE ENGLAND CLASS OF FIFTH ORDER RUNGE - KUTTA SCHEMES. MATH. CENTRE REPORT NW 24/74. 1SECTION : 5.2.1.1.1.1.B (DECEMBER 1975) PAGE 4 EXAMPLE OF USE: THE SOLUTION AT T = 1 AND T = -1 OF THE SYSTEM DX / DT = Y - Z, DY / DT = X * X + 2 * Y + 4 * T, DZ / DT = X * X + 5 * X + Z * Z + 4 * T, WITH X = Y = 0 AND Z = 2 AT T = 0, CAN BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "REAL" T, TE; "ARRAY" Y[1 : 3], DATA[1 : 6]; "PROCEDURE" RHS(T, Y); "VALUE" T; "REAL" T; "ARRAY" Y; "BEGIN" "REAL" XX, YY, ZZ; XX:= Y[1]; YY:= Y[2]; ZZ:= Y[3]; Y[1]:= YY - ZZ; Y[2]:= XX * XX + 2 * YY + 4 * T; Y[3]:= XX * (XX + 5) + 2 * ZZ + 4 * T "END" RHS; "PROCEDURE" INFO; "IF" T = TE "THEN" "BEGIN" "REAL" ET, T2, AEX, AEY, AEZ, REX, REY, REZ; ET:= EXP(T); T2:= 2 * T; REX:= -ET * SIN(T2); AEX:= REX - Y[1]; REX:= ABS(AEX / REX); REY:= ET * ET * (8 + 2 * T2 - SIN(2 * T2)) / 8 - T2 - 1; REZ:= ET * (SIN(T2) + 2 * COS(T2)) + REY; AEY:= REY - Y[2]; REY:= ABS(AEY / REY); AEZ:= REZ - Y[3]; REZ:= ABS(AEZ / REZ); OUTPUT(61, "(""(" T = ")", +D, //, "(" RELATIVE AND ABSOLUTE ERRORS IN X, Y AND Z :")", //, "(" RE(X) RE(Y) RE(Z) AE(X) AE(Y) AE(Z)")", //, 6(B,-.2D"+D), //, "(" NUMBER OF INTEGRATION STEPS PERFORMED :")",4ZD,/, "(" NUMBER OF INTEGRATION STEPS SKIPPED :")",4ZD,/, "(" NUMBER OF INTEGRATION STEPS REJECTED :")",4ZD,///")", T, REX, REY, REZ, ABS(AEX), ABS(AEY), ABS(AEZ), DATA[4], DATA[6], DATA[5]) "END" INFO; TE:= 1; LEFT: Y[1]:= Y[2]:= 0; Y[3]:= 2; T:=0; DATA[1]:= DATA[2]:= "-5; RKE(T, TE, 3, Y, RHS, DATA, "TRUE", INFO); "IF" TE = 1 "THEN" "BEGIN" TE:= -1; "GOTO" LEFT "END" "END" 1SECTION : 5.2.1.1.1.1.B (DECEMBER 1975) PAGE 5 THIS PROGRAM DELIVERS: T = +1 RELATIVE AND ABSOLUTE ERRORS IN X, Y AND Z : RE(X) RE(Y) RE(Z) AE(X) AE(Y) AE(Z) .37"-6 .15"-5 .13"-5 .91"-6 .13"-4 .11"-4 NUMBER OF INTEGRATION STEPS PERFORMED : 9 NUMBER OF INTEGRATION STEPS SKIPPED : 0 NUMBER OF INTEGRATION STEPS REJECTED : 5 T = -1 RELATIVE AND ABSOLUTE ERRORS IN X, Y AND Z : RE(X) RE(Y) RE(Z) AE(X) AE(Y) AE(Z) .22"-6 .52"-7 .19"-6 .75"-7 .55"-7 .77"-7 NUMBER OF INTEGRATION STEPS PERFORMED : 10 NUMBER OF INTEGRATION STEPS SKIPPED : 0 NUMBER OF INTEGRATION STEPS REJECTED : 7 SOURCE TEXT(S): 0"CODE" 33033; "PROCEDURE" RKE (X, XE, N, Y, DER, DATA, FI, OUT); "VALUE" FI, N; "INTEGER" N; "REAL" X, XE; "BOOLEAN" FI; "ARRAY" Y, DATA; "PROCEDURE" DER, OUT; "BEGIN" "INTEGER" J; "REAL" XT, H, HMIN, INT, HL, HT, ABSH, FHM, DISCR, TOL, MU, MU1, FH, E1, E2; "BOOLEAN" LAST, FIRST, REJECT; "ARRAY" K0, K1, K2, K3, K4[1:N]; "IF" FI "THEN" "BEGIN" DATA[3]:= XE - X; DATA[4]:= DATA[5]:= DATA[6]:= 0 "END"; ABSH:= H:= ABS(DATA[3]); "IF" XE < X "THEN" H:= - H; INT:= ABS(XE - X); HMIN:= INT * DATA[1] + DATA[2]; E1:= 12 * DATA[1] / INT; E2:= 12 * DATA[2] / INT; FIRST:= "TRUE"; REJECT:= "FALSE"; "IF" FI "THEN" "BEGIN" LAST:= "TRUE"; "GOTO" STEP "END"; TEST: ABSH:= ABS(H); "IF" ABSH < HMIN "THEN" "BEGIN" H:= SIGN (XE - X) * HMIN; ABSH:= HMIN "END"; "IF" H >= XE - X "EQV" H >= 0 "THEN" "BEGIN" LAST:= "TRUE"; H:= XE - X; ABSH:= ABS(H) "END" "ELSE" LAST:= "FALSE"; "COMMENT" 1SECTION : 5.2.1.1.1.1.B (DECEMBER 1975) PAGE 6 ; STEP: "IF" ^ REJECT "THEN" "BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K0[J]:= Y[J]; DER(X, K0) "END"; HT:= .184262134833347 * H; XT:= X + HT; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K1[J]:= K0[J] * HT + Y[J]; DER(XT, K1); HT:= .690983005625053"-1 * H; XT:= 4 * HT + X; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K2[J]:= (3 * K1[J] + K0[J]) * HT + Y[J]; DER(XT, K2); XT:= .5 * H + X; HT:= .1875 * H; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K3[J]:=((1.74535599249993 * K2[J] - K1[J]) * 2.23606797749979 + K0[J]) * HT + Y[J]; DER(XT, K3); XT:= .723606797749979 * H + X; HT:= .4 * H; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K4[J]:= (((.517595468166681 * K0[J] - K1[J]) * .927050983124840 + K2[J]) * 1.46352549156242 + K3[J]) * HT + Y[J]; DER(XT, K4); XT:= "IF" LAST "THEN" XE "ELSE" X + H; HT:= 2 * H; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" K1[J]:= ((((2 * K4[J] + K2[J]) * .412022659166595 + K1[J]) * 2.23606797749979 - K0[J]) * .375 - K3[J]) * HT + Y[J]; DER(XT, K1); REJECT:= "FALSE"; FHM:= 0; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" DISCR:= ABS((1.6 * K3[J] - K2[J] - K4[J]) * 5 + K0[J] + K1[J]); TOL:= ABS(K0[J]) * E1 + E2; REJECT:= DISCR > TOL "OR" REJECT; FH:= DISCR / TOL; "IF" FH > FHM "THEN" FHM:= FH "END"; MU:= 1 / (1 + FHM) + .45; "IF" REJECT "THEN" "BEGIN" DATA[5]:= DATA[5] + 1; "IF" ABSH <= HMIN "THEN" "BEGIN" DATA[6]:= DATA[6] + 1; HL:= H; REJECT:= "FALSE"; FIRST:= "TRUE"; "GOTO" NEXT "END"; H:= MU * H; "GOTO" TEST "END"; "IF" FIRST "THEN" "BEGIN" FIRST:= "FALSE"; HL:= H; H:= MU * H; "GOTO" ACC "END"; FH:= MU * H / HL + MU - MU1; HL:= H; H:= FH * H; ACC: MU1:= MU; HT:= HL / 12; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" Y[J]:= ((K2[J] + K4[J]) * 5 + K0[J] + K1[J]) * HT + Y[J]; NEXT: DATA[3]:= HL; DATA[4]:= DATA[4] + 1; X:= XT; OUT; "IF" X ^= XE "THEN" "GOTO" TEST "END" RKE; "EOP" 1SECTION : 5.2.1.1.1.1.C (FEBRUARY 1979) PAGE 1 PROCEDURE : RK4A. AUTHOR:J.A.ZONNEVELD. CONTRIBUTORS:M.BAKKER AND I.BRINK. INSTITUTE: MATHEMATICAL CENTRE, RECEIVED: 730715. BRIEF DESCRIPTION: RK4A SOLVES AN INITIAL VALUE PROBLEM FOR A SINGLE FIRST ORDER ORDINARY DIFFERENTIAL EQUATION DY / DX = F(X,Y), WHERE F(X,Y) MAY BECOME LARGE, E.G. IN THE NEIGHBOURHOOD OF A SINGULARITY. RK4A INTERCHANGES THE DEPENDENT AND INDEPENDENT VARIABLE. THE EQUATION IS UPPOSED TO BE NON-STIFF. KEYWORDS: INITIAL VALUE PROBLEM, SINGLE FIRST ORDER ORDINARY DIFFERENTIAL EQUATION LARGE DERIVATIVE VALUES. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" RK4A(X,XA,B,Y,YA,FXY,E,D,FI,XDIR,POS); "VALUE" FI,XDIR,POS; "BOOLEAN" FI,XDIR,POS; "REAL" X,XA,B,Y,YA,FXY; "ARRAY" E,D; "CODE" 33016; 1SECTION : 5.2.1.1.1.1.C (FEBRUARY 1979) PAGE 2 THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE ; UPON COMPLETION OF A CALL X IS EQUAL TO THE MOST RECENT VALUE OF THE INDEPENDENT VARIABLE; XA: ; ENTRY: THE INITIAL VALUE OF X; B: ; B DEPENDS ON X AND Y. THE EQUATION B=0 FULFILLED WITHIN THE TOLERANCES E[4] AND E[5] SPECIFIES THE END OF THE INTEGRATION INTERVAL. AT THE END OF EACH INTEGRATION STEP B IS EVALUATED AND IS TESTED FOR CHANGE OF SIGN; Y: ; THE DEPENDENT VARIABLE; YA: ; ENTRY: THE VALUE OF Y AT X=XA; FXY: ; FXY GIVES THE VALUE OF DY/DX; FXY USES X AND Y AS JENSEN PARAMETERS. E: ; "ARRAY" E[0:5]; ENTRY: E[0], E[2] : RELATIVE TOLERANCES, FOR X AND Y RESPECTIVELY; E[1], E[3] : ABSOLUTE TOLERANCES, FOR X AND Y RESPECTIVELY; E[4], E[5] : TOLERANCES USED IN THE DETERMINATION OF THE ZERO OF B; D: ; "ARRAY" D[0:4]; AFTER COMPLETION OF EACH STEP WE HAVE : IF D[0]>0 THEN X IS THE INTEGRATION VARIABLE; IF D[0]<0 THEN Y IS THE INTEGRATION VARIABLE; D[1] IS THE NUMBER OF STEPS SKIPPED; D[2] IS THE STEPSIZE; D[3] IS EQUAL TO THE LAST VALUE OF X; D[4] IS EQUAL TO THE LAST VALUE OF Y; FI: ; IF FI="TRUE" THEN THE INTEGRATION IS STARTED WITH INITIAL VALUES X=XA,Y=YA. IF FI="FALSE" THEN THE INTEGRATION IS STARTED WITH X=D[3], Y=D[4]; XDIR, POS : ; IF FI="TRUE" THEN THE INTEGRATION STARTS IN SUCH A WAY THAT IF POS="TRUE" AND XDIR="TRUE" THEN X INCREASES, IF POS="TRUE" AND XDIR="FALSE" THEN Y INCREASES, IF POS="FALSE" AND XDIR="TRUE" THEN X DECREASES, IF POS="FALSE" AND XDIR="FALSE" THEN Y DECREASES. 1SECTION : 5.2.1.1.1.1.C (FEBRUARY 1979) PAGE 3 PROCEDURES USED : ZEROIN = CP34150. METHOD AND PERFORMANCE: RK4A IS BASED ON AN EXPLICIT, 5-TH ORDER RUNGE-KUTTA METHOD AND INTERCHANGES THE DEPENDENT AND INDEPENDENT INTEGRATION VARIABLE. IF ABS(F(X,Y)) < 1, X IS USED AS INTEGRATION VARIABLE,OTHERWISE Y. THE PROCEDURE IS PROVIDED WITH STEP SIZE AND ERROR CONTROL. FOR DETAILS,E.G. HOW TO USE ARRAY E AND HOW TO SPECIFY THE ENDPOINT SEE [1] ( RK4A IS A SLIGHTLY ADAPTED VERSION OF RK4). REFERENCES: [1]J.A.ZONNEVELD, AUTOMATIC NUMERICAL INTEGRATION. MATHEMATICAL CENTRE TRACT 8(1970). EXAMPLE OF USE: THE SOLUTION OF THE DIFFERENTIAL EQUATION DY/DX=1-2*(X**2+Y), X>=0, Y=0 , X =0, IS REPRESENTED BY THE PARABOLA Y=X*(1-X); WE WOULD LIKE TO FIND THE VALUE OF X FOR WHICH THE CURVE OF THE SOLUTION INTERSECTS THE LINE Y+X=0. THE SOLUTION CAN BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "COMMENT" INTEGRATION OF DY/DX=1-2*(Y+X**2), X>=0, Y(0)=0,UNTIL THE CONDITION Y+X=0 IS SATISFIED; "REAL" X,Y;"ARRAY" D[0:4],E[0:5]; E[0]:=E[1]:=E[2]:=E[3]:=E[4]:=E[5]:="-6; RK4A(X,0.0,X+Y,Y,0.0,1-2*(X*X+Y),E,D,"TRUE","TRUE","TRUE"); OUTPUT(61,"("10B"("X=")"+D.9D,10B"("EXACTLY:")"2B+D.9D/, 10B"("Y=")"+D.9D/,10B"("Y-X*(1-X)=")"+.10D")",X,2, Y,Y-X*(1-X)) "END" THE PROGRAM PRINTS THE FOLLOWING RESULTS: X=1.9999998554 EXACTLY: 2.0000000000 Y=-1.9999995347427 Y-X*(1-X)=0.0000000313 1SECTION : 5.2.1.1.1.1.C (AUGUST 1974) PAGE 4 SOURCE TEXT(S): 0"CODE" 33016 ; "PROCEDURE" RK4A(X, XA, B, Y, YA, FXY, E, D, FI, XDIR, POS); "VALUE" FI, XDIR, POS; "BOOLEAN" FI, XDIR, POS; "REAL" X, XA, B, Y, YA, FXY; "ARRAY" E, D; "BEGIN" "INTEGER" I; "BOOLEAN" IV, FIRST, FIR, REJ; "REAL" K0, K1, K2, K3, K4, K5, FHM, ABSH, DISCR, S, XL, COND0, S1, COND1, YL, HMIN, H, ZL, TOL, HL, MU, MU1; "ARRAY" E1[1:2]; "PROCEDURE" RKSTEP(X, XL, H, Y, YL, ZL, FXY, D); "VALUE" XL, YL, ZL, H; "REAL" X, XL, H, Y, YL, ZL, FXY; "INTEGER" D; "BEGIN" "IF" D = 2 "THEN" "GOTO" INTEGRATE; "IF" D = 3 "THEN" "BEGIN" X:= XL; Y:= YL; K0:= FXY * H "END" "ELSE" "IF" D = 1 "THEN" K0:= ZL * H "ELSE" K0:= K0 * MU; X:= XL + H / 4.5; Y:= YL + K0 / 4.5; K1:= FXY * H; X:= XL + H / 3; Y:= YL + (K0 + K1 * 3) / 12; K2:= FXY * H; X:= XL + H * .5; Y:= YL + (K0 + K2 * 3) / 8; K3:= H * FXY; X:= XL + H * .8; Y:= YL + (K0 * 53 - K1 * 135 + K2 * 126 + K3 * 56) / 125; K4:= FXY * H; "IF" D <= 1 "THEN" "BEGIN" X:= XL + H; Y:= YL + (K0 * 133 - K1 * 378 + K2 * 276 + K3 * 112 + K4 * 25) / 168; K5:= FXY * H; DISCR:= ABS(K0 * 21 - K2 * 162 + K3 * 224 - K4 * 125 + K5 * 42) / 14; "GOTO" END "END"; INTEGRATE: X:= XL + H; Y:= YL + ( - K0 * 63 + K1 * 189 - K2 * 36 - K3 * 112 + K4 * 50) / 28; K5:= FXY * H; Y:= YL + (K0 * 35 + K2 * 162 + K4 * 125 + K5 * 14) / 336; END: "END" RKSTEP; "COMMENT" 1SECTION : 5.2.1.1.1.1.C (AUGUST 1974) PAGE 5 ; "REAL" "PROCEDURE" FZERO; "BEGIN" "IF" IV "THEN" "BEGIN" "IF" S = XL "THEN" FZERO:= COND0 "ELSE" "IF" S = S1 "THEN" FZERO:= COND1 "ELSE" "BEGIN" RKSTEP(X, XL, S - XL, Y, YL, ZL, FXY, 3); FZERO:= B "END" "END" "ELSE" "BEGIN" "IF" S = YL "THEN" FZERO:= COND0 "ELSE" "IF" S = S1 "THEN" FZERO:= COND1 "ELSE" "BEGIN" RKSTEP(Y, YL, S - YL, X, XL, ZL, 1 / FXY, 3); FZERO:= B "END" "END" "END" FZERO; "IF" FI "THEN" "BEGIN" D[3]:= XA; D[4]:= YA; D[0]:= 1 "END"; D[1]:= 0; X:= XL:= D[3]; Y:= YL:= D[4]; IV:= D[0] > 0; FIRST:= FIR:= "TRUE"; HMIN:= E[0] + E[1]; H:= E[2] + E[3]; "IF" H < HMIN "THEN" HMIN:= H; CHANGE: ZL:= FXY; "IF" ABS(ZL) <= 1 "THEN" "BEGIN" "IF" "NOT"IV "THEN" "BEGIN" D[2]:= H:= H / ZL; D[0]:= 1; IV:= FIRST:= "TRUE" "END"; "IF" FIR "THEN" "GOTO" A; I:= 1; "GOTO" AGAIN "END" "ELSE" "BEGIN" "IF" IV "THEN" "BEGIN" "IF" "NOT"FIR "THEN" D[2]:= H:= H * ZL; D[0]:= - 1; IV:= "FALSE"; FIRST:= "TRUE" "END"; "IF" FIR "THEN" "BEGIN" H:= E[0] + E[1]; A: "IF" ("IF" FI "THEN" ("IF" IV "EQV" XDIR "THEN" H "ELSE" H * ZL) < 0 "EQV" POS "ELSE" H * D[2] < 0) "THEN" H:= - H "END"; I:= 1 "END"; "COMMENT" 1SECTION : 5.2.1.1.1.1.C (AUGUST 1974) PAGE 6 ; AGAIN: ABSH:= ABS(H); "IF" ABSH < HMIN "THEN" "BEGIN" H:= SIGN(H) * HMIN; ABSH:= HMIN "END"; "IF" IV "THEN" "BEGIN" RKSTEP(X, XL, H, Y, YL, ZL, FXY, I); TOL:= E[2] * ABS(K0) + E[3] * ABSH "END" "ELSE" "BEGIN" RKSTEP(Y, YL, H, X, XL, 1 / ZL, 1 / FXY, I); TOL:= E[0] * ABS(K0) + E[1] * ABSH "END"; REJ:= DISCR > TOL; MU:= TOL / (TOL + DISCR) + .45; "IF" REJ "THEN" "BEGIN" "IF" ABSH <= HMIN "THEN" "BEGIN" "IF" IV "THEN" "BEGIN" X:= XL + H; Y:= YL + K0 "END" "ELSE" "BEGIN" X:= XL + K0; Y:= YL + H "END"; D[1]:= D[1] + 1; FIRST:= "TRUE"; "GOTO" NEXT "END"; H:= H * MU; I:= 0; "GOTO" AGAIN "END" REJ; "IF" FIRST "THEN" "BEGIN" FIRST:= FIR; HL:= H; H:= MU * H; "GOTO" ACCEPT "END"; FHM:= MU * H / HL + MU - MU1; HL:= H; H:= FHM * H; ACCEPT: "IF" IV "THEN" RKSTEP(X, XL, HL, Y, YL, ZL, FXY, 2) "ELSE" RKSTEP(Y, YL, HL, X, XL, ZL, 1 / FXY, 2); MU1:= MU; NEXT: "IF" FIR "THEN" "BEGIN" FIR:= "FALSE"; COND0:= B; "IF" "NOT"(FI "OR" REJ) "THEN" H:= D[2] "END" "ELSE" "BEGIN" D[2]:= H; COND1:= B; "IF" COND0 * COND1 <= 0 "THEN" "GOTO" ZERO; COND0:= COND1 "END"; D[3]:= XL:= X; D[4]:= YL:= Y; "GOTO" CHANGE; ZERO: E1[1]:= E[4]; E1[2]:= E[5]; S1:= "IF" IV "THEN" X "ELSE" Y; S:= "IF" IV "THEN" XL "ELSE" YL ; ZEROIN(S,S1,FZERO,ABS(E1[1]*S)+ABS(E1[2])) ; S1:= "IF" IV "THEN" X "ELSE" Y ; "IF" IV "THEN" RKSTEP(X, XL, S - XL, Y, YL, ZL, FXY, 3) "ELSE" RKSTEP(Y, YL, S - YL, X, XL, ZL, 1 / FXY, 3); D[3]:= X; D[4]:= Y "END" RK4A; "EOP" 1SECTION : 5.2.1.1.1.1.D (FEBRUARY 1979) PAGE 1 PROCEDURE : RK4NA. AUTHOR:J.A.ZONNEVELD. CONTRIBUTORS:M.BAKKER AND I.BRINK. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730715. BRIEF DESCRIPTION: RK4NA SOLVES AN INITIAL VALUE PROBLEM FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DY / DX = F(X,Y), OF WHICH THE DERIVATIVE COMPONENTS ARE SUPPOSED TO BECOME LARGE, E.G. IN THE NEIGHBOURHOOD OF SINGULARITIES. RK4NA INTERCHANGES THE INDEPENDENT VARIABLE AND ONE DEPENDENT VARIABLE. THE SYSTEM IS SUPPOSED TO BE NON-STIFF. KEYWORDS: INITIAL VALUE PROBLEM, SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS, LARGE DERIVATIVE COMPONENTS. 1SECTION : 5.2.1.1.1.1.D (FEBRUARY 1979) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" RK4NA(X,XA,B,FXJ,J,E,D,FI,N,L,POS); "VALUE" FI,N,L,POS; "INTEGER" J,N,L; "BOOLEAN" FI,POS; "REAL" B,FXJ; "ARRAY" X,XA,E,D; "CODE" 33017; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; "ARRAY" X[0:N]; X[0] IS THE INDEPENDENT VARIABLE, X[1],...,X[N] ARE THE DEPENDENT VARIABLES; EXIT:THE SOLUTION AT B=0; XA: ; "ARRAY" XA[0:N]; ENTRY: THE INITIAL VALUES OF X[0],...,X[N]; B: ; B DEPENDS ON X[0],...,X[N]; IF THE EQUATION B=0 IS SATISFIED WITHIN A CERTAIN TOLERANCE,THE INTEGRATION IS TERMINATED; B IS EVALUATED AND TESTED FOR CHANGE OF SIGN AT THE END OF EACH STEP; FOR THE TOLERANCE SEE PARAMETER E. FXJ: ; FXJ DEPENDS ON X[0],...,X[N] AND J, DEFINING THE RIGHT HAND SIDE OF THE DIFFERENTIAL EQUATION; AT EACH CALL IT DELIVERS :DX[J]/DX[0]; J: ; J IS USED AS A JENSEN PARAMETER FOR FXJ; E: ; "ARRAY" E[0:2*N+3]; ENTRY: E[2*J] AND E[2*J+1] , 0<=J<=N , ARE THE RELATIVE AND THE ABSOLUTE TOLERANCE , RESPECTIVELY, ASSOCIATED WITH X[J]; E[2*N+2] AND E[2*N+3] ARE THE RELATIVE AND ABSOLUTE TOLERANCE USED IN THE DETERMINATION OF THE ZERO OF B; D: ; "ARRAY" D[0:N+3]; AFTER COMPLETION OF EACH STEP WE HAVE : ENTIER(D[0]+.5) IS THE NUMBER OF STEPS SKIPPED; D[2] IS THE STEP LENGTH; D[J+3] IS THE LAST VALUE OF X[J], J=0,...,J=N; 1SECTION : 5.2.1.1.1.1.D (DECEMBER 1975) PAGE 3 FI: IF FI="TRUE" THEN THE INTEGRATION IS STARTED WITH INITIAL CONDITIONS X[J]=XA[J];IF FI="FALSE" THEN THE INTEGRATION IS CONTINUED WITH X[J]=D[J+3]; N: ; THE NUMBER OF EQUATIONS; L: ; AN INTEGER TO BE SUPPLIED BY THE USER, 0<=L<=N (SEE POS); POS: IF FI="TRUE" THEN THE INTEGRATION STARTS IN SUCH A WAY THAT X[L] INCREASES IF POS="TRUE" AND X[L] DECREASES IF POS="FALSE"; IF FI="FALSE" THEN POS IS OF NO SIGNIFICANCE. PROCEDURES USED : ZEROIN = CP34150. REQUIRED CENTRAL MEMORY : CIRCA 9*N MEMORY PLACES. METHOD AND PERFORMANCE : RK4NA IS BASED ON AN EXPLICIT, 5-TH ORDER RUNGE-KUTTA METHOD AND INTERCHANGES VARIABLES. THE PROCEDURE IS PROVIDED WITH STEPSIZE AND ERROR CONTROL. FOR DETAILS, E.G. HOW TO USE ARRAY E, HOW TO SPECIFY THE ENDPOINT, HOW TO USE L AND POS, SEE [1] ( RK4NA IS A SLIGHTLY ADAPTED VERSION OF RK4N ). REFERENCES: [1].J.A.ZONNEVELD. AUTOMATIC NUMERICAL INTEGRATION. MATHEMATICAL CENTRE TRACT 8 (1970). 1SECTION : 5.2.1.1.1.1.D (FEBRUARY 1979) PAGE 4 EXAMPLE OF USE: THE PERIOD OF THE SOLUTION OF THE VAN DER POL EQUATION DX[1]/DT = X[2] DX[2]/DT = 10*(1-X[1]**2)*X[2]-X[1], T>=0, CAN BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "COMMENT" VAN DER POL; "REAL" X0; "PROCEDURE" PRINT(X);"ARRAY" X; OUTPUT(61,"("/,4(+ZD.10D3B)")",X[0],X[1],X[2],X0); "INTEGER" J,K;"BOOLEAN" FIRST; "ARRAY" E[0:7],XA,X[0:2],D[0:5]; "FOR" K:=0,1,2,3,4,5 "DO" E[K]:=.1"-6; E[6]:=E[7]:="-8 ; OUTPUT(61,"(""("VAN DER POL")",/BB"("EPS=")"D.10D,/BB "("THE VALUES OF X[0],X[1],X[2],P,RESPECTIVELY")"")",E[0]); X0:=XA[0]:=XA[2]:=0;XA[1]:=2;J:=0;PRINT(XA); FIRST:="TRUE"; "FOR" J:=1,2,3,4 "DO" "BEGIN" RK4NA(X,XA,X[2],"IF" K=1 "THEN" X[2] "ELSE" 10*(1-X[1]**2)*X[2]-X[1],K,E,D,FIRST,2,0,"TRUE"); X0:=X[0]-X0;PRINT(X);FIRST:="FALSE"; X0:=X[0] "END" "END" THE PROGRAM PRINTS THE FOLLOWING RESULTS: VAN DER POL EPS=0.0000001000 THE VALUES OF X[0],X[1],X[2],P,RESPECTIVELY: +0.00000000 +2.00000000 +0.00000000 +0.00000000 +9.32386570 -2.01428560 +0.00000000 +9.32386570 +18.86305411 +2.01428557 +0.00000000 +9.53918840 +28.40224194 -2.01428858 -0.00000000 +9.53918783 +37.94143003 +2.01428558 +0.00000000 +9.53918809 1SECTION : 5.2.1.1.1.1.D (AUGUST 1974) PAGE 5 SOURCE TEXT(S): 0"CODE" 33017 ; "PROCEDURE" RK4NA(X, XA, B, FXJ, J, E, D, FI, N, L, POS); "VALUE" FI, N, L, POS; "INTEGER" J, N, L; "BOOLEAN" FI, POS; "REAL" B, FXJ; "ARRAY" X, XA, E, D; "BEGIN" "INTEGER" I, IV, IV0; "BOOLEAN" FIR, FIRST, REJ; "REAL" H, COND0, COND1, FHM, ABSH, TOL, FH, MAX, X0, X1, S, HMIN, HL, MU, MU1; "ARRAY" XL, DISCR, Y[0:N], K[0:5,0:N], E1[1:2]; "PROCEDURE" RKSTEP(H, D); "VALUE" H, D; "INTEGER" D; "REAL" H; "BEGIN" "INTEGER" I; "PROCEDURE" F(T); "VALUE" T; "INTEGER" T; "BEGIN" "INTEGER" I; "REAL" P; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" Y[J]:= FXJ; P:= H / Y[IV]; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" K[T,I]:= Y[I] * P "END" "END" F; "IF" D = 2 "THEN" "GOTO" INTEGRATE; "IF" D = 3 "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I]; F(0) "END" "ELSE" "IF" D = 1 "THEN" "BEGIN" "REAL" P; P:= H / Y[IV]; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" K[0,I]:= P * Y[I] "END" "END" "ELSE" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" K[0,I]:= K[0,I] * MU "END"; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H "ELSE" K[0,I]) / 4.5; F(1); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H * 4 "ELSE" (K[0,I] + K[1,I] * 3)) / 12; F(2); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H * .5 "ELSE" (K[0,I] + K[2,I] * 3) / 8); F(3); "COMMENT" 1SECTION : 5.2.1.1.1.1.D (AUGUST 1974) PAGE 6 ; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H * .8 "ELSE" (K[0,I] * 53 - K[1,I] * 135 + K[2,I] * 126 + K[3,I] * 56) / 125); F(4); "IF" D <= 1 "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H "ELSE" (K[0,I] * 133 - K[1,I] * 378 + K[2,I] * 276 + K[3,I] * 112 + K[4,I] * 25) / 168); F(5); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" DISCR[I]:= ABS(K[0,I] * 21 - K[2,I] * 162 + K[3,I] * 224 - K[4,I] * 125 + K[5,I] * 42) / 14 "END"; "GOTO" END "END"; INTEGRATE: "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ("IF" I = IV "THEN" H "ELSE" ( - K[0,I] * 63 + K[1,I] * 189 - K[2,I] * 36 - K[3,I] * 112 + K[4,I] * 50) / 28); F(5); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" X[I]:= XL[I] + (K[0,I] * 35 + K[2,I] * 162 + K[4,I] * 125 + K[5,I] * 14) / 336 "END" ; END .. "END" RKSTEP ; "REAL" "PROCEDURE" FZERO; "BEGIN" "IF" S = X0 "THEN" FZERO:= COND0 "ELSE" "IF" S = X1 "THEN" FZERO:= COND1 "ELSE" "BEGIN" RKSTEP(S - XL[IV], 3); FZERO:= B "END" "END" FZERO; "IF" FI "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= XA[I]; D[0]:= D[2]:= 0 "END"; D[1]:= 0; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I]:= D[I + 3]; IV:= D[0]; H:= D[2]; FIRST:= FIR:= "TRUE"; Y[0]:= 1; "GOTO" CHANGE; AGAIN: ABSH:= ABS(H); "IF" ABSH < HMIN "THEN" "BEGIN" H:= "IF" H > 0 "THEN" HMIN "ELSE" - HMIN; ABSH:= ABS(H) "END"; RKSTEP(H, I); REJ:= "FALSE"; FHM:= 0; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" "BEGIN" TOL:= E[2 * I] * ABS(K[0,I]) + E[2 * I + 1] * ABSH; REJ:= TOL < DISCR[I] "OR" REJ; FH:= DISCR[I] / TOL; "IF" FH > FHM "THEN" FHM:= FH "END" "END"; "COMMENT" 1SECTION : 5.2.1.1.1.1.D (AUGUST 1974) PAGE 7 ; MU:= 1 / (1 + FHM) + .45; "IF" REJ "THEN" "BEGIN" "IF" ABSH <= HMIN "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" I ^= IV "THEN" X[I]:= XL[I] + K[0,I] "ELSE" X[I]:= XL[I] + H "END"; D[1]:= D[1] + 1; FIRST:= "TRUE"; "GOTO" NEXT "END"; H:= H * MU; I:= 0; "GOTO" AGAIN "END"; "IF" FIRST "THEN" "BEGIN" FIRST:= FIR; HL:= H; H:= MU * H; "GOTO" ACCEPT "END"; FH:= MU * H / HL + MU - MU1; HL:= H; H:= FH * H; ACCEPT: RKSTEP(HL, 2); MU1:= MU; NEXT: "IF" FIR "THEN" "BEGIN" FIR:= "FALSE"; COND0:= B; "IF" "NOT"(FI "OR" REJ) "THEN" H:= D[2] "END" "ELSE" "BEGIN" D[2]:= H; COND1:= B; "IF" COND0 * COND1 <= 0 "THEN" "GOTO" ZERO; COND0:= COND1 "END"; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= XL[I]:= X[I]; CHANGE: IV0:= IV; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" Y[J]:= FXJ; MAX:= ABS(Y[IV]); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" "IF" ABS(Y[I]) > MAX "THEN" "BEGIN" MAX:= ABS(Y[I]); IV:= I "END" "END"; "IF" IV0 ^= IV "THEN" "BEGIN" FIRST:= "TRUE"; D[0]:= IV; D[2]:= H:= Y[IV] / Y[IV0] * H "END"; X0:= XL[IV]; "IF" FIR "THEN" "BEGIN" HMIN:= E[0] + E[1]; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" H:= E[2 * I] + E[2 * I + 1]; "IF" H < HMIN "THEN" HMIN:= H "END"; H:= E[2 * IV] + E[2 * IV + 1]; "IF" (FI "AND" (Y[L]/Y[IV]*H<0 "EQV" POS)) "OR" ("NOT" FI "AND" D[2]*H<0) "THEN" H:= -H "END"; I:= 1; "GOTO" AGAIN; ZERO: E1[1]:= E[2 * N + 2]; E1[2]:= E[2 * N + 3]; X1:=X[IV] ; S:=X0 ; ZEROIN(S,X1,FZERO,ABS(E1[1]*S) + ABS(E1[2])) ; X0:=S ; X1:=X[IV]; RKSTEP(X0 - XL[IV], 3); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= X[I] "END" RK4NA; "EOP" 1SECTION : 5.2.1.1.1.1.E (FEBRUARY 1979) PAGE 1 PROCEDURE : RK5NA. AUTHOR: J.A.ZONNEVELD. CONTRIBUTORS: M.BAKKER AND I.BRINK. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730715. BRIEF DESCRIPTION: RK5NA SOLVES AN INITIAL VALUE PROBLEM FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DY / DX = F(X,Y), OF WHICH THE DERIVATIVE COMPONENTS ARE SUPPOSED TO BECOME LARGE, E.G. IN THE NEIGHBOURHOOD OF SINGULARITIES. RK5NA INTRODUCES THE ARC LENGTH AS INTEGRATION VARIABLE. THE SYSTEM IS SUPPOSED TO BE NON-STIFF. KEYWORDS: INITIAL VALUE PROBLEM. SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. LARGE DERIVATIVE COMPONENTS. 1SECTION : 5.2.1.1.1.1.E (FEBRUARY 1979) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" RK5NA(X, XA, B, FXJ, J, E, D, FI, N, L, POS); "VALUE" FI, N, L, POS; "INTEGER" J, N, L; "BOOLEAN" FI, POS; "REAL" B, FXJ; "ARRAY" X, XA, E, D; "CODE" 33018; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; "ARRAY" X[0 : N]; THE DEPENDENT VARIABLES; X[0], ..., X[N] CAN BE USED AS JENSEN PARAMETERS; XA: ; "ARRAY" XA[0 : N]; ENTRY: THE INITIAL VALUES OF X[J], J = 0, ..., N; B: ; B DEPENDS ON X[0], ..., X[N]; IF,WITHIN SOME TOLERANCE,B = 0 THEN THE INTEGRATION IS TER- MINATED; SEE ALSO THE EXPLANATION OF THE PARAMETER E; FXJ: ; THE RIGHT HAND SIDE OF THE DIFFERENTIAL EQUATION; FXJ DEPENDS ON X[0], ..., X[N], J, GIVING THE VALUE OF DX[J] / DX[0]; J: ; J IS USED AS A JENSEN PARAMETER TO DENOTE,IN THE ACTUAL PARAMETER CORRESPONDING TO FXJ,THE NUMBER OF THE FUNCTION REQUIRED; E: ; "ARRAY" E[0 : 2 * N + 3]; ENTRY: E[2 * J] AND E[2 * J + 1] ARE THE RELATIVE AND THE ABSOLUTE TOLERANCE,RESPECTIVELY,ASSOCIATED WITH X[J],J = 0,..., N, WHILE E[2 * N + 2] AND E[2 * N + 3] ARE THE ONES ASSOCIATED WITH B; D: ; "ARRAY" D[1 : N + 3]; AFTER COMPLETION OF EACH STEP WE HAVE: ABS(D[1]) THE ARC LENGTH, D[2] THE STEP LENGTH, D[I + 3] THE LATEST VALUE OF X[I], I = 0, ..., N; FI: ; IF FI = "TRUE" THEN THE INTEGRATION IS STARTED WITH INITIAL CONDITIONS X[I] = XA[I], I = 0, ..., N; IF FI = "FALSE" THEN THE INTEGRATION IS CONTINUED WITH X[I] = D[I + 3]; N: ; THE NUMBER OF EQUATIONS; 1SECTION : 5.2.1.1.1.1.E (FEBRUARY 1979) PAGE 3 L: ; 1 <= L <= N; SEE THE EXPLANATION OF POS; POS: ; IF FI = "TRUE" THEN THE INTEGRATION STARTS IN SUCH A WAY THAT X[L] INCREASES IF POS = "TRUE" AND DECREASES IF POS = "FALSE". IF FI = "FALSE" THEN POS IS IGNORED. PROCEDURES USED : ZEROIN = CP34150. REQUIRED CENTRAL MEMORY : CIRCA 9 * N MEMORY PLACES. METHOD AND PERFORMANCE : RK5NA IS BASED ON A 5-TH ORDER RUNGE-KUTTA METHOD AND INTEGRATES THE SYSTEM DX[J] / DX[0] = F(J,X[0],..,X[N]) / F(0,X[0],..,X[N]). THE ARC LENGTH IS INTRODUCED AS INTEGRATION VARIABLE. THE INTEGRATION PROCESS IS TERMINATED IF SOME CONDITION ON X[0], .. ,X[N], TO BE SUPPLIED BY THE USER, IS SATISFIED. RK5NA USES STEPLENGTH AND ERROR CONTROL. DETAILS ABOUT THE PROCEDURE AND THE UNDERLYING THEORY ARE GIVEN IN [1] ( RK5NA IS A SLIGHTLY ADAPTED VERSION OF RK5N). REFERENCES: [1]. J.A.ZONNEVELD, AUTOMATIC NUMERICAL INTEGRATION, MATH.CENTRE TRACT 8 (1970) . 1SECTION : 5.2.1.1.1.1.E (AUGUST 1974) PAGE 4 EXAMPLE OF USE: THE VAN DER POL EQUATION IN THE PHASE PLANE DX[1] / DX[0] = (10*(1-X[0]**2)*X[1]-X[0])/X[1] CAN BE INTEGRATED BY THE PROCEDURE RK5NA; THE STARTING VALUES ARE X[0] = 2, X[1] = 0. THE INTEGRATION PROCEEDS UNTIL THE NEXT ZERO OF X[1],THEN IT CONTINUES UNTIL THE NEXT ZERO AND SO ON UNTIL THE FOURTH ZERO IS ENCOUNTERED.IN THE EXAMPLE THE OUTPUT IS GIVEN FOR THE TOLERANCES E[0]=E[1]=E[2]=E[3]= "-6, E[4]=E[5]= "-10. THE PROGRAM READS AS FOLLOWS: "BEGIN" "COMMENT" VAN DER POL IN THE PHASE PLANE; "INTEGER" J,K; "BOOLEAN" FIRST; "ARRAY" E[0:5],XA,X[0:1],D[1:4]; "PROCEDURE" PRINT(X); "ARRAY" X; "BEGIN" OUTPUT(61,"("/B"("X[0]=")"+D.10D, 10B"("X[1]=")"+D.10D,10B"("S=")"3D.10D")",X[0], X[1],ABS(D[1])) "END"; "FOR" K:=0,1,2,3 "DO" E[K]:="-6 ; E[4]:=E[5]:="-10 ; D[1]:=0; XA[0]:=2; XA[1]:=0; J:=0; PRINT(XA); AA: FIRST:=J=0; RK5NA(X,XA,X[1],"IF" K=0 "THEN" X[1] "ELSE" 10*(1-X[0]**2)*X[1]-X[0],K,E,D,FIRST,1,1,"FALSE");;J:=J+1; PRINT(X); "IF" J<4 "THEN" "GO TO" AA "END" RESULTS: X[0]=+2.0000000000 X[1]=+0.0000000000 S=000.0000000000 X[0]=-2.0142853657 X[1]=-0.0000000012 S=029.3873834087 X[0]=+2.0142853659 X[1]=+0.0000000001 S=058.7884331939 X[0]=-2.0142853659 X[1]=-0.0000000000 S=088.1894829781 X[0]=+2.0142853659 X[1]=+0.0000000000 S=117.5905327623 1SECTION : 5.2.1.1.1.1.E (AUGUST 1974) PAGE 5 SOURCE TEXT(S): "CODE" 33018 ; "PROCEDURE" RK5NA(X, XA, B, FXJ, J, E, D, FI, N, L, POS); "VALUE" FI, N, L, POS; "INTEGER" J, N, L; "BOOLEAN" FI, POS; "REAL" B, FXJ; "ARRAY" X, XA, E, D; "BEGIN" "INTEGER" I; "BOOLEAN" FIRST, FIR, REJ; "REAL" FHM, S, S0, COND0, S1, COND1, H, ABSH, TOL, FH, HL, MU, MU1; "ARRAY" Y, XL, DISCR[0:N], K[0:5,0:N], E1[1:2]; "REAL" "PROCEDURE" SUM(J,A,B,XJ) ; "INTEGER" J,A,B ; "REAL" XJ ; "BEGIN" "REAL" S ; S:= 0 ; "FOR" J:=A "STEP" 1 "UNTIL" B "DO" S:=S+XJ ; SUM:= S "END" SUM ; "PROCEDURE" RKSTEP(H, D); "VALUE" H, D; "INTEGER" D; "REAL" H; "BEGIN" "INTEGER" I; "PROCEDURE" F(T); "VALUE" T; "INTEGER" T; "BEGIN" "INTEGER" I; "REAL" P; "FOR" J:= 0 "STEP" 1 "UNTIL" N "DO" Y[J]:= FXJ; P:= H / SQRT(SUM(I, 0, N, Y[I] ** 2)); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" K[T,I]:= Y[I] * P "END" F; "IF" D = 2 "THEN" "GOTO" INTEGRATE; "IF" D = 1 "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" K[0,I]:= K[0,I] * MU; "GOTO" A "END"; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I]; F(0); A: "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + K[0,I] / 4.5; F(1); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + (K[0,I] + K[1,I] * 3) / 12; F(2); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + (K[0,I] + K[2,I] * 3) / 8; F(3); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + (K[0,I] * 53 - K[1,I] * 135 + K[2,I] * 126 + K[3,I] * 56) / 125; F(4); "IF" D <= 1 "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + (K[0,I] * 133 - K[1,I] * 378 + K[2,I] * 276 + K[3,I] * 112 + K[4,I] * 25) / 168; F(5); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" DISCR[I]:= ABS(K[0,I] * 21 - K[2,I] * 162 + K[3,I] * 224 - K[4,I] * 125 + K[5,I] * 42) / 14; "GOTO" END "END"; INTEGRATE: "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + ( - K[0,I] * 63 + K[1,I] * 189 - K[2,I] * 36 - K[3,I] * 112 + K[4,I] * 50) / 28; F(5); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I] + (K[0,I] * 35 + K[2,I] * 162 + K[4,I] * 125 + K[5,I] * 14) / 336; END: "END" RKSTEP; "COMMENT" 1SECTION : 5.2.1.1.1.1.E (AUGUST 1974) PAGE 6 ; "REAL" "PROCEDURE" FZERO; "BEGIN" "IF" S = S0 "THEN" FZERO:= COND0 "ELSE" "IF" S = S1 "THEN" FZERO:= COND1 "ELSE" "BEGIN" RKSTEP(S - S0, 3); FZERO:= B "END" "END" FZERO; "IF" FI "THEN" "BEGIN" "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= XA[I]; D[1]:= D[2]:= 0 "END"; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" X[I]:= XL[I]:= D[I + 3]; S:= D[1]; FIRST:= FIR:= "TRUE"; H:= E[0] + E[1]; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" ABSH:= E[2 * I] + E[2 * I + 1]; "IF" H > ABSH "THEN" H:= ABSH "END"; "IF" FI "THEN" "BEGIN" J:= L; "IF" FXJ * H < 0 "EQV" POS "THEN" H:= - H "END" "ELSE" "IF" D[2] * H < 0 "THEN" H:= - H; I:= 0; AGAIN: RKSTEP(H, I); REJ:= "FALSE"; FHM:= 0; ABSH:= ABS(H); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" "BEGIN" TOL:= E[2 * I] * ABS(K[0,I]) + E[2 * I + 1] * ABSH; REJ:= TOL < DISCR[I] "OR" REJ; FH:= DISCR[I] / TOL; "IF" FH > FHM "THEN" FHM:= FH "END"; MU:= 1 / (1 + FHM) + .45; "IF" REJ "THEN" "BEGIN" H:= H * MU; I:= 1; "GOTO" AGAIN "END"; "IF" FIRST "THEN" "BEGIN" FIRST:= FIR; HL:= H; H:= MU * H "END" "ELSE" "BEGIN" FH:= MU * H / HL + MU - MU1; HL:= H; H:= FH * H "END"; ACCEPT: RKSTEP(HL, 2); MU1:= MU; S:= S + HL; "IF" FIR "THEN" "BEGIN" COND0:= B; FIR:= "FALSE"; "IF" "NOT"FI "THEN" H:= D[2] "END" "ELSE" "BEGIN" D[2]:= H; COND1:= B; "IF" COND0 * COND1 <= 0 "THEN" "GOTO" ZERO; COND0:= COND1 "END"; "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= XL[I]:= X[I]; D[1]:= S0:= S; I:= 0; "GOTO" AGAIN; ZERO: E1[1]:= E[2 * N + 2]; E1[2]:= E[2 * N + 3]; S1:=S ; S:=S0 ; ZEROIN(S,S1,FZERO,ABS(E1[1]*S)+ABS(E1[2])) ; RKSTEP(S - S0, 3); "FOR" I:= 0 "STEP" 1 "UNTIL" N "DO" D[I + 3]:= X[I]; D[1]:= S "END" RK5NA; "EOP" 1SECTION : 5.2.1.1.1.1.F (FEBRUARY 1979) PAGE 1 PROCEDURE : MULTISTEP. AUTHOR: P.W.HEMKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730515. BRIEF DESCRIPTION: MULTISTEP SOLVES AN INITIAL VALUE PROBLEM, FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DY = F(Y). IN PARTICULAR THIS PROCEDURE IS SUITABLE FOR THE INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS. IT CAN ALSO BE USED FOR NON-STIFF PROBLEMS. KEYWORDS: INITIAL VALUE PROBLEM, SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. STIFF EQUATIONS, 1SECTION : 5.2.1.1.1.1.F (FEBRUARY 1979) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "BOOLEAN" "PROCEDURE" MULTISTEP(X,XEND,Y,HMIN,HMAX,YMAX,EPS, FIRST,SAVE,DERIV,AVAILABLE,JACOBIAN,STIFF,N,OUT); "VALUE" HMIN,HMAX,EPS,XEND,N,STIFF; "BOOLEAN" FIRST,AVAILABLE,STIFF; "INTEGER" N; "REAL" X,XEND,HMIN,HMAX,EPS; "ARRAY" Y,YMAX,SAVE,JACOBIAN; "PROCEDURE" DERIV,OUT; "CODE" 33080; MULTISTEP DELIVERS THE FOLLOWING BOOLEAN VALUE: IF DIFFICULTIES ARE ENCOUNTERED DURING THE INTEGRATION (I.E. SAVE[-1] ^= 0 "OR" SAVE[-2] ^= 0 ) MULTISTEP IS SET TO "FALSE", OTHERWISE MULTISTEP IS SET "TRUE". THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE X. CAN BE USED IN DERIV, AVAILABLE ETC.; ENTRY: THE INITIAL VALUE X0; EXIT : THE FINAL VALUE 'XEND'; XEND: ; THE FINAL VALUE OF X ( XEND >= X ); Y: ; "ARRAY" Y[1:6*N]; THE DEPENDENT VARIABLE; ENTRY Y[1:N] : THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT X = X0; EXIT Y[1:N] : THE FINAL VALUES OF THE SOLUTION AT X = XEND; HMIN,HMAX: ; ENTRY : MINIMAL RESP. MAXIMAL STEPLENGTH ALLOWED; YMAX: ; "ARRAY" YMAX[1:N]; ENTRY: THE ABSOLUTE LOCAL ERROR BOUND DIVIDED BY EPS EXIT : YMAX[1] GIVES THE MAXIMAL VALUE OF THE ENTRY VALUE OF YMAX[I] AND THE VALUES OF ABS(Y[I]) DURING INTEGRATION; EPS: ; THE RELATIVE LOCAL ERROR BOUND; FIRST: ; IF FIRST = "TRUE" THEN THE PROCEDURE STARTS THE INTEGRATION WITH A FIRST ORDER ADAMS METHOD AND A STEPLENGTH EQUAL TO HMIN. UPON COMPLETION OF A CALL FIRST:= "FALSE"; IF FIRST = "FALSE" THEN THE PROCEDURE CONTINUES INTEGRATION; 1SECTION : 5.2.1.1.1.1.F (DECEMBER 1979) PAGE 3 SAVE: ; "ARRAY" SAVE[-38:6*N]; IN THIS ARRAY THE PROCEDURE STORES INFORMATION WHICH CAN BE USED IN A CONTINUING CALL WITH FIRST="FALSE"; BESIDES THE FOLLOWING MESSAGES ARE DELIVERED: SAVE[ 0]=0 : AN ADAMS METHOD HAS BEEN USED; 1 : THE PROCEDURE SWITCHED TO GEARS METHOD; SAVE[-1]=0 : NO ERROR MESSAGE; 1 : WITH THE HMIN SPECIFIED THE PROCEDURE CANNOT HANDLE THE NONLINEARITY (DECREASE HMIN! ); SAVE[-2] NUMBER OF TIMES THAT THE REQUESTED LOCAL ERROR BOUND WAS EXCEEDED; SAVE[-3] IF SAVE[-2] IS NONZERO THEN SAVE[-3] GIVES AN ESTIMATE OF THE MAXIMAL LOCAL ERROR BOUND, OTHERWISE SAVE[-3]=0; DERIV: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIV(DF); "ARRAY" DF; THIS PROCEDURE SHOULD DELIVER DY[I]/DX IN DF[I]; AVAILABLE: ; IF AN ANALYTICAL EXPRESSION OF THE JACOBIAN MATRIX IS NOT AVAILABLE THIS EXPRESSION IS SET TO "FALSE"; OTHERWISE THIS EXPRESSION IS SET TO "TRUE" AND THE EVALUATION OF THIS BOOLEAN EXPRESSION MUST EFFECT THE FOLLOWING SIDE-EFFECT: THE ENTRIES OF THE JACOBIAN MATRIX D(DY[I]/DX)/DY[J] ARE DELIVERED IN THE ARRAY ELEMENTS JACOBIAN[I,J]; JACOBIAN: ; "ARRAY" JACOBIAN[1:N,1:N]; AT EACH EVALUATION OF THE BOOLEAN EXPRESSION AVAILABLE WITH THE RESULT AVAILABLE:="TRUE", THE JACOBIAN MATRIX HAS TO BE ASSIGNED TO THIS ARRAY (SEE THE EXAMPLE OF USE); STIFF: ; IF STIFF = "TRUE" THE PROCEDURE SKIPS AN ATTEMPT TO SOLVE THE PROBLEM WITH ADAMS-BASHFORTH- OR ADAMS-MOULTON METHODS, DIRECTLY USING GEARS METHOD; N: ; THE NUMBER OF EQUATIONS; OUT: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" OUT(H,K); "VALUE" H,K; "REAL" H; "INTEGER" K; AT THE END OF EACH ACCEPTED STEP OF THE INTEGRATION PROCESS THIS PROCEDURE IS CALLED. THE LAST STEPLENGTH USED (H) AND THE ORDER OF THE METHOD (K) ARE DELIVERED. AT EACH CALL OF THE PROCEDURE OUT, THE CURRENT VALUES OF THE INDEPENDENT VARIABLE (X) AND OF THE SOLUTION (Y[I](X) ) ARE AVAILABLE FOR USE. MOREOVER, IN THE NEIGHBOURHOOD OF THE CURRENT VALUE OF X, ANY VALUE OF Y[I](X SPECIFIED) CAN BE COMPUTED BY MEANS OF THE FOLLOWING INTERPOLATION FORMULA Y[I](X SPECIFIED) = SUM(J,0,K, Y[I+J*N] * ((X SPECIFIED - X)/H) ** J ). 1SECTION : 5.2.1.1.1.1.F (DECEMBER 1979) PAGE 4 PROCEDURES USED: MATVEC = CP34011 , DEC = CP34300 , SOL = CP34051 . REQUIRED CENTRAL MEMORY: CIRCA N * ( 2 * N + 5 ) MEMORY PLACES. METHOD AND PERFORMANCE : MULTISTEP IS BASED ON TWO LINEAR MULTISTEP METHODS. FOR STIFF PROBLEMS IT USES THE BACKWARD DIFFERENTIATION METHODS, FOR FOR NON-STIFF PROBLEMS THE ADAMS-BASHFORTH-MOULTON METHODS. MULTISTEP IS PROVIDED WITH ORDER, STEPSIZE AND ERROR CONTROL. REFERENCES: [1].P.W.HEMKER. AN ALGOL 60 PROCEDURE FOR THE SOLUTION OF STIFF DIFFERENTIAL EQUATIONS. MATH. CENTRE, AMSTERDAM. REPORT MR 128/71; EXAMPLE OF USE: THE SOLUTION AT X=1 AND AT X=10 OF THE DIFFERENTIAL EQUATIONS: DY[1]/DX = 0.04 * (1-Y[1]-Y[2]) - Y[1] * ("4*Y[2] + 3"7*Y[1]) DY[2]/DX = 3"7 Y[1]**2 WITH THE INITIAL CONDITIONS AT X = 0 : Y[1] = 0 AND Y[2] = 0 MAY BE OBTAINED BY THE FOLLOWING PROGRAM: 1SECTION : 5.2.1.1.1.1.F (AUGUST 1974) PAGE 5 "BEGIN" "BOOLEAN" FIRST; "INTEGER" I,J,CF,CJ,CA; "REAL" X,XEND,HMIN,EPS,R; "ARRAY" Y[1:12],YMAX[1:2],D[-40:12],JAC[1:2,1:2]; "PROCEDURE" DER (F);"ARRAY" F; "BEGIN" "REAL" R; CF:=CF+1; F[2]:= R:= 3"7*Y[1]*Y[1]; F[1]:= 0.04*(1-Y[1]-Y[2]) - "4*Y[1]*Y[2] - R; "END" F; "BOOLEAN" "PROCEDURE" AVAIL; "BEGIN" "REAL" R; CJ:= CJ+1; AVAIL:= "TRUE"; JAC[2,1]:= R:= 6"7*Y[1]; JAC[1,1]:= -0.04 - "4*Y[2] - R; JAC[1,2]:= -0.04 - "4*Y[1]; JAC[2,2]:= 0 "END" JAC AVAIL; "PROCEDURE" OUT(H,K); "VALUE" H, K; "REAL" H; "INTEGER" K; CA:= CA+1; LABEL: OUTPUT(61,"("/,"("HMIN,EPS?")",/")"); INREAL(70,HMIN); INREAL(70,EPS); "IF" HMIN<0 "THEN" "GOTO" ESCAPE; FIRST:= "TRUE"; CA:=CF:=CJ:=0; X:=0; Y[1]:= Y[2]:= 0; YMAX[1]:= 0.0001; YMAX[2]:= 1; "FOR" XEND:= 1, 10 "DO" "BEGIN" MULTISTEP(X,XEND,Y,HMIN,5,YMAX,EPS,FIRST,D,DER,AVAIL, JAC,"TRUE",2,OUT); OUTPUT(61,"("3(5ZD,2B),2(+.13D"+2D,2B),/")", CA,CF,CJ,Y[1],Y[2]); "END"; "GOTO" LABEL; ESCAPE: "END" IT DELIVERS WITH HMIN = "-10 AND EPS = "-9: 240 648 2 +.3074626[602000]"-04 +.3350951[493111]"-01 315 902 3 +.16233909[62091]"-04 +.15861383[92015]"+00 (NON-SIGNIFICANT DIGITS ARE PLACED BETWEEN [ ] ). 1SECTION : 5.2.1.1.1.1.F (DECEMBER 1979) PAGE 6 SOURCE TEXT(S): 0"CODE" 33080; "BOOLEAN" "PROCEDURE" MULTISTEP(X,XEND,Y,HMIN,HMAX,YMAX,EPS, FIRST,SAVE,DERIV,AVAILABLE,JACOBIAN,STIFF,N,OUT); "VALUE" HMIN,HMAX,EPS,XEND,N,STIFF; "BOOLEAN" FIRST,AVAILABLE,STIFF; "INTEGER" N; "REAL" X,XEND,HMIN,HMAX,EPS; "ARRAY" Y,YMAX,SAVE,JACOBIAN; "PROCEDURE" DERIV,OUT; "BEGIN" "OWN" "BOOLEAN" ADAMS,WITH JACOBIAN; "OWN" "INTEGER" M,SAME,KOLD; "OWN" "REAL" XOLD,HOLD,A0,TOLUP,TOL,TOLDWN,TOLCONV; "BOOLEAN" EVALUATE,EVALUATED,DECOMPOSE,DECOMPOSED,CONV; "INTEGER" I,J,L,K,KNEW,FAILS; "REAL" H, CH, CHNEW,ERROR,DFI,C; "ARRAY" A[0:5],DELTA,LAST DELTA,DF[1:N],JAC[1:N, 1:N],AUX[1:3]; "INTEGER" "ARRAY" P[1:N]; "REAL" "PROCEDURE" NORM2(AI); "REAL" AI; "BEGIN" "REAL" S,A; S:= 1.0"-100; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" A:= AI/YMAX[I]; S:= S + A * A "END"; NORM2:= S "END" NORM2; "PROCEDURE" RESET; "BEGIN" "IF" CH < HMIN/HOLD "THEN" CH:= HMIN/HOLD "ELSE" "IF" CH > HMAX/HOLD "THEN" CH:= HMAX/HOLD; X:= XOLD; H:= HOLD * CH; C:= 1; "FOR" J:= 0 "STEP" M "UNTIL" K*M "DO" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" Y[J+I]:= SAVE[J+I] * C; C:= C * CH "END"; DECOMPOSED:= "FALSE" "END" RESET; "COMMENT" 1SECTION : 5.2.1.1.1.1.F (AUGUST 1974) PAGE 7 ; "PROCEDURE" METHOD; "BEGIN" I:= -39; "IF" ADAMS "THEN" "BEGIN" "FOR" C:= 1,1,144,4,0,.5,1,.5,576,144,1,5/12,1, .75,1/6,1436,576,4,.375,1,11/12,1/3,1/24, 2844,1436,1,251/720,1,25/24,35/72, 5/48,1/120,0,2844,0.1 "DO" "BEGIN" I:= I+ 1; SAVE[I]:= C "END" "END" "ELSE" "BEGIN" "FOR" C:= 1,1,9,4,0,2/3,1,1/3,36,20.25,1,6/11, 1,6/11,1/11,84.028,53.778,0.25,.48,1,.7,.2,.02, 156.25, 108.51, .027778, 120/274, 1, 225/274, 85/274, 15/274, 1/274, 0, 187.69, .0047361 "DO" "BEGIN" I:= I + 1; SAVE[I]:= C "END" "END" "END" METHOD; "PROCEDURE" ORDER; "BEGIN" C:= EPS * EPS; J:= (K-1) * (K + 8)/2 - 38; "FOR" I:= 0 "STEP" 1 "UNTIL" K "DO" A[I]:= SAVE[I+J]; TOLUP := C * SAVE[J + K + 1]; TOL := C * SAVE[J + K + 2]; TOLDWN := C * SAVE[J + K + 3]; TOLCONV:= EPS/(2 * N * (K + 2)); A0:= A[0]; DECOMPOSE:= "TRUE"; "END" ORDER; "PROCEDURE" EVALUATE JACOBIAN; "BEGIN" EVALUATE:= "FALSE"; DECOMPOSE:= EVALUATED:= "TRUE"; "IF" AVAILABLE "THEN" "ELSE" "BEGIN" "REAL" D; "ARRAY" FIXY,FIXDY,DY[1:N]; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" FIXY[I]:= Y[I]; DERIV(FIXDY); "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" D:= "IF" EPS > ABS(FIXY[J]) "THEN" EPS * EPS "ELSE" EPS * ABS(FIXY[J]); Y[J]:= Y[J] + D; DERIV(DY); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" JACOBIAN[I,J]:= (DY[I]-FIXDY[I])/D; Y[J]:= FIXY[J] "END" "END" "END" EVALUATE JACOBIAN; "COMMENT" 1SECTION : 5.2.1.1.1.1.F (DECEMBER 1979) PAGE 8 ; "PROCEDURE" DECOMPOSE JACOBIAN; "BEGIN" DECOMPOSE:= "FALSE"; DECOMPOSED:= "TRUE"; C:= -A0 * H; "FOR" J:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" JAC[I,J]:= JACOBIAN[I,J] * C; JAC[J,J]:= JAC[J,J] + 1 "END"; AUX[2]:=1.0"-12; DEC(JAC,N,AUX,P) "END" DECOMPOSE JACOBIAN; "PROCEDURE" CALCULATE STEP AND ORDER; "BEGIN" "REAL" A1,A2,A3; A1:= "IF" K <= 1 "THEN" 0 "ELSE" 0.75 * (TOLDWN/NORM2(Y[K*M+I])) ** (0.5/K); A2:= 0.80 * (TOL/ERROR) ** (0.5/(K + 1)); A3:= "IF" K >= 5 "OR" FAILS ^= 0 "THEN" 0 "ELSE" 0.70 * (TOLUP/NORM2(DELTA[I] - LAST DELTA[I])) ** (0.5/(K+2)); "IF" A1 > A2 "AND" A1 > A3 "THEN" "BEGIN" KNEW:= K-1; CHNEW:= A1 "END" "ELSE" "IF" A2 > A3 "THEN" "BEGIN" KNEW:= K ; CHNEW:= A2 "END" "ELSE" "BEGIN" KNEW:= K+1; CHNEW:= A3 "END" "END" CALCULATE STEP AND ORDER; "IF" FIRST "THEN" "BEGIN" FIRST:= "FALSE"; M:= N; "FOR" I:= -1,-2,-3 "DO" SAVE[I]:= 0; OUT(0,0); ADAMS:= "NOT" STIFF; WITH JACOBIAN:= "NOT" ADAMS; "IF" WITH JACOBIAN "THEN" EVALUATE JACOBIAN; METHOD; NEW START: K:= 1; SAME:= 2; ORDER; DERIV(DF); H:= "IF" "NOT" WITH JACOBIAN "THEN" HMIN "ELSE" SQRT(2 * EPS/SQRT(NORM2 (MATVEC(1,N,I,JACOBIAN,DF)))); "IF" H > HMAX "THEN" H:= HMAX "ELSE" "IF" H < HMIN "THEN" H:= HMIN; XOLD:= X; HOLD:= H; KOLD:= K; CH:= 1; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" SAVE[I]:= Y[I]; SAVE[M+I]:= Y[M+I]:= DF[I] * H "END"; OUT(0,0) "END" "ELSE" "BEGIN" WITH JACOBIAN:= "NOT" ADAMS; CH:= 1; K:=KOLD; RESET; ORDER; DECOMPOSE:= WITH JACOBIAN "END"; FAILS:= 0; "COMMENT" 1SECTION : 5.2.1.1.1.1.F (AUGUST 1974) PAGE 9 ; "FOR" L:= 0 "WHILE" X < XEND "DO" "BEGIN" "IF" X + H <= XEND "THEN" X:= X + H "ELSE" "BEGIN" H:= XEND-X; X:= XEND; CH:= H/HOLD; C:= 1; "FOR" J:= M "STEP" M "UNTIL" K*M "DO" "BEGIN" C:= C* CH; "FOR" I:= J+1 "STEP" 1 "UNTIL" J+N "DO" Y[I]:= Y[I] * C "END"; SAME:= "IF" SAME<3 "THEN" 3 "ELSE" SAME+1; "END"; "COMMENT" PREDICTION; "FOR" L:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" "FOR" I:= L "STEP" M "UNTIL" (K-1)*M+L "DO" "FOR" J:= (K-1)*M+L "STEP" -M "UNTIL" I "DO" Y[J]:= Y[J] + Y[J+M]; DELTA[L]:= 0 "END"; EVALUATED:= "FALSE"; "COMMENT" CORRECTION AND ESTIMATION LOCAL ERROR; "FOR" L:= 1,2,3 "DO" "BEGIN" DERIV(DF); "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" DF[I]:= DF[I] * H - Y[M+I]; "IF" WITH JACOBIAN "THEN" "BEGIN" "IF" EVALUATE "THEN" EVALUATE JACOBIAN; "IF" DECOMPOSE "THEN" DECOMPOSE JACOBIAN; SOL(JAC,N,P,DF) "END"; CONV:= "TRUE"; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" DFI:= DF[I]; Y[ I]:= Y[ I] + A0 * DFI; Y[M+I]:= Y[M+I] + DFI; DELTA[I]:= DELTA[I] + DFI; CONV:= CONV "AND" ABS(DFI) < TOLCONV * YMAX[I] "END"; "IF" CONV "THEN" "BEGIN" ERROR:= NORM2(DELTA[I]); "GOTO" CONVERGENCE "END" "END"; "COMMENT" 1SECTION : 5.2.1.1.1.1.F (DECEMBER 1979) PAGE 10 ; "COMMENT" ACCEPTANCE OR REJECTION; "IF" "NOT" CONV "THEN" "BEGIN" "IF" "NOT" WITH JACOBIAN "THEN" "BEGIN" EVALUATE:= WITH JACOBIAN:= SAME >= K "OR" H<1.1 * HMIN; "IF" "NOT" WITH JACOBIAN "THEN" CH:= CH/4; "END" "ELSE" "IF" "NOT" DECOMPOSED "THEN" DECOMPOSE:= "TRUE" "ELSE" "IF" "NOT" EVALUATED "THEN" EVALUATE := "TRUE" "ELSE" "IF" H > 1.1 * HMIN "THEN" CH:= CH/4 "ELSE" "IF" ADAMS "THEN" "GOTO" TRY CURTISS "ELSE" "BEGIN" SAVE[-1]:= 1; "GOTO" RETURN "END"; RESET "END" "ELSE" CONVERGENCE: "IF" ERROR > TOL "THEN" "BEGIN" FAILS:= FAILS + 1; "IF" H > 1.1 * HMIN "THEN" "BEGIN" "IF" FAILS > 2 "THEN" "BEGIN" "IF" ADAMS "THEN" "BEGIN" ADAMS:= "FALSE"; METHOD "END"; KOLD:= 0; RESET; "GOTO" NEW START "END" "ELSE" "BEGIN" CALCULATE STEP AND ORDER; "IF" KNEW ^= K "THEN" "BEGIN" K:= KNEW; ORDER "END"; CH:= CH * CHNEW; RESET "END" "END" "ELSE" "BEGIN" "IF" ADAMS "THEN" TRY CURTISS: "BEGIN" ADAMS:= "FALSE"; METHOD "END" "ELSE" "IF" K = 1 "THEN" "BEGIN" "COMMENT" VIOLATE EPS CRITERION; C:= EPS * SQRT(ERROR/TOL); "IF" C > SAVE[-3] "THEN" SAVE[-3]:= C; SAVE[-2]:= SAVE[-2] + 1; SAME:= 4; "GOTO" ERROR TEST OK "END"; K:= KOLD:= 1; RESET; ORDER; SAME:= 2 "END" "END" "ELSE" ERROR TEST OK: "BEGIN" "COMMENT" 1SECTION : 5.2.1.1.1.1.F (AUGUST 1974) PAGE 11 ; FAILS:= 0; "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" "BEGIN" C:= DELTA[I]; "FOR" L:= 2 "STEP" 1 "UNTIL" K "DO" Y[L*M+I]:= Y[L*M+I] + A[L] * C; "IF" ABS(Y[I]) > YMAX[I] "THEN" YMAX[I]:= ABS(Y[I]) "END"; SAME:= SAME-1; "IF" SAME= 1 "THEN" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" LAST DELTA[I]:= DELTA[I] "END" "ELSE" "IF" SAME= 0 "THEN" "BEGIN" CALCULATE STEP AND ORDER; "IF" CHNEW > 1.1 "THEN" "BEGIN" DECOMPOSED:= "FALSE"; "IF" K ^= KNEW "THEN" "BEGIN" "IF" KNEW > K "THEN" "BEGIN" "FOR" I:= 1 "STEP" 1 "UNTIL" N "DO" Y[KNEW*M+I] := DELTA[I] * A[K]/KNEW "END"; K:= KNEW; ORDER "END"; SAME:= K+1; "IF" CHNEW * H > HMAX "THEN" CHNEW:= HMAX/H; H:= H * CHNEW; C:= 1; "FOR" J:= M "STEP" M "UNTIL" K*M "DO" "BEGIN" C:= C * CHNEW; "FOR" I:= J+1 "STEP" 1 "UNTIL" J+N "DO" Y[I]:= Y[I] * C "END" "END" "ELSE" SAME:= 10 "END"; "IF" X ^= XEND "THEN" "BEGIN" XOLD:= X; HOLD:= H; KOLD:= K; CH:= 1; "FOR" I:= K * M + N "STEP" -1 "UNTIL" 1 "DO" SAVE[I]:= Y[I]; OUT(H,K) "END" "END" CORRECTION AND ESTIMATION LOCAL ERROR; "END" STEP; RETURN: SAVE[0]:= "IF" ADAMS "THEN" 0 "ELSE" 1; MULTISTEP:= SAVE[-1]= 0 "AND" SAVE[-2]=0 "END" MULTISTEP; "EOP" 1SECTION : 5.2.1.1.1.1.G (FEBRUARY 1979) PAGE 1 PROCEDURE : DIFFSYS. AUTHORS : R.BULIRSCH AND J.STOER. CONTRIBUTOR: K.DEKKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 731231. BRIEF DESCRIPTION: DIFFSYS SOLVES AN INITIAL VALUE PROBLEM ,FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DY / DX = F(X,Y). THE METHOD IS RECOMMENDED IF HIGH ACCURACY IS DESIRED. DIFFSYS IS NOT SUITED FOR STIFF EQUATIONS. KEYWORDS: INITIAL VALUE PROBLEMS, SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. 1SECTION : 5.2.1.1.1.1.G (FEBRUARY 1979) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE DIFFSYS READS: "PROCEDURE" DIFFSYS(X,XE,N,Y,DERIVATIVE,AETA,RETA,S,H0,OUTPUT); "VALUE" N; "INTEGER" N; "REAL" X,XE,AETA,RETA,H0; "ARRAY" Y,S; "PROCEDURE" DERIVATIVE,OUTPUT; "CODE" 33180; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE ; ENTRY: THE INITIAL VALUE X0; EXIT : THE FINAL VALUE XE; XE: ; THE FINAL VALUE OF X (XE>=X); N: ; THE NUMBER OF EQUATIONS; Y: ; "REAL" "ARRAY" Y[1:N]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT X=X0; EXIT : THE FINAL VALUES OF THE SOLUTION AT X=XE; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(X,Y,DY); "REAL" X; "ARRAY" Y,DY; THIS PROCEDURE SHOULD DELIVER THE RIGHT HAND SIDE OF THE I-TH DIFFERENTIAL EQUATION AT THE POINT (X,Y) AS DY[I], I=1,...,N; AETA: ; REQUIRED ABSOLUTE PRECISION IN THE INTEGRATION PROCESS; RETA: ; REQUIRED RELATIVE PRECISION IN THE INTEGRATION PROCESS; S: ; "REAL" "ARRAY" S[1:N]; THE ARRAY S IS USED TO CONTROL THE ACCURACY OF THE COMPUTED VALUES OF Y; ENTRY: IT IS ADVISABLE TO SET S[I]=0, I=1,...,N; EXIT : THE MAXIMUM VALUE OF ABS(Y[I]), ENCOUNTERED DURING INTEGRATION, IF THIS VALUE EXCEEDS THE VALUE OF S[I] ON ENTRY; H0: ; THE INITIAL STEP TO BE TAKEN; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS; "PROCEDURE" OUTPUT; THIS PROCEDURE IS CALLED AT THE END OF EACH INTEGRATION STEP ; THE USER CAN ASK FOR OUTPUT OF SOME PARAMETERS , FOR EXAMPLE X, Y, S. 1SECTION : 5.2.1.1.1.1.G (FEBRUARY 1979) PAGE 3 PROCEDURES USED: NONE. REQUIRED CENTRAL MEMORY : CIRCA 28* N MEMORY PLACES. METHOD AND PERFORMANCE: THE PROCEDURE DIFFSYS IS A SLIGHT MODIFICATION OF THE ALGORITHM PUBLISHED BY BULIRSCH AND STOER (SEE REF[1]) . BY THIS MODIFICATION INTEGRATION FROM X0 UNTIL XE CAN BE PERFORMED BY ONE CALL OF DIFFSYS. A NUMBER OF INTEGRATION STEPS ARE TAKEN, STARTING WITH THE INITIAL STEP H0. IN EACH INTEGRATION STEP A NUMBER OF SOLUTIONS ARE COMPUTED BY MEANS OF THE MODIFIED MIDPOINT RULE . EXTRAPOLATION IS USED TO IMPROVE THESE SOLUTIONS,UNTIL THE REQUIRED ACCURACY IS MET. AN INTEGRATION STEP IS REJECTED, IF THE ACCURACY REQUIREMENTS ARE NOT FULFILLED AFTER NINE EXTRAPOLATION STEPS . IN THESE CASES THE INTEGRATION STEP IS REJECTED , AND INTEGRATION IS TRIED AGAIN WITH THE INTEGRATION STEP HALVED. THE ALGORITHM IS FOR EACH STEP A VARIABLE ORDER METHOD (THE HIGHEST ORDER IS 14 ), AND USES A VARIABLE NUMBER OF FUNCTION EVALUATIONS, DEPENDING ON THE ORDER (MINIMUM IS 3, MAXIMUM IS 217). THE ALGORITHM IS LESS SENSITIVE TO TOO SMALL VALUES OF THE INITIAL STEPSIZE THAN THE ORIGINAL ALGORITHM ( SEE REF [2] ); HOWEVER BAD GUESSES REQUIRE STILL SOME MORE COMPUTATIONS. REFERENCES: [1]. R.BULIRSCH AND J.STOER. NUMERICAL TREATMENT OF ORDINARY DIFFERENTIAL EQUATIONS BY EXTRAPOLATION METHODS. NUMERISCHE MATHEMATIK, VOLUME 8, PAGE 1-13, 1965. [2]. PHYLLIS FOX. A COMPARATIVE STUDY OF COMPUTER PROGRAMS FOR INTEGRATING DIFFERENTIAL EQUATIONS. COMMUNICATIONS OF THE A.C.M., VOLUME 15, PAGE 941-948, 1972. [3]. T.E.HULL, W.H.ENRIGHT, B.M.FELLEN AND A.E.SEDGWICK. COMPARING NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS. SIAM JOURNAL ON NUMERICAL ANALYSIS,VOLUME 9,PAGE 603-635,1972. 1SECTION : 5.2.1.1.1.1.G (AUGUST 1974) PAGE 4 EXAMPLE OF USE: THE FOLLOWING PROGRAM ILLUSTRATES THE COSTS AND THE ACCURACIES WHICH ARE OBTAINED WHEN SOLVING A SYSTEM OF DIFFERENTIAL EQUATIONS ARISING FROM THE RESTRICTED PROBLEM OF THREE BODIES ( SEE REF[1] ). THE SOLUTION IS A CLOSED ORBIT WITH PERIOD T=6.192169331396. "BEGIN" "INTEGER" PASSES,K; "REAL" X,XE,TIME,TOL,H0; "REAL" "ARRAY" Y,S[1:4]; "PROCEDURE" DER(X,Y,DY); "REAL" X; "ARRAY" Y,DY; "BEGIN" "REAL" MU,MU1,Y1,Y2,Y3,Y4,S1,S2; MU:=1/82.45; MU1:=1-MU; PASSES:=PASSES+1; Y1:=Y[1]; Y2:=DY[1]:=Y[2]; Y3:=Y[3]; Y4:=DY[3]:=Y[4]; S1:=(Y1+MU)**2+Y3**2; S2:=(Y1-MU1)**2+Y3**2; S1:=S1*SQRT(S1); S2:=S2*SQRT(S2); DY[2]:=Y1+2*Y4-MU1*(Y1+MU)/S1-MU*(Y1-MU1)/S2; DY[4]:=Y3-2*Y2-MU1*Y3/S1-MU*Y3/S2 "END"; "PROCEDURE" OUT; "BEGIN" K:=K+1; "IF" X>=XE "THEN" OUTPUT(61,"("2(-5ZD),2(4B+Z.3DB3DB3DB3D),-5ZD.3D,/")",K, PASSES,Y[1],Y[3],CLOCK-TIME) "END"; OUTPUT(61,"(""(" THIS LINE AND THE FOLLOWING TEXT IS ")" "("PRINTED BY THIS PROGRAM")",//, "(" THE RESULTS WITH DIFFSYS - H0=.2 - ARE: ")",/, "(" K DER.EV. Y[1] Y[3] ")", "(" TIME")",/")"); "FOR" TOL:="-4,"-6,"-8,"-10,"-12 "DO" "BEGIN" PASSES:=K:=0; X:=0; XE:=6.192169331396; Y[1]:=1.2; Y[2]:=Y[3]:=0; Y[4]:=-1.04935750983; S[1]:=S[2]:=S[3]:=S[4]:=0; H0:=.2; TIME:=CLOCK; DIFFSYS(X,XE,4,Y,DER,TOL,TOL,S,H0,OUT); "END" "END" THIS LINE AND THE FOLLOWING TEXT IS PRINTED BY THIS PROGRAM: THE RESULTS WITH DIFFSYS - H0=.2 - ARE: K DER.EV. Y[1] Y[3] TIME 30 2591 +1.320 357 347 741 -.032 645 454 836 5.686 33 3414 +1.200 078 037 878 -.000 053 906 067 7.455 37 4213 +1.200 003 282 801 -.000 002 363 741 9.267 44 4618 +1.199 999 999 711 -.000 000 000 095 10.242 56 6299 +1.200 000 000 003 -.000 000 000 090 13.827 1SECTION : 5.2.1.1.1.1.G (AUGUST 1974) PAGE 5 SOURCE TEXT: 0"CODE" 33180; "PROCEDURE" DIFFSYS(X,XE,N,Y,DERIVATIVE,AETA,RETA,S,H0,OUTPUT); "VALUE" N; "INTEGER" N; "REAL" X,XE,AETA,RETA,H0; "ARRAY" Y,S; "PROCEDURE" DERIVATIVE,OUTPUT; "BEGIN" "REAL" A,B,B1,C,G,H,U,V,TA,FC; "INTEGER" I,J,K,KK,JJ,L,M,R,SR; "ARRAY" YA,YL,YM,DY,DZ[1:N],DT[1:N,0:6],D[0:6],YG,YH[0:7,1:N]; "BOOLEAN" KONV,B0,BH,LAST; LAST:="FALSE"; H:=H0; NEXT: "IF" H*1.1>=XE-X "THEN" "BEGIN" LAST:="TRUE"; H0:=H; H:=XE-X+"-13 "END"; DERIVATIVE(X,Y,DZ); BH:="FALSE"; "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" YA[I]:=Y[I]; ANF: A:=H+X; FC:=1.5; B0:="FALSE"; M:=1; R:=2; SR:=3; JJ:=-1; "FOR" J:=0 "STEP" 1 "UNTIL" 9 "DO" "BEGIN" "IF" B0 "THEN" "BEGIN" D[1]:=16/9; D[3]:=64/9; D[5]:=256/9 "END" "ELSE" "BEGIN" D[1]:=9/4; D[3]:=9; D[5]:=36 "END"; KONV:="TRUE"; "IF" J>6 "THEN" "BEGIN" L:=6; D[6]:=64; FC:=.6*FC "END" "ELSE" "BEGIN" L:=J; D[L]:=M*M "END"; M:=M*2; G:=H/M; B:=G*2; "IF" BH "AND" J<8 "THEN" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" YM[I]:=YH[J,I]; YL[I]:=YG[J,I] "END" "END" "ELSE" "BEGIN" "COMMENT" 1SECTION : 5.2.1.1.1.1.G (AUGUST 1974) PAGE 6 ; KK:=(M-2)/2; M:=M-1; "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" YL[I]:=YA[I]; YM[I]:=YA[I]+G*DZ[I] "END"; "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" DERIVATIVE(X+K*G,YM,DY); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" U:=YL[I]+B*DY[I]; YL[I]:=YM[I]; YM[I]:=U; U:=ABS(U); "IF" U>S[I] "THEN" S[I]:=U "END"; "IF" K=KK "AND" K^=2 "THEN" "BEGIN" JJ:=JJ+1; "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" YH[JJ,I]:=YM[I]; YG[JJ,I]:=YL[I] "END" "END" "END" "END"; DERIVATIVE(A,YM,DY); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" V:=DT[I,0]; TA:=C:=DT[I,0]:=(YM[I]+YL[I]+G*DY[I])/2; "FOR" K:=1 "STEP" 1 "UNTIL" L "DO" "BEGIN" B1:=D[K]*V; B:=B1-C; U:=V; "IF" B^=0 "THEN" "BEGIN" B:=(C-V)/B; U:=C*B; C:=B1*B "END"; V:=DT[I,K]; DT[I,K]:=U; TA:=U+TA "END"; "IF" ABS(Y[I]-TA)>RETA*S[I]+AETA "THEN" KONV:="FALSE"; Y[I]:=TA "END"; "IF" KONV "THEN" "GOTO" END; D[2]:=4; D[4]:=16; B0:=^B0; M:=R; R:=SR; SR:=M*2 "END"; BH:=^BH; LAST:="FALSE"; H:=H/2; "GOTO" ANF; END: H:=FC*H; X:=A; OUTPUT; "IF" "NOT" LAST "THEN" "GOTO" NEXT; "END" DIFFSYS; "EOP" 1SECTION : 5.2.1.1.1.1.H (FEBRUARY 1979) PAGE 1 PROCEDURE : ARK. AUTHOR: P.A. BEENTJES. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 740510. BRIEF DESCRIPTION: ARK SOLVES AN INITIAL VALUE PROBLEM, FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS . ARK IS RECOMMENDED FOR THE INTEGRATION OF SEMI-DISCRETE PARABOLIC AND HYPERBOLIC INITIAL-BOUNDARY PROBLEMS. KEYWORDS: INITIAL VALUE PROBLEM, SEMI-DISCRETE PARABOLIC AND HYPERBOLIC PROBLEM. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" ARK (T, TE, M0, M, U, DERIVATIVE, DATA, OUT); "INTEGER" M0, M; "REAL" T, TE; "ARRAY" U, DATA; "PROCEDURE" DERIVATIVE, OUT; "CODE" 33061; ARK : INTEGRATES THE SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS DU / DT = H(T, U), U = U0 AT T = T0. THE MEANING OF THE FORMAL PARAMETERS IS: T: ; THE INDEPENDENT VARIABLE T; ENTRY: THE INITIAL VALUE T0; EXIT : THE FINAL VALUE TE; TE: ; THE FINAL VALUE OF T (TE >= T); M0,M: ; INDICES OF THE FIRST AND LAST EQUATION OF THE SYSTEM; U: ; "ARRAY" U[M0 : M]; ENTRY: THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT T = T0; EXIT : THE VALUES OF THE SOLUTION AT T = TE; 1SECTION : 5.2.1.1.1.1.H (FEBRUARY 1979) PAGE 2 DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(T, V); "REAL" T; "ARRAY" V; THIS PROCEDURE PERFORMS AN EVALUATION OF THE RIGHT HAND SIDE OF THE SYSTEM WITH DEPENDENT VARIABLES V[M0 : M] AND INDEPENDENT VARIABLE T; UPON COMPLETION OF DERIVATIVE, THE RIGHT HAND SIDE SHOULD BE OVERWRITTEN ON V[M0 : M]; DATA: ; "ARRAY" DATA[1 : 10 + DATA[1]]; IN ARRAY DATA ONE SHOULD GIVE: DATA[1]: THE NUMBER OF EVALUATIONS OF H(T, U) PER INTEGRATION STEP(DATA[1] >= DATA[2]); DATA[2]: THE ORDER OF ACCURACY OF THE METHOD (DATA[2]<=3); DATA[3]: STABILITY BOUND(SEE REFERENCE [3]); DATA[4]: THE SPECTRAL RADIUS OF THE JACOBIAN MATRIX WITH RESPECT TO THOSE EIGENVALUES, WHICH ARE LOCATED IN THE NON-POSITIVE HALF PLANE; DATA[5]: THE MINIMAL STEPSIZE; DATA[6]: THE ABSOLUTE TOLERANCE; DATA[7]: THE RELATIVE TOLERANCE; IF BOTH DATA[6] AND DATA[7] ARE NEGATIVE, THE INTEGRATION IS PERFORMED WITH A CONSTANT STEP DATA[5]; DATA[8]: DATA[8] SHOULD BE 0 IF ARK IS CALLED FOR A FIRST TIME; FOR CONTINUED INTEGRATION DATA[8] SHOULD NOT BE CHANGED; DATA[11], ..., DATA[10 + DATA[1]]: POLYNOMIAL COEFFICIENTS (SEE REFERENCE [3]); AFTER EACH STEP THE FOLLOWING BY-PRODUCTS ARE DELIVERED: DATA[8]: THE NUMBER OF INTEGRATION STEPS PERFORMED; DATA[9]: AN ESTIMATION OF THE LOCAL ERROR LAST MADE; DATA[10]: INFORMATIVE MESSAGES: DATA[10] = 0: NO DIFFICULTIES; DATA[10] = 1: MINIMAL STEPLENGTH EXCEEDS THE STEPLENGTH PRESCRIBED BY STABILITY THEORY, I.E. DATA[5] > DATA[3] / DATA[4]; (TERMINATION OF ARK); DECREASE MINIMAL STEPLENGTH; IF NECESSARY, DATA[I],I = 4(1)7, CAN BE UPDATED(AFTER EACH STEP) BY MEANS OF PROCEDURE OUT; OUT: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" OUT; AFTER EACH INTEGRATION STEP PERFORMED INFORMATION CAN BE OBTAINED OR UPDATED BY THIS PROCEDURE, E.G. THE VALUES OF T, U[M0 : M] AND DATA[I], I = 4(1)10. 1SECTION : 5.2.1.1.1.1.H (FEBRUARY 1979) PAGE 3 DATA AND RESULTS: FOR THE INDICES M0 AND M THE FOLLOWING REMARKS CAN BE MADE: WHEN THE METHOD OF LINES IS APPLIED TO HYPERBOLIC DIFFERENTIAL EQUATIONS THE NUMBER OF RELEVANT ORDINARY DIFFERENTIAL EQUATIONS DECREASES DURING THE INTEGRATION PROCESS; IN PROCEDURE ARK THIS MAY BE REALIZED BY INTEGERS M0 AND M, WHICH ARE DEFINED AS FUNCTIONS OF THE NUMBER OF RIGHT HAND SIDE EVALUATIONS. A SELECTION OF POSSIBLE ENTRIES FOR ARRAY DATA (DEPENDENT ON THE KIND OF INITIAL VALUE PROBLEM) IS GIVEN IN REFERENCE [4],SECTION 8. PROCEDURES USED: INIVEC = CP31010, MULVEC = CP31020, DUPVEC = CP31030, VECVEC = CP34010, ELMVEC = CP34020, DECSOL = CP34301. REQUIRED CENTRAL MEMORY : CIRCA 75 + 2 * (M - M0) MEMORY PLACES. METHOD AND PERFORMANCE: ARK IS AN IMPLEMENTATION OF LOW ORDER STABILIZED RUNGE KUTTA METHODS (SEE REFERENCE [1]); AUTOMATIC STEPSIZE CONTROL IS PROVIDED BUT STEP-REJECTION HAS BEEN EXCLUDED IN ORDER TO SAVE STORAGE; BECAUSE OF ITS LIMITED STORAGE REQUIREMENTS AND ADAPTIVE STABILITY FACILITIES THE METHOD IS WELL SUITED FOR THE SOLUTION OF INITIAL BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS; NUMERICAL RESULTS, OBTAINED WITH A SLIGHTLY DIFFERENT IMPLEMENTATION CAN BE FOUND IN REFERENCE [2]. REFERENCES: [1]. P.J. VAN DER HOUWEN. STABILIZED RUNGE KUTTA METHOD WITH LIMITED STORAGE REQUIREMENTS. MATH. CENTR. REPORT TW 124/71; [2]. P.A. BEENTJES. AN ALGOL 60 VERSION OF STABILIZED RUNGE KUTTA METHODS (DUTCH). MATH. CENTR. REPORT NR 23/72; 1SECTION : 5.2.1.1.1.1.H (DECEMBER 1975) PAGE 4 [3]. P.J. VAN DER HOUWEN, J. KOK. NUMERICAL SOLUTION OF A MINIMAX PROBLEM. MATH. CENTR. REPORT TW 123/71; [4]. P.J. VAN DER HOUWEN ET AL. ONE STEP METHODS FOR LINEAR INITIAL VALUE PROBLEMS, I.I.I. NUMERICAL EXAMPLES, MATH. CENTR. REPORT TW 130/71. EXAMPLE OF USE: THE VALUES OF 1. Y(1) AND Y(2) OF THE INITIAL VALUE PROBLEM DY / DX = Y - 2 * X / Y, Y(0) = 1 AND 2. U(.6, 0) OF THE CAUCHY PROBLEM (SEE REFERENCE [2]): DU / DT = .5 * DU / DX, U(0, X) = EXP(-X * X) MAY BE OBTAINED BY THE FOLLOWING PROGRAM: "BEGIN" "INTEGER" M0, M, I; "REAL" T, TE, DAT; "ARRAY" Y[1 : 1], U[-150 : 150], DATA[1 : 14]; "PROCEDURE" DER1(T, V); "REAL" T; "ARRAY" V; V[1]:= V[1] - 2 * T / V[1]; "PROCEDURE" DER2(T, V); "REAL" T; "ARRAY" V; "BEGIN" "INTEGER" J; "REAL" V1, V2, V3; V2:= V[M0]; M0:= M0 + 1; M:= M - 1; V3:= V[M0]; "FOR" J:= M0 "STEP" 1 "UNTIL" M "DO" "BEGIN" V1:= V2; V2:= V3; V3:= V[J + 1]; V[J]:= 250 * (V3 - V1) / 3 "END" "END" DER2; 1SECTION : 5.2.1.1.1.1.H (DECEMBER 1975) PAGE 5 "PROCEDURE" OUT1; "IF" T = TE "THEN" "BEGIN" "IF" T = 1 "THEN" OUTPUT(61, "("/, "(" PROBLEM 1")", //, "(" X NUMBER OF INTEGRATION STEPS Y(COMPUTED) Y(EXACT)")", //")"); OUTPUT(61, "("ZD, 13ZD,12B,2(-3ZD.7D),"("...")", /")", T, DATA[8], Y[1], SQRT(2 * T + 1)); TE:= 2 "END" OUT1; "PROCEDURE" OUT2; "IF" T = .6 "THEN" OUTPUT(61, "("//, "(" PROBLEM 2")", //, "(" NUMBER OF DERIVATIVE CALLS")", "(" U(.6, 0)COMPUTED U(.6, 0)EXACT")", //, 13ZD, 2(-10Z.7D), "("...")"")", DATA[1] * DATA[8], U[0], EXP(-.09)); I:= 1; "FOR" DAT:= 3, 3, 1, 1, "-3, "-6, "-6, 0, 0, 0, 1, .5, 1 / 6 "DO" "BEGIN" DATA[I]:= DAT; I:= I + 1 "END"; T:= 0; Y[1]:= 1; TE:= 1; ARK(T, TE, 1, 1, Y, DER1, DATA, OUT1); I:= 1; "FOR" DAT:= 4, 3, SQRT(8), 500 / 3, DATA[3] / DATA[4], -1, -1, 0, 0, 0, 1, .5, 1 / 6, 1 / 24 "DO" "BEGIN" DATA[I]:= DAT; I:= I + 1 "END"; M0:= -150; M:= 150; T:= 0; U[0]:= 1; "FOR" I:= 1 "STEP" 1 "UNTIL" M "DO" U[I]:= U[-I]:= EXP(-(.003 * I) ** 2); ARK(T, .6, M0, M, U, DER2, DATA, OUT2) "END" THIS PROGRAM DELIVERS: PROBLEM 1 X NUMBER OF INTEGRATION STEPS Y(COMPUTED) Y(EXACT) 1 38 1.7320535 1.7320508... 2 56 2.2360928 2.2360680... PROBLEM 2 NUMBER OF DERIVATIVE CALLS U(.6, 0)COMPUTED U(.6, 0)EXACT 144 .9139326 .9139312... 1SECTION : 5.2.1.1.1.1.H (FEBRUARY 1979) PAGE 6 SOURCE TEXT(S): 0"CODE" 33061; "PROCEDURE" ARK (T, TE, M0, M, U, DERIVATIVE, DATA, OUT); "INTEGER" M0, M; "REAL" T, TE; "ARRAY" U, DATA; "PROCEDURE" DERIVATIVE, OUT; "BEGIN" "INTEGER" P, N, Q; "OWN" "REAL" EC0, EC1, EC2, TAU0, TAU1, TAU2, TAUS, T2; "REAL" THETANM1, TAU, BETAN, QINV, ETA; "ARRAY" MU, LAMBDA[1:DATA[1]], THETHA[0:DATA[1]], RO, R[M0:M]; "BOOLEAN" START, STEP1, LAST; "PROCEDURE" INITIALIZE; "BEGIN" "INTEGER" I, J, K, L, N1; "REAL" S, THETA0; "ARRAY" ALFA[1:8, 1:DATA[1]+1], TH[1:8], AUX[1:3]; "REAL" "PROCEDURE" LABDA(I, J); "VALUE" I, J; "INTEGER" I, J; LABDA:= "IF" P < 3 "THEN" ("IF" J =I-1 "THEN" MUI(I) "ELSE" 0) "ELSE" "IF" P =3 "THEN" ("IF" I =N "THEN" ("IF" J=0 "THEN" .25 "ELSE" "IF" J =N - 1 "THEN" .75 "ELSE" 0) "ELSE" "IF" J =0 "THEN" ("IF" I =1 "THEN" MUI(1) "ELSE" .25) "ELSE" "IF" J =I - 1 "THEN" LAMBDA[I] "ELSE" 0) "ELSE" 0; "REAL" "PROCEDURE" MUI(I); "VALUE" I; "INTEGER" I; MUI:= "IF" I =N "THEN" 1 "ELSE" "IF" I < 1 ! I > N "THEN" 0 "ELSE" "IF" P < 3 "THEN" LAMBDA[I] "ELSE" "IF" P =3 "THEN" LAMBDA[I] + .25 "ELSE" 0; "REAL" "PROCEDURE" SUM(I, A, B, X); "VALUE" B; "INTEGER" I, A, B; "REAL" X; "BEGIN" "REAL" S; S:= 0; "FOR" I:= A "STEP" 1 "UNTIL" B "DO" S:= S + X; SUM:= S "END" SUM; "COMMENT" 1SECTION : 5.2.1.1.1.1.H (DECEMBER 1979) PAGE 7 ; N:= DATA[1]; P:= DATA[2]; EC1:= EC2 := 0; BETAN:= DATA[3]; THETANM1:= "IF" P=3 "THEN" .75 "ELSE" 1; THETA0:= 1 - THETANM1; S:= 1; "FOR" J:= N - 1 "STEP" - 1 "UNTIL" 1 "DO" "BEGIN" S:= - S * THETA0 + DATA[N + 10 - J]; MU[J]:= DATA[N + 11 - J] / S; LAMBDA[J]:= MU[J] - THETA0 "END"; "FOR" I:= 1 "STEP" 1 "UNTIL" 8 "DO" "FOR" J:= 0 "STEP" 1 "UNTIL" N "DO" ALFA[I, J + 1]:= "IF" I = 1 "THEN" 1 "ELSE" "IF" J = 0 "THEN" 0 "ELSE" "IF" I = 2 ! I = 4 ! I = 8 "THEN" MUI(J) ** ENTIER((I + 2) / 3) "ELSE" "IF" (I = 3 ! I = 6) & J > 1 "THEN" SUM(L, 1, J-1, LABDA(J, L) * MUI(L) ** ENTIER(I / 3)) "ELSE" "IF" I = 5 & J > 2 "THEN" SUM(L, 2, J - 1, LABDA(J, L) * SUM(K, 1, L - 1, LABDA(L, K) * MUI(K))) "ELSE" "IF" I = 7 & J > 1 "THEN" SUM(L, 1, J - 1, LABDA(J, L) * MUI(L)) * MUI(J) "ELSE" 0; N1:="IF" N < 4 "THEN" N + 1 "ELSE" "IF" N < 7 "THEN" 4 "ELSE" 8; I:= 1; "FOR" S:= 1, .5, 1 / 6, 1 / 3, 1 / 24, 1 / 12, .125, .25 "DO" "BEGIN" TH[I]:= S; I:= I + 1 "END"; "IF" P = 3 & N < 7 "THEN" TH[1]:= TH[2]:= 0; AUX[2]:= " - 14; DECSOL(ALFA, N1, AUX, TH); INIVEC(0, N, THETHA, 0); DUPVEC(0, N1 - 1, 1, THETHA, TH); "IF" ^ (P = 3 & N < 7) "THEN" "BEGIN" THETHA[0]:= THETHA[0] - THETA0; THETHA[N - 1]:= THETHA[N - 1] - THETANM1; Q:= P + 1 "END" "ELSE" Q:= 3; QINV:= 1 / Q; START:= DATA[8] = 0; DATA[10]:= 0; LAST:= "FALSE"; DUPVEC(M0, M, 0, R, U); DERIVATIVE(T, R) "END" INITIALIZE 1SECTION : 5.2.1.1.1.1.H (DECEMBER 1975) PAGE 8 ; "PROCEDURE" LOCAL ERROR CONSTRUCTION(I); "VALUE" I; "INTEGER" I; "BEGIN" "IF" THETHA[I] ^= 0 "THEN" ELMVEC(M0, M, 0, RO, R, THETHA[I]); "IF" I = N "THEN" "BEGIN" DATA[9]:= SQRT(VECVEC(M0, M, 0, RO, RO))* TAU; EC0:= EC1; EC1:= EC2; EC2:= DATA[9] / TAU ** Q "END" "END" LEC; "PROCEDURE" STEPSIZE; "BEGIN" "REAL" TAUACC, TAUSTAB, AA, BB, CC, EC; ETA:= SQRT(VECVEC(M0, M, 0, U, U)) * DATA[7] + DATA[6]; "IF" ETA > 0 "THEN" "BEGIN" "IF" START "THEN" "BEGIN" "IF" DATA[8] = 0 "THEN" "BEGIN" TAUACC:= DATA[5]; STEP1:= "TRUE" "END" "ELSE" "IF" STEP1 "THEN" "BEGIN" TAUACC:= (ETA / EC2) ** QINV; "IF" TAUACC > 10 * TAU2 "THEN" TAUACC:= 10 * TAU2 "ELSE" STEP1:= "FALSE" "END" "ELSE" "BEGIN" BB:= (EC2 - EC1) / TAU1; CC:= - BB * T2 + EC2; EC:= BB * T + CC; TAUACC:= "IF" EC < 0 "THEN" TAU2 "ELSE" (ETA / EC) ** QINV; START:= "FALSE" "END" "END" "ELSE" "BEGIN" AA:= ((EC0 - EC1) / TAU0 + (EC2 - EC1) / TAU1) / (TAU1 + TAU0); BB:= (EC2 - EC1) / TAU1 - (2 * T2 - TAU1) * AA; CC:= - (AA * T2 + BB) * T2 + EC2; EC:= (AA * T + BB) * T + CC; TAUACC:= "IF" EC < 0 "THEN" TAUS "ELSE" (ETA / EC) ** QINV; "IF" TAUACC > 2 * TAUS "THEN" TAUACC:= 2 * TAUS; "IF" TAUACC < TAUS / 2 "THEN" TAUACC:= TAUS / 2 "END" "END" "ELSE" TAUACC:= DATA[5]; "IF" TAUACC < DATA[5] "THEN" TAUACC:= DATA[5]; TAUSTAB:= BETAN / DATA[4]; "IF" TAUSTAB < DATA[5] "THEN" "BEGIN" DATA[10]:= 1; "GOTO" ENDARK "END"; TAU:= "IF" TAUACC > TAUSTAB "THEN" TAUSTAB "ELSE" TAUACC; TAUS:= TAU; "IF" TAU >= TE - T "THEN" "BEGIN" TAU:= TE - T; LAST:= "TRUE" "END"; TAU0:= TAU1; TAU1:= TAU2; TAU2:= TAU "END" STEPSIZE 1SECTION : 5.2.1.1.1.1.H (DECEMBER 1975) PAGE 9 ; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" "INTEGER" I, J; "REAL" MT, LT; MULVEC(M0, M, 0, RO, R, THETHA[0]); "IF" P = 3 "THEN" ELMVEC(M0, M, 0, U, R, .25 * TAU); "FOR" I:= 1 "STEP" 1 "UNTIL" N - 1 "DO" "BEGIN" MT:= MU[I] * TAU; LT:= LAMBDA[I] * TAU; "FOR" J:= M0 "STEP" 1 "UNTIL" M "DO" R[J]:= LT * R[J] + U[J]; DERIVATIVE(T + MT, R); LOCAL ERROR CONSTRUCTION(I) "END"; ELMVEC(M0, M, 0, U, R, THETANM1 * TAU); DUPVEC(M0, M, 0, R, U); DERIVATIVE(T + TAU, R); LOCAL ERROR CONSTRUCTION(N); T2:= T; "IF" LAST "THEN" "BEGIN" LAST:= "FALSE"; T:= TE "END" "ELSE" T:= T + TAU; DATA[8]:= DATA[8]+1 "END" DIFSCH; INITIALIZE; NEXT STEP: STEPSIZE; DIFFERENCE SCHEME; OUT; "IF" T ^= TE "THEN" "GOTO" NEXT STEP; ENDARK: "END" ARK; "EOP" 1SECTION : 5.2.1.1.1.1.I (FEBRUARY 1979) PAGE 1 PROCEDURE : EFRK. AUTHOR: K.DEKKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 740710. BRIEF DESCRIPTION: EFRK SOLVES AN INITIAL VALUE PROBLEM FOR A SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS DU / DT = H(T,U) . EFRK IS A SPECIAL PURPOSE PROCEDURE FOR STIFF EQUATIONS WITH A KNOWN, CLUSTERED EIGENVALUE SPECTRUM. KEYWORDS: INITIAL VALUE PROBLEM, SYSTEM OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS, STIFF EQUATION. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE EFRK READS: "PROCEDURE" EFRK ( T,TE, M0,M, U, SIGMA, PHI, DIAMETER, DERIVATIVE, K, STEP, R, L, BETA, THIRDORDER, TOL, OUTPUT ); "VALUE" R,L; "INTEGER" M0,M,K,R,L; "REAL" T,TE,SIGMA,PHI,DIAMETER,STEP,TOL; "ARRAY" U,BETA; "BOOLEAN" THIRDORDER; "PROCEDURE" DERIVATIVE,OUTPUT; "CODE" 33070; THE MEANING OF THE FORMAL PARAMETERS IS: T: ; THE INDEPENDENT VARIABLE T; ENTRY: THE INITIAL VALUE T0; EXIT : THE FINAL VALUE TE; TE: ; THE FINAL VALUE OF T (TE>=T); M0: ; THE INDEX OF THE FIRST EQUATION; M: ; THE INDEX OF THE LAST EQUATION; 1SECTION : 5.2.1.1.1.1.I (FEBRUARY 1979) PAGE 2 U: ; "REAL" "ARRAY" U[M0:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT T = T0; EXIT : THE VALUES OF THE SOLUTION AT T = TE; SIGMA: ; THE MODULUS OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED , FOR EXAMPLE AN APPROXIMATION OF THE CENTRE OF THE LEFT HAND CLUSTER; PHI: ; THE ARGUMENT OF THE CENTRE OF THE LEFT HAND CLUSTER; IN THE CASE OF TWO COMPLEX CONJUGATED CLUSTERS , THE ARGUMENT OF THE CENTRE IN THE SECOND QUADRANT SHOULD BE TAKEN; DIAMETER: ; THE DIAMETER OF THE LEFT HAND CLUSTER OF EIGENVALUES OF THE JACOBIAN MATRIX OF THE SYSTEM OF DIFFERENTIAL EQUATIONS; IN CASE OF NON-LINEAR EQUATIONS DIAMETER SHOULD HAVE SUCH A VALUE THAT THE VARIATION OF THE EIGENVALUES IN THIS CLUSTER IN THE PERIOD ( T ,T+STEP ) IS LESS THAN HALF THE DIAMETER; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(T,U); "REAL" T; "ARRAY" U; THIS PROCEDURE SHOULD DELIVER THE VALUE OF H(T,U) IN THE POINT (T,U) IN THE ARRAY U; K: ; COUNTS THE NUMBER OF INTEGRATION STEPS TAKEN; FOR EXAMPLE, K MAY BE USED IN THE EXPRESSION FOR TE; ENTRY: AN (ARBIRARY) CHOSEN VALUE K0, E.G. K0=0; EXIT : K0 + THE NUMBER OF INTEGRATION STEPS PERFORMED; STEP: ; THE STEPSIZE CHOSEN WILL BE AT MOST EQUAL TO STEP ; THIS STEPSIZE MAY BE REDUCED BY STABILITY CONSTRAINTS, IMPOSED BY A POSITIVE DIAMETER , OR BY CONSIDERATIONS OF INTERNAL STABILITY (SEE REF[1], PAGE 11); R: ; R + L: THE NUMBER OF EVALUATIONS OF H(T, U) ON WHICH THE RUNGE-KUTTA SCHEME IS BASED; FOR R=1,2,>=3 FIRST, SECOND AND THIRD ORDER ACCURACY MAY BE OBTAINED BY AN APPROPRIATE CHOICE OF THE ARRAY BETA; L: ; ENTRY: IF PHI = 4*ARCTAN(1): THE ORDER OF THE EXPONENTIAL FITTING, ELSE TWICE THE ORDER OF THE EXPONENTIAL FITTING; NOTE THAT L SHOULD BE EVEN IN THE LATTER CASE; BETA: ; "REAL" "ARRAY" BETA[0:R+L]; ENTRY: THE ELEMENTS BETA[I] , I=0, ... ,R SHOULD HAVE THE VALUE OF THE R+1 FIRST COEFFICIENTS OF THE STABILITY POLYNOMIAL; 1SECTION : 5.2.1.1.1.1.I (FEBRUARY 1979) PAGE 3 THIRDORDER: ; IF THIRD ORDER ACCURACY IS DESIRED , THIRDORDER SHOULD HAVE THE VALUE "TRUE" , IN COMBINATION WITH APPROPRIATE CHOICES OF R (R>=3) AND THE ARRAY BETA ( BETA[I]=1/I!, I=0,1,2,3 ); IN ALL OTHER CASES THIRDORDER MUST HAVE THE VALUE "FALSE"; TOL: ; AN UPPERBOUND FOR THE ROUNDING ERRORS IN THE COMPUTATIONS IN ONE RUNGE-KUTTA STEP ; IN SOME CASES ( E.G. LARGE VALUES OF SIGMA AND R ) TOL WILL CAUSE A DECREASE OF THE STEPSIZE; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS; "PROCEDURE" OUTPUT; THIS PROCEDURE IS CALLED AT THE END OF EACH INTEGRATION STEP ; THE USER CAN ASK FOR OUTPUT OF SOME PARAMETERS , FOR EXAMPLE T, K, U, AND COMPUTE NEW VALUES FOR SIGMA, PHI, AND DIAMETER. PROCEDURES USED: INIVEC= CP31010, ELMVEC= CP34020, DEC = CP34300, SOL = CP34051. REQUIRED CENTRAL MEMORY : CIRCA 30 + (M - M0) + L * (5 + L). METHOD AND PERFORMANCE: EFRK IS BASED ON EXPLICIT RUNGE-KUTTA METHODS OF ORDER 1, 2 AND 3, WHICH MAKE USE OF EXPONENTIAL FITTING. AUTOMATIC ERROR CONTROL IS NOT PROVIDED. A DETAILED DESCRIPTION OF THE METHOD AND SOME NUMERICAL EXAMPLES ARE GIVEN IN REF[1]. REF [3], PAGE 170 REPRESENTS A BRIEF SURVEY. A COMPARATIVE TEST OVER A LARGE CLASS OF DIFFERENTIAL EQUATIONS IS GIVEN IN REF [4]. FROM THESE RESULTS IT APPEARS THAT CALLS WITH THIRDORDER = "TRUE" ARE LESS ADVISABLE. 1SECTION : 5.2.1.1.1.1.I (FEBRUARY 1979) PAGE 4 REFERENCES: [1]. K. DEKKER. AN ALGOL 60 VERSION OF EXPONENTIALLY FITTED RUNGE-KUTTA METHODS (DUTCH). NR 25 (1972), MATHEMATICAL CENTRE. [2]. T. J. DEKKER, P. W. HEMKER AND P. J. VAN DER HOUWEN. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 1 (DUTCH). MC SYLLABUS 15.1, (1972) MATHEMATICAL CENTRE. [3]. P. A. BEENTJES, K. DEKKER, H. C. HEMKER, S.P.N. VAN KAMPEN AND G. M. WILLEMS. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 2 (DUTCH). MC SYLLABUS 15.2, (1973) MATHEMATICAL CENTRE. [4]. P. A. BEENTJES, K. DEKKER, H. C. HEMKER AND M. V. VELDHUIZEN. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 3 (DUTCH). MC SYLLABUS 15.3, (1974) MATHEMATICAL CENTRE. EXAMPLE OF USE: CONSIDER THE SYSTEM OF DIFFERENTIAL EQUATIONS: DY[1]/DX = -Y[1] + Y[1] * Y[2] + .99 * Y[2] DY[2]/DX = -1000 * ( -Y[1] + Y[1] * Y[2] + Y[2] ) WITH THE INITIAL CONDITIONS AT X = 0: Y[1] = 1 AND Y[2] = 0. (SEE REF[2], PAGE 11). THE SOLUTION AT X = 50 IS APPROXIMATELY: Y[1] = .765 878 320 487 AND Y[2] = .433 710 353 5768. THE FOLLOWING PROGRAM SHOWS SOME DIFFERENT CALLS OF THE PROCEDURE EFRK: 1SECTION : 5.2.1.1.1.1.I (AUGUST 1974) PAGE 5 "BEGIN" "INTEGER" K,R,L,PASSES; "REAL" X,SIGMA,PHI,TIME,STEP,DIAMETER; "REAL" "ARRAY" Y[1:2],BETA[0:6]; "PROCEDURE" DER(X,Y); "REAL" X; "ARRAY" Y; "BEGIN" "REAL" Y1,Y2; Y1:=Y[1]; Y2:=Y[2]; Y[1]:=(Y1+.99)*(Y2-1)+.99; Y[2]:=1000*((1+Y1)*(1-Y2)-1); PASSES:=PASSES+1 "END"; "PROCEDURE" OUT; "BEGIN" "REAL" S; S:=(-1000*Y[1]-1001+Y[2])/2; SIGMA:=ABS(S-SQRT(S*S+10*(Y[2]-1))); DIAMETER:=2*STEP*ABS(1000*(1.99*Y[2]-2*Y[1]*(1-Y[2]))); "IF" X=50 "THEN" OUTPUT(61,"("4BD,2BD,2(-5ZD),2(4B+.3DB3DB3D),-5ZD.3D,/")", R,L,K,PASSES,Y[1],Y[2],CLOCK-TIME) "END"; OUTPUT(61,"(""(" THIS LINE AND THE FOLLOWING TEXT IS ")" "("PRINTED BY THIS PROGRAM")",//, "(" THE RESULTS WITH EFRK ARE:")",/, "(" R L K DER.EV. Y[1] Y[2]")" "(" TIME")",/")"); PHI:=4*ARCTAN(1); BETA[0]:=BETA[1]:=1; "FOR" R:=1,2,3 "DO" "FOR" L:=1,2,3 "DO" "BEGIN" "FOR" K:=2 "STEP" 1 "UNTIL" R "DO" BETA[K]:=BETA[K-1]/K; "FOR" STEP:=1,.1 "DO" "BEGIN" PASSES:=K:=0; X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; OUT; EFRK(X,50.0,1,2,Y,SIGMA,PHI,DIAMETER,DER,K,STEP,R,L,BETA, R>=3,"-4,OUT); "END"; OUTPUT(61,"("/")"); "END"; "END" 1SECTION : 5.2.1.1.1.1.I (AUGUST 1974) PAGE 6 THIS LINE AND THE FOLLOWING TEXT IS PRINTED BY THIS PROGRAM: THE RESULTS WITH EFRK ARE: R L K DER.EV. Y[1] Y[2] TIME 1 1 237 474 +.765 812 555 +.433 689 306 1.395 1 1 501 1002 +.765 847 870 +.433 700 619 3.381 1 2 52 156 +.765 570 874 +.433 615 119 0.465 1 2 501 1503 +.765 848 220 +.433 700 709 4.200 1 3 52 208 +.765 571 278 +.433 615 202 0.531 1 3 500 2000 +.765 848 512 +.433 700 827 4.879 2 1 3317 9951 +.765 878 320 +.433 710 353 21.808 2 1 1050 3150 +.765 878 321 +.433 710 330 7.153 2 2 174 696 +.765 878 335 +.433 710 335 1.385 2 2 501 2004 +.765 878 323 +.433 709 211 4.915 2 3 57 285 +.765 881 339 +.433 817 185 0.642 2 3 501 2505 +.765 878 323 +.433 709 725 5.756 3 1 7010 28040 +.765 878 320 +.433 710 354 55.298 3 1 3255 13020 +.765 878 320 +.433 710 374 25.772 3 2 949 4745 +.765 878 319 +.433 711 893 8.499 3 2 1384 6920 +.765 862 498 +.449 724 830 13.452 3 3 917 5502 +.765 878 018 +.434 105 184 9.143 3 3 1166 6996 +.765 861 696 +.433 705 641 15.512 SOURCE TEXT(S): 0"CODE" 33070; "PROCEDURE" EFRK(T,TE,M0,M,U,SIGMA,PHI,DIAMETER,DERIVATIVE,K,STEP,R,L, BETA,THIRDORDER,TOL,OUTPUT); "VALUE" R,L; "INTEGER" M0,M,K,R,L; "REAL" T,TE,SIGMA,PHI,DIAMETER,STEP,TOL; "ARRAY" U,BETA; "BOOLEAN" THIRDORDER; "PROCEDURE" DERIVATIVE,OUTPUT; "BEGIN" "INTEGER" N; "REAL" THETA0,THETANM1,H,B,B0,PHI0,PHIL,PI,COSPHI,SINPHI,EPS,BETAR; "BOOLEAN" FIRST,LAST,COMPLEX,CHANGE; "INTEGER" "ARRAY" P[1:L]; "REAL" "ARRAY" MU,LABDA[0:R+L-1],PT[0:R],FAC,BETAC[0:L-1],RL[M0:M], A[1:L,1:L],AUX[0:3]; "COMMENT" 1SECTION : 5.2.1.1.1.1.I (AUGUST 1974) PAGE 7 ; "PROCEDURE" FORM CONSTANTS; "BEGIN" "INTEGER" I; FIRST:="FALSE"; FAC[0]:=1; "FOR" I:=1 "STEP" 1 "UNTIL" L-1 "DO" FAC[I]:=I*FAC[I-1]; PT[R]:=L*FAC[L-1]; "FOR" I:=1 "STEP" 1 "UNTIL" R "DO" PT[R-I]:=PT[R-I+1]*(L+I)/I "END" FORM CONSTANTS; "PROCEDURE" FORM BETA; "BEGIN" "INTEGER" I,J; "REAL" BB,C,D; "IF" FIRST "THEN" FORM CONSTANTS; "IF" L=1 "THEN" "BEGIN" C:=1-EXP(-B); "FOR" J:=1 "STEP" 1 "UNTIL" R "DO" C:=BETA[J]-C/B; BETA[R+1]:=C/B "END" "ELSE" "IF" B>40 "THEN" "BEGIN" "FOR" I:=R+1 "STEP" 1 "UNTIL" R+L "DO" "BEGIN" C:=0; "FOR" J:=0 "STEP" 1 "UNTIL" R "DO" C:=BETA[J]*PT[J]/(I-J)-C/B; BETA[I]:=C/B/FAC[L+R-I]/FAC[I-R-1] "END"; "END" "ELSE" "BEGIN" D:=C:=EXP(-B); BETAC[L-1]:=D/FAC[L-1]; "FOR" I:=1 "STEP" 1 "UNTIL" L-1 "DO" "BEGIN" C:=B*C/I; D:=D+C; BETAC[L-1-I]:=D/FAC[L-1-I] "END"; BB:=1; "FOR" I:=R+1 "STEP" 1 "UNTIL" R+L "DO" "BEGIN" C:=0; "FOR" J:=0 "STEP" 1 "UNTIL" R "DO" C:=(BETA[J]-("IF" JL-2 "THEN" 2 "ELSE" L-2*I; COSPHIL :=COSIPHI*COSPHI-SINIPHI*SINPHI; SINIPHI:=COSIPHI*SINPHI+SINIPHI*COSPHI; COSIPHI:=COSPHIL; "FOR" J:=L "STEP" -2 "UNTIL" C3 "DO" "BEGIN" D[J]:=D[J]+ZI*C2*SINIPHI; D[J-1]:=D[J-1]-ZI*C2*COSIPHI; C2:=C2*C1; C1:=C1-1 "END"; ZI:=ZI*B1 "END" "END" RIGHT HAND SIDE; "IF" PHI0^=PHIL "THEN" ELEMENTS OF MATRIX; RIGHTHANDSIDE; SOL(A,L,P,D); "FOR" I:=1 "STEP" 1 "UNTIL" L "DO" BETA[R+I]:=D[L+1-I]*B1 "END" SOLOFCOMEQ; "COMMENT" 1SECTION : 5.2.1.1.1.1.I (AUGUST 1974) PAGE 9 ; "PROCEDURE" COEFFICIENT; "BEGIN" "INTEGER" J,K; "REAL" C; B0:=B; PHI0:=PHI; "IF" B>=1 "THEN" "BEGIN" "IF" COMPLEX "THEN" SOLUTION OF COMPLEX EQUATIONS "ELSE" FORM BETA "END"; LABDA[0]:=MU[0]:=0; "IF" THIRDORDER "THEN" "BEGIN" THETA0:=.25; THETANM1:=.75; "IF" B<1 "THEN" "BEGIN" C:=MU[N-1]:=2/3; LABDA[N-1]:=5/12; "FOR" J:=N-2 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" C:=MU[J]:=C/(C-.25)/(N-J+1); LABDA[J]:=C-.25 "END" "END" "ELSE" "BEGIN" C:=MU[N-1]:=BETA[2]*4/3; LABDA[N-1]:=C-.25; "FOR" J:=N-2 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" C:=MU[J]:=C/(C-.25)*BETA[N-J+1]/BETA[N-J]/ ("IF" JDIAMETER; "IF" DIAMETER>0 "THEN" HSTAB:=(SIGMA**2/(DIAMETER*(DIAMETER*.25+D)))**(L*.5/R)/ BETAR/SIGMA "ELSE" HSTAB:=H; D:= "IF" THIRDORDER "THEN" (2*TOL/EPS/BETA[R])**(1/(N-1))* 4**((L-1)/(N-1)) "ELSE" (TOL/EPS)**(1/R)/BETAR; HSTABINT:= ABS(D/SIGMA); "IF" H>HSTAB "THEN" H:=HSTAB; "IF" H>HSTABINT "THEN" H:=HSTABINT; "IF" T+H>TE*(1-K*EPS) "THEN" "BEGIN" LAST:="TRUE"; H:=TE-T "END"; B:=H*SIGMA; D:=DIAMETER*.1*H; D:=D*D; "IF" HD) "END" STEPSIZE; "PROCEDURE" DIFFERENCESCHEME ; "BEGIN" "INTEGER" I,J; "REAL" MT,LT,THT; I:=-1; NEXTTERM: I:=I+1; MT:=MU[I]*H; LT:=LABDA[I]*H; "FOR" J:=M0 "STEP" 1 "UNTIL" M "DO" RL[J]:=U[J]+LT*RL[J]; DERIVATIVE(T+MT,RL); "IF" I=0 "OR" I=N-1 "THEN" "BEGIN" THT:="IF" I=0 "THEN" THETA0*H "ELSE" THETANM1*H; ELMVEC(M0,M,0,U,RL,THT) "END"; "IF" I; THE INDEPENDENT VARIABLE X; ENTRY: THE INITIAL VALUE X0; EXIT : THE END VALUE XE; XE: ; THE END VALUE OF X; M: ; THE NUMBER OF DIFFERENTIAL EQUATIONS; Y: ; "ARRAY" Y[1 : M]; THE DEPENDENT VARIABLE; DURING THE INTEGRATION PROCESS THE COMPUTED SOLUTION AT THE POINT X IS ASSIGNED TO THE ARRAY Y; ENTRY: THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM; 1SECTION : 5.2.1.1.1.2.A (AUGUST 1974) PAGE 2 DELTA: ; DELTA DENOTES THE REAL PART OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED; ALTERNATIVES: DELTA = (AN ESTIMATE OF) THE REAL PART OF THE, IN ABSOLUTE VALUE, LARGEST EIGENVALUE OF THE JACOBIAN MATRIX OF THE SYSTEM; DELTA < -10**14, IN ORDER TO OBTAIN ASYMPTOTIC STABILITY; DELTA = 0, IN ORDER TO OBTAIN A HIGHER ORDER OF ACCURACY IN CASE OF LINEAR OR ALMOST LINEAR EQUATIONS; DERIVATIVE: ; "PROCEDURE" DERIVATIVE(A); "ARRAY" A; WHEN IN EFSIRK DERIVATIVE IS CALLED, A[I] CONTAINS THE VALUES OF Y[I]; UPON COMPLETION OF A CALL OF DERIVATIVE, THE ARRAY A SHOULD CONTAIN THE VALUES OF F(Y); NOTE THAT THE VARIABLE X SHOULD NOT BE USED IN DERIVATIVE, BECAUSE THE DIFFERENTIAL EQUATION IS SUPPOSED TO BE AUTONOMOUS; JACOBIAN: ; "PROCEDURE" JACOBIAN(J, Y); "ARRAY" J, Y; WHEN IN EFSIRK JACOBIAN IS CALLED THE ARRAY Y CONTAINS THE VALUES OF THE DEPENDENT VARIABLE; UPON COMPLETION OF A CALL OF JACOBIAN THE ARRAY J SHOULD CONTAIN THE VALUES OF THE JACOBIAN MATRIX OF F(Y); J: ; J[1 : M, 1 : M]; J IS AN AUXILLIARY ARRAY WHICH IS USED IN THE PROCEDURE JACOBIAN; N: ; AN INTEGER WHICH COUNTS THE INTEGRATION STEPS; AETA, RETA: ; REQUIRED ABSOLUTE AND RELATIVE LOCAL ACCURACY; HMIN, HMAX: ; MINIMAL AND MAXIMAL STEPSIZE BY WHICH THE INTEGRATION IS PERFORMED; LINEAR: ; IF LINEAR = "TRUE" THE PROCEDURE JACOBIAN WILL ONLY BE CALLED IF N = 1; THE INTEGRATION WILL THEN BE PERFORMED WITH A STEPSIZE HMAX; THE CORRESPONDING REDUCTION OF COMPUTING TIME CAN BE EXPLOITED IN CASE OF LINEAR OR ALMOST LINEAR EQUATIONS; OUTPUT: ; "PROCEDURE" OUTPUT; IN OUTPUT ONE MAY PRINT THE VALUES OF E.G. X, Y[I], J[K, L] AND N. 1SECTION : 5.2.1.1.1.2.A (AUGUST 1974) PAGE 3 DATA AND RESULTS: SEE REF[2] AND [3]. PROCEDURES USED: VECVEC = CP34010, MATVEC = CP34011, MATMAT = CP34013, GSSELM = CP34231, SOLELM = CP34061. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA M * M + 5 * M. RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO BE SOLVED LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE PROCEDURE EFSIRK IS AN EXPONENTIALLY FITTED, A-STABLE, SEMI-IMPLICIT RUNGE-KUTTA METHOD OF THIRD ORDER (SEE REF[1] AND [2]). THE ALGORITHM USES FOR EACH STEP TWO FUNCTION EVALUATIONS AND IF LINEAR = "FALSE" ONE EVALUATION OF THE JACOBIAN MATRIX. THE STEPSIZE IS NOT DETERMINED BY THE ACCURACY OF THE NUMERICAL SOLUTION, BUT BY THE AMOUNT BY WHICH THE GIVEN DIFFERENTIAL EQUATION DIFFERS FROM A LINEAR EQUATION (SEE REF[2]). THE PROCEDURE DOES NOT REJECT INTEGRATION STEPS. REFERENCES: [1].P.J. VAN DER HOUWEN. ONE-STEP METHODS WITH ADAPTIVE STABILITY FUCNTIONS FOR THE INTEGRATION OF DIFFERENTIAL EQUATIONS. LECTURE NOTES OF THE CONFERENCE ON NUMERISCHE, INSBESONDERE APPROXIMATIONSTHEORETISCHE BEHANDLUNG VON FUNKTIONALGLEICHUNGEN. OBERWOLFACH, DECEMBER, 3 - 12, 1972. [2].SYLLABUS COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 2 (DUTCH). MATH.CENTR. SYLLABUS 15.2/73. [3].SYLLABUS COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 3 (DUTCH). MATH.CENTR. SYLLABUS 15.3/73. TO APPEAR IN 1973. 1SECTION : 5.2.1.1.1.2.A (AUGUST 1974) PAGE 4 EXAMPLE OF USE: WE CONSIDER THE DIFFERENTIAL EQUATION DY / DX = -EXP(X) * (Y - LN(X)) + 1 / X, ON THE INTERVAL [0.01, 8], WITH INITIAL VALUE Y(0.01) = LN(0.01) AND ANALYTICAL SOLUTION Y(X) = LN(X); FOR THE FIT POINT WE USE THE EIGENVALUE OF THE JACOBIAN MATRIX, I.E. DELTA = -EXP(X); "BEGIN" "PROCEDURE" DER(Y); "ARRAY" Y; "BEGIN" "REAL" Y2; Y2:= Y[2]; DELTA:= -EXP(Y2); LNX:= LN(Y2); Y[1]:= (Y[1] - LNX) * DELTA + 1 / Y2; Y[2]:= 1 "END" DER; "PROCEDURE" JAC(J, Y); "ARRAY" J, Y; "BEGIN" "REAL" Y2; Y2:= Y[2]; J[1, 1]:= DELTA; J[1, 2]:= (Y[1] - LNX - 1 / Y2) * DELTA - 1 / (Y2 * Y2); J[2, 1]:= J[2, 2]:= 0 "END" JAC; "PROCEDURE" OUTP; "IF" X = XE "THEN" "BEGIN" "REAL" Y1; Y1:= Y[1]; LNX:= LN(X); OUTPUT(61, "(""("N = ")", 2ZD, "(" X = ")", +D.D, "(" Y(X) = ")", +D.5D, "(" DELTA = ")", +3ZD.2D, /, "("ABS. ERR. = ")", .2D"+2D, "(" REL. ERR. = ")", .2D"+2D, //")", N, X, Y1, DELTA, ABS(Y1 - LNX), ABS((Y1 - LNX) / LNX)); "IF" X = 0.4 "THEN" XE:= 8 "END" OUTP; "INTEGER" N; "REAL" X, XE, DELTA, LNX; "ARRAY" Y[1 : 2], J[1 : 2, 1 : 2]; XE:= 0.4; X:= 0.01; Y[1]:= LN(0.01); Y[2]:= X; EFSIRK(X, XE, 2, Y, DELTA, DER, JAC, J, N, "-2, "-2, 0.005, 1.5, "FALSE", OUTP) "END" THIS PROGRAM DELIVERS: N = 10 X = +0.4 Y(X) = -0.91099 DELTA = -1.44 ABS. ERR. = .53"-02 REL. ERR. = .58"-02 N = 98 X = +8.0 Y(X) = +2.07911 DELTA = -2980.02 ABS. ERR. = .33"-03 REL. ERR. = .16"-03 1SECTION : 5.2.1.1.1.2.A (AUGUST 1974) PAGE 5 SOURCE TEXT(S): 0"CODE" 33160; "PROCEDURE" EFSIRK(X, XE, M, Y, DELTA, DERIVATIVE, JACOBIAN, J, N, AETA, RETA, HMIN, HMAX, LINEAR, OUTPUT); "VALUE" M; "INTEGER" M, N; "REAL" X, XE, DELTA, AETA, RETA, HMIN, HMAX; "PROCEDURE" DERIVATIVE, JACOBIAN, OUTPUT; "BOOLEAN" LINEAR; "ARRAY" Y, J; "BEGIN" "INTEGER" K, L; "REAL" STEP, H, MU0, MU1, MU2, THETA0, THETA1, NU1, NU2, NU3, YK, FK, C1, C2, D; "ARRAY" F, K0, LABDA[1 : M], J1[1 : M,1 : M], AUX[1 : 7]; "INTEGER" "ARRAY" RI, CI[1 : M]; "BOOLEAN" LIN; "REAL" "PROCEDURE" STEPSIZE; "BEGIN" "REAL" DISCR, ETA, S; "IF" LINEAR "THEN" S:= H:= HMAX "ELSE" "IF" N = 1 "OR" HMIN = HMAX "THEN" S:= H:= HMIN "ELSE" "BEGIN" ETA:= AETA + RETA * SQRT(VECVEC(1, M, 0, Y, Y)); C1:= NU3 * STEP; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" LABDA[K]:= LABDA[K] + C1 * F[K] - Y[K]; DISCR:= SQRT(VECVEC(1, M, 0, LABDA, LABDA)); S:= H:= (ETA / (0.75 * (ETA + DISCR)) + 0.33) * H; "IF" H < HMIN "THEN" S:= H:= HMIN "ELSE" "IF" H > HMAX "THEN" S:= H:= HMAX "END"; "IF" X + S > XE "THEN" S:= XE - X; LIN:= STEP = S "AND" LINEAR; STEPSIZE:= S "END" STEPSIZE; "PROCEDURE" COEFFICIENT; "BEGIN" "REAL" Z1, E, ALPHA1, A, B; "OWN" "REAL" Z2; Z1:= STEP * DELTA; "IF" N = 1 "THEN" Z2:= Z1 + Z1; "IF" ABS(Z2 - Z1) > " - 6 * ABS(Z1) "OR" Z2 > - 1 "THEN" "BEGIN" A:= Z1 * Z1 + 12; B:= 6 * Z1; "IF" ABS(Z1) < 0.1 "THEN" ALPHA1:= (Z1 * Z1 / 140 - 1) * Z1 / 30 "ELSE" "IF" Z1 < - "14 "THEN" ALPHA1:= 1 / 3 "ELSE" "IF" Z1 < - 33 "THEN" ALPHA1:= (A + B) / (3 * Z1 * (2 + Z1)) "ELSE" "BEGIN" E:= "IF" Z1 < 230 "THEN" EXP(Z1) "ELSE" "100; ALPHA1:= ((A - B) * E - A - B) / (((2 - Z1) * E - 2 - Z1) * 3 * Z1) "END"; "COMMENT" 1SECTION : 5.2.1.1.1.2.A (AUGUST 1974) PAGE 6 ; MU2:= (1 / 3 + ALPHA1) * 0.25; MU1:= - (1 + ALPHA1) * 0.5; MU0:= (6 * MU1 + 2) / 9; THETA0:= 0.25; THETA1:= 0.75; A:= 3 * ALPHA1; NU3:= (1 + A) / (5 - A) * 0.5; A:= NU3 + NU3; NU1:= 0.5 - A; NU2:= (1 + A) * 0.75; Z2:= Z1 "END" "END" COEFFICIENT; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" DERIVATIVE(F); STEP:= STEPSIZE; "IF" "NOT" LINEAR "OR" N = 1 "THEN" JACOBIAN(J, Y); "IF" "NOT" LIN "THEN" "BEGIN" COEFFICIENT; C1:= STEP * MU1; D:= STEP * STEP * MU2; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" "FOR" L:= 1 "STEP" 1 "UNTIL" M "DO" J1[K,L]:= D * MATMAT(1, M, K, L, J, J) + C1 * J[K,L]; J1[K,K]:= J1[K,K] + 1 "END"; GSSELM(J1, M, AUX, RI, CI) "END"; C1:= STEP * STEP * MU0; D:= STEP * 2 / 3; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" K0[K]:= FK:= F[K]; LABDA[K]:= D * FK + C1 * MATVEC(1, M, K, J, F) "END"; SOLELM(J1, M, RI, CI, LABDA); "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" F[K]:= Y[K] + LABDA[K]; DERIVATIVE(F); C1:= THETA0 * STEP; C2:= THETA1 * STEP; D:= NU1 * STEP; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" "BEGIN" YK:= Y[K]; FK:= F[K]; LABDA[K]:= YK + D * FK + NU2 * LABDA[K]; Y[K]:= F[K]:= YK + C1 * K0[K] + C2 * FK "END" "END" DIFFERENCE SCHEME; AUX[2]:= "-14; AUX[4]:= 8; "FOR" K:= 1 "STEP" 1 "UNTIL" M "DO" F[K]:= Y[K]; N:= 0; OUTPUT; STEP:= 0; NEXT STEP: N:= N + 1; DIFFERENCE SCHEME; X:= X + STEP; OUTPUT; "IF" X < XE "THEN" "GOTO" NEXT STEP "END" EFSIRK; "EOP" 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 1 AUTHOR: K.DEKKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 1973/07/31. BRIEF DESCRIPTION: EFERK SOLVES INITIAL VALUE PROBLEMS , GIVEN AS AN AUTONOMOUS SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS, BY MEANS OF AN EXPONENTIALLY FITTED, EXPLICIT RUNGE KUTTA METHOD OF THIRD ORDER, WHICH INVOLVES THE USE OF THE JACOBIAN MATRIX. AUTOMATIC STEP CONTROL IS PROVIDED. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS. KEYWORDS: DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEMS, STIFF EQUATIONS, EXPONENTIAL FITTING, EXPLICIT RUNGE KUTTA METHODS. 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE EFERK READS: "PROCEDURE" EFERK(X,XE,M,Y,SIGMA,PHI,DERIVATIVE,J,JACOBIAN, K,L,AUT,AETA,RETA,HMIN,HMAX,LINEAR,OUTPUT); "VALUE" L; "INTEGER" M,K,L; "REAL" X,XE,SIGMA,PHI,AETA,RETA,HMIN,HMAX; "ARRAY" Y,J; "BOOLEAN" AUT,LINEAR; "PROCEDURE" DERIVATIVE,JACOBIAN,OUTPUT; "CODE" 33120; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE X; CAN BE USED IN DERIVATIVE, JACOBIAN, OUTPUT, ETC.; ENTRY: THE INITIAL VALUE X0; EXIT : THE FINAL VALUE XE; XE: ; THE FINAL VALUE OF X (XE>=X); M: ; THE NUMBER OF EQUATIONS; Y: ; "REAL" "ARRAY" Y[1:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SYSTEM OF DIFFERENTIAL EQUATIONS: Y[I] AT X=X0; EXIT : THE FINAL VALUES OF THE SOLUTION: Y[I] AT X=XE; SIGMA: ; THE MODULUS OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED, FOR EXAMPLE THE LARGEST NEGATIVE EIGENVALUE OF THE JACOBIAN MATRIX OF THE SYSTEM OF DIFFERENTIAL EQUATIONS; PHI: ; THE ARGUMENT OF THE COMPLEX POINT AT WHICH EXPONENTIAL FITTING IS DESIRED; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(Y); "ARRAY" Y; THIS PROCEDURE SHOULD DELIVER THE RIGHT HAND SIDE OF THE I-TH DIFFERENTIAL EQUATION AT THE POINT (Y) AS Y[I]; J: ; "REAL" "ARRAY" J[1:M,1:M]; THE JACOBIAN MATRIX OF THE SYSTEM; THE ARRAY J SHOULD BE UPDATED IN THE PROCEDURE JACOBIAN; 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 3 JACOBIAN: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; IN THIS PROCEDURE THE JACOBIAN AT THE POINT (Y) HAS TO BE ASSIGNED TO THE ARRAY J; K: ; COUNTS THE NUMBER OF INTEGRATION STEPS TAKEN; FOR EXAMPLE, MAY BE USED IN THE EXPRESSION FOR XE; L: ; ENTRY: IF PHI = 4*ARCTAN(1): THE ORDER OF THE EXPONENTIAL FITTING, ELSE TWICE THE ORDER OF THE EXPONENTIAL FITTING; AUT: ; IF THE SYSTEM HAS BEEN WRITTEN IN AUTONOMOUS FORM BY ADDING THE EQUATION DY[M]/DX = 1 TO THE SYSTEM , THEN AUT MAY HAVE THE VALUE "FALSE", ELSE AUT SHOULD HAVE THE VALUE "TRUE"; AETA: ; REQUIRED ABSOLUTE PRECISION IN THE INTEGRATION PROCESS; AETA HAS TO BE POSITIVE; RETA: ; REQUIRED RELATIVE PRECISION IN THE INTEGRATION PROCESS; RETA HAS TO BE POSITIVE; HMIN: ; THE STEPLENGTH CHOSEN WILL BE AT LEAST EQUAL TO HMIN; HMAX: ; THE STEPLENGTH CHOSEN WILL BE AT MOST EQUAL TO HMAX; LINEAR: ; THE PROCEDURE JACOBIAN IS CALLED ONLY IF LINEAR="FALSE" OR K=0; SO IF THE SYSTEM IS LINEAR , LINEAR MAY HAVE THE VALUE "TRUE"; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS; "PROCEDURE" OUTPUT; THIS PROCEDURE IS CALLED AT THE END OF EACH INTEGRATION STEP ; THE USER CAN ASK FOR OUTPUT OF SOME PARAMETERS , FOR EXAMPLE X, K, Y. DATA AND RESULTS: SEE EXAMPLE OF USE, AND REF[4]. PROCEDURES USED: VECVEC= CP34010, MATVEC= CP34011, DEC = CP34300, SOL = CP34051. 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 4 REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 30 + 4 * M + L * (5+L). RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO SOLVE. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE PROCEDURE EFERK IS AN EXPONENTIALLY FITTED, SEMI-EXPLICIT RUNGE KUTTA METHOD OF THIRD ORDER ( SEE REF [1] AND [3] ) . THE ALGORITHM USES FOR EACH STEP TWO FUNCTION EVALUATIONS AND IF LINEAR = "FALSE" ONE EVALUATION OF THE JACOBIAN MATRIX.THE STEPSIZE IS DETERMINED BY AN ESTIMATION OF THE LOCAL TRUNCATION ERROR BASED ON THE RESIDUAL FUNCTION (SEE REF[3]). INTEGRATION STEPS ARE NOT REJECTED. REFERENCES: [1]. P.J.VAN DER HOUWEN. ONE-STEP METHODS WITH ADAPTIVE STABILITY FUNCTIONS FOR THE INTEGRATION OF DIFFERENTIAL EQUATIONS. LECTURES NOTES OF THE CONFERENCE ON NUMERISCHE, INSBESONDERE APPROXIMATIONSTHEORETISCHE BEHANDLUNG VON FUNKTIONAL- GLEICHUNGEN. OBERWOLFACH, DECEMBER, 3 -12, 1972. [2]. T.J.DEKKER, P.W.HEMKER AND P.J.VAN DER HOUWEN. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 1 (DUTCH). MC SYLLABUS 15.1, (1972) MATHEMATICAL CENTRE. [3]. P.A.BEENTJES, K.DEKKER, H.C.HEMKER, S.P.N.VAN KAMPEN AND G.M.WILLEMS. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 2 (DUTCH). MC SYLLABUS 15.2, (1973) MATHEMATICAL CENTRE. [4]. (TO APPEAR). COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 3 (DUTCH). MC SYLLABUS 15.3, (1973) MATHEMATICAL CENTRE. EXAMPLE OF USE: CONSIDER THE SYSTEM OF DIFFERENTIAL EQUATIONS: DY[1]/DX = -Y[1] + Y[1] * Y[2] + .99 * Y[2] DY[2]/DX = -1000 * ( -Y[1] + Y[1] * Y[2] + Y[2] ) WITH THE INITIAL CONDITIONS AT X = 0: Y[1] = 1 AND Y[2] = 0. (SEE REF[2], PAGE 11). THE SOLUTION AT X = 50 IS APPROXIMATELY: Y[1] = .765 878 320 487 AND Y[2] = .433 710 353 5768. THE FOLLOWING PROGRAM SHOWS SOME DIFFERENT CALLS OF THE PROCEDURE EFERK,AND ILLUSTRATES THE ACCURACIES WHICH MAY BE OBTAINED BY THEM: 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 5 "BEGIN" "INTEGER" K,PASSES,PASJAC; "REAL" X,SIGMA,PHI,TIME,TOL; "REAL" "ARRAY" Y[1:2],J[1:2,1:2]; "PROCEDURE" DER(Y); "ARRAY" Y; "BEGIN" "REAL" Y1,Y2; Y1:=Y[1]; Y2:=Y[2]; Y[1]:=(Y1+.99)*(Y2-1)+.99; Y[2]:=1000*((1+Y1)*(1-Y2)-1); PASSES:=PASSES+1 "END"; "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; "BEGIN" J[1,1]:=Y[2]-1; J[1,2]:=.99+Y[1]; J[2,1]:=1000*(1-Y[2]); J[2,2]:=-1000*(1+Y[1]); SIGMA:=ABS(J[2,2]+J[1,1]-SQRT((J[2,2]-J[1,1])**2+ 4*J[2,1]*J[1,2]))/2; PASJAC:=PASJAC+1 "END" JACOBIAN; "PROCEDURE" OUT; "IF" X=50 "THEN" OUTPUT(61,"("3(-5ZD),2(4B+.3DB3DB3D),-5ZD.3D,/")",K,PASSES, PASJAC,Y[1],Y[2],CLOCK-TIME); OUTPUT(61,"(""(" THIS LINE AND THE FOLLOWING TEXT IS ")" "("PRINTED BY THIS PROGRAM")",//, "(" THE RESULTS WITH EFERK -FIRST ORDER FIT- ARE:")",/, "(" K DER.EV. JAC.EV. Y[1] Y[2]")" "(" TIME")",/")"); PHI:=4*ARCTAN(1); "FOR" TOL:=1,"-1,"-2,"-3 "DO" "BEGIN" PASSES:=PASJAC:=0; X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; EFERK(X,50.0,2,Y,SIGMA,PHI,DER,J,JACOBIAN,K,1,"TRUE",TOL, TOL,"-6,50.0,"FALSE",OUT); "END"; "END" THIS LINE AND THE FOLLOWING TEXT IS PRINTED BY THIS PROGRAM: THE RESULTS WITH EFERK -FIRST ORDER FIT- ARE: K DER.EV. JAC.EV. Y[1] Y[2] TIME 93 186 93 +.765 883 211 +.428 752 781 1.170 105 210 105 +.765 878 445 +.433 569 561 1.296 147 294 147 +.765 878 317 +.433 708 489 1.834 266 532 266 +.765 878 320 +.433 710 229 3.297 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 6 SOURCE TEXT(S): 0"CODE" 33120; "PROCEDURE" EFERK(X,XE,M,Y,SIGMA,PHI,DERIVATIVE,J,JACOBIAN, K,L,AUT,AETA,RETA,HMIN,HMAX,LINEAR,OUTPUT); "VALUE" L; "INTEGER" M,K,L; "REAL" X,XE,SIGMA,PHI,AETA,RETA,HMIN,HMAX; "ARRAY" Y,J; "BOOLEAN" AUT,LINEAR; "PROCEDURE" DERIVATIVE,JACOBIAN,OUTPUT; "BEGIN" "INTEGER" M1,I; "REAL" H,B,B0,PHI0,COSPHI,SINPHI,ETA,DISCR,FAC,PI; "BOOLEAN" CHANGE,LAST; "INTEGER" "ARRAY" P[1:L]; "REAL" "ARRAY" BETA,BETHA[0:L],BETAC[0:L+3],K0,D,D1,D2[1:M], A[1:L,1:L],AUX[1:3]; "REAL" "PROCEDURE" SUM(I,L,U,T); "VALUE" L,U; "INTEGER" I,L,U; "REAL" T; "BEGIN" "REAL" S; S:=0; "FOR" I:=L "STEP" 1 "UNTIL" U "DO" S:=S+T; SUM:=S "END"; "PROCEDURE" FORMBETA; "IF" L=1 "THEN" "BEGIN" BETHA[1]:=(.5-(1-(1-EXP(-B))/B)/B)/B; BETA[1]:=(1/6-BETHA[1])/B "END" "ELSE" "IF" L=2 "THEN" "BEGIN" "REAL" E,EMIN1; E:=EXP(-B); EMIN1:=E-1; BETHA[1]:=(1-(3+E+4*EMIN1/B)/B)/B; BETHA[2]:=(.5-(2+E+3*EMIN1/B)/B)/B/B; BETA[2]:=(1/6-BETHA[1])/B/B; BETA[1]:=(1/3-(1.5-(4+E+5*EMIN1/B)/B)/B)/B "END" "ELSE" "BEGIN" "REAL" B0,B1,B2,A0,A1,A2,A3,C,D; BETAC[L-1]:=C:=D:=EXP(-B)/FAC; "FOR" I:=L-1 "STEP" -1 "UNTIL" 1 "DO" "BEGIN" C:=I*B*C/(L-I); BETAC[I-1]:=D:=D*I+C "END"; B2:=.5-BETAC[2]; B1:=(1-BETAC[1])*(L+1)/B; B0:=(1-BETAC[0])*(L+2)*(L+1)*.5/B/B; A3:=1/6-BETAC[3]; A2:=B2*(L+1)/B; A1:=B1*(L+2)*.5/B; A0:=B0*(L+3)/3/B; D:=L/B; "FOR" I:=1 "STEP" 1 "UNTIL" L "DO" "BEGIN"BETA[I]:=(A3/I-A2/(I+1)+A1/(I+2)-A0/(I+3))*D+BETAC[I+3]; BETHA[I]:=(B2/I-B1/(I+1)+B0/(I+2))*D+BETAC[I+2]; D:=D*(L-I)/I/B; "END" "END" FORMBETA; "COMMENT" 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 7 ; "PROCEDURE" SOLUTIONOFCOMPLEXEQUATIONS; "IF" L=2 "THEN" "BEGIN" "REAL" COS2PHI,COSA,SINA,E,ZI; PHI0:=PHI; COSPHI:=COS(PHI0); SINPHI:=SIN(PHI0); E:=EXP(B*COSPHI); ZI:=B*SINPHI-3*PHI0; SINA:=("IF" ABS(SINPHI)<"-6 "THEN" -E*(B+3) "ELSE" E*SIN(ZI)/SINPHI); COS2PHI:=2*COSPHI*COSPHI-1; BETHA[2]:=(.5+(2*COSPHI+(1+2*COS2PHI+SINA)/B)/B)/B/B; SINA:=("IF" ABS(SINPHI)<"-6 "THEN" E*(B+4) "ELSE" SINA*COSPHI-E*COS(ZI)); BETHA[1]:=-(COSPHI+(1+2*COS2PHI+(4*COSPHI*COS2PHI+SINA) /B)/B)/B; BETA[1]:=BETHA[2]+2*COSPHI*(BETHA[1]-1/6)/B; BETA[2]:=(1/6-BETHA[1])/B/B "END" "ELSE" "BEGIN" "INTEGER" J,C1; "REAL" C2,E,ZI,COSIPHI,SINIPHI,COSPHIL; "REAL" "ARRAY" D[1:L]; "PROCEDURE" ELEMENTS OF MATRIX; "BEGIN" PHI0:=PHI; COSPHI:=COS(PHI0); SINPHI:=SIN(PHI0); COSIPHI:=1; SINIPHI:=0; "FOR" I:=0 "STEP" 1 "UNTIL" L-1 "DO" "BEGIN" C1:=4+I; C2:=1; "FOR" J:=L-1 "STEP" -2 "UNTIL" 1 "DO" "BEGIN" A[J,L-I]:=C2*COSIPHI; A[J+1,L-I]:=C2*SINIPHI; C2:=C2*C1; C1:=C1-1 "END"; COSPHIL:=COSIPHI*COSPHI-SINIPHI*SINPHI; SINIPHI:=COSIPHI*SINPHI+SINIPHI*COSPHI; COSIPHI:=COSPHIL "END"; AUX[2]:=0; DEC(A,L,AUX,P) "END" EL OF MAT; "PROCEDURE" RIGHT HAND SIDE; "BEGIN" E:=EXP(B*COSPHI); ZI:=B*SINPHI-4*PHI0; COSIPHI:=E*COS(ZI); SINIPHI:=E*SIN(ZI); ZI:=1/B/B/B; "FOR" J:=L "STEP" -2 "UNTIL" 2 "DO" "BEGIN" D[J]:=ZI*SINIPHI; D[J-1]:=ZI*COSIPHI; COSPHIL:=COSIPHI*COSPHI-SINIPHI*SINPHI; SINIPHI:=COSIPHI*SINPHI+SINIPHI*COSPHI; COSIPHI:=COSPHIL; ZI:=ZI*B "END"; "COMMENT" 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 8 ; SINIPHI:=2*SINPHI*COSPHI; COSIPHI:=2*COSPHI*COSPHI-1; COSPHIL:=COSPHI*(2*COSIPHI-1); D[L]:=D[L]+SINPHI*(1/6+(COSPHI+(1+2*COSIPHI*(1+2*COSPHI/B)) /B)/B); D[L-1]:=D[L-1]-COSPHI/6-(.5*COSIPHI+(COSPHIL+(2*COSIPHI* COSIPHI-1)/B)/B)/B; D[L-2]:=D[L-2]+SINPHI*(.5+(2*COSPHI+(2*COSIPHI+1)/B)/B); D[L-3]:=D[L-3]-.5*COSPHI-(COSIPHI+COSPHIL/B)/B; "IF" L<5 "THEN" "GOTO" END; D[L-4]:=D[L-4]+SINPHI+SINIPHI/B; D[L-5]:=D[L-5]-COSPHI-COSIPHI/B; "IF" L<7 "THEN" "GOTO" END; D[L-6]:=D[L-6]+SINPHI; D[L-7]:=D[L-7]-COSPHI; END: "END" RHS; "IF" PHI0^=PHI "THEN" ELEMENTS OF MATRIX; RIGHT HAND SIDE; SOL(A,L,P,D); ZI:=1/B; "FOR" I:=1 "STEP" 1 "UNTIL" L "DO" "BEGIN" BETA[I]:=D[L+1-I]*ZI; BETHA[I]:=(I+3)*BETA[I]; ZI:=ZI/B "END" "END" SOLOFEQCOM; "PROCEDURE" COEFFICIENT; "BEGIN" B0:=B:=ABS(H*SIGMA); "IF" B>=.1 "THEN" "BEGIN" "IF" PHI^=PI "AND" L=2 "OR" ABS(PHI-PI)>.01 "THEN" SOLUTION OF COMPLEX EQUATIONS "ELSE" FORMBETA "END" "ELSE" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" L "DO" "BEGIN" BETHA[I]:=BETA[I-1]; BETA[I]:=BETA[I-1]/(I+3); "END" "END" "END" COEFFICIENT; "PROCEDURE" LOCAL ERROR BOUND; ETA:=AETA+RETA*SQRT(VECVEC(1,M1,0,Y,Y)); "PROCEDURE" STEPSIZE; "BEGIN" LOCAL ERROR BOUND; "IF" K=0 "THEN" "BEGIN" DISCR:=SQRT(VECVEC(1,M1,0,D,D)); H:=ETA/DISCR "END" "ELSE" "BEGIN" DISCR:=H*SQRT(SUM(I,1,M1,(D[I]-D2[I])**2))/ETA; H:=H*("IF" LINEAR "THEN" 4/(4+DISCR)+.5 "ELSE" 4/(3+DISCR)+1/3) "END"; "COMMENT" 1SECTION : 5.2.1.1.1.2.B (AUGUST 1974) PAGE 9 ; "IF" HHMAX "THEN" H:=HMAX; B:=ABS(H*SIGMA); CHANGE:=ABS(1-B/B0)>.05 "OR" PHI^=PHI0; "IF" 1.1*H>=XE-X "THEN" "BEGIN" CHANGE:=LAST:="TRUE"; H:=XE-X "END"; "IF" "NOT" CHANGE "THEN" H:=H*B0/B "END" STEPSIZE; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" "INTEGER" K; "REAL" BETAI,BETHAI; "IF" M1; THE INDEPENDENT VARIABLE X; ENTRY: THE INITIAL VALUE X0; EXIT : THE FINAL VALUE XE; XE: ; THE FINAL VALUE OF X (XE>=X); M: ; THE NUMBER OF EQUATIONS; Y: ; "ARRAY" Y[1:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SYSTEM OF DIFFERENTIAL EQUATIONS: Y[I] AT X=X0; EXIT : THE FINAL VALUES OF THE SOLUTION: Y[I] AT X=XE; 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 2 SIGMA: ; THE MODULUS OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED, FOR EXAMPLE THE LARGEST NEGATIVE EIGENVALUE OF THE JACOBIAN OF THE SYSTEM OF DIFFERENTIAL EQUATIONS; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(Y); "ARRAY" Y; THIS PROCEDURE SHOULD DELIVER THE RIGHT HAND SIDE OF THE I-TH DIFFERENTIAL EQUATION AT THE POINT (Y) AS Y[I]; J: ; "ARRAY" J[1:M,1:M]; THE JACOBIAN MATRIX OF THE SYSTEM; THE ARRAY J SHOULD BE UPDATED IN THE PROCEDURE JACOBIAN; JACOBIAN: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; IN THIS PROCEDURE (AN APPROXIMATION OF) THE JACOBIAN HAS TO BE ASSIGNED TO THE ARRAY J; ITMAX: ; AN UPPERBOUND FOR THE NUMBER OF ITERATIONS IN NEWTON'S PROCESS, USED TO SOLVE THE IMPLICIT EQUATIONS; HMIN: ; MINIMAL STEPSIZE BY WHICH THE INTEGRATION IS PERFORMED; HMAX: ; MAXIMAL STEPSIZE BY WHICH THE INTEGRATION IS PERFORMED; AETA: ; REQUIRED ABSOLUTE PRECISION IN THE INTEGRATION PROCESS; RETA: ; REQUIRED RELATIVE PRECISION IN THE INTEGRATION PROCESS; IF BOTH AETA AND RETA HAVE NEGATIVE VALUES , INTEGRATION WILL BE PERFORMED WITH A STEPSIZE EQUAL TO HMAX , WHICH MAY BE VARIATED BY USER ; IN THIS CASE THE ABSOLUTE VALUES OF AETA AND RETA WILL CONTROL THE NEWTON ITERATION; INFO: ; "ARRAY" INFO[1:9]; DURING INTEGRATION AND UPON EXIT THIS ARRAY CONTAINS THE FOLLOWING INFORMATION: INFO[1]: NUMBER OF INTEGRATION STEPS TAKEN; INFO[2]: NUMBER OF DERIVATIVE EVALUATIONS USED; INFO[3]: NUMBER OF JACOBIAN EVALUATIONS USED; INFO[4]: NUMBER OF INTEGRATION STEPS EQUAL TO HMIN TAKEN ; INFO[5]: NUMBER OF INTEGRATION STEPS EQUAL TO HMAX TAKEN ; INFO[6]: MAXIMAL NUMBER OF ITERATIONS TAKEN IN THE NEWTON PROCESS; INFO[7]: LOCAL ERROR TOLERANCE; INFO[8]: ESTIMATED LOCAL ERROR; INFO[9]: MAXIMUM VALUE OF THE ESTIMATED LOCAL ERROR; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS; "PROCEDURE" OUTPUT; THIS PROCEDURE IS CALLED AT THE END OF EACH INTEGRATION STEP ; THE USER CAN ASK FOR OUTPUT OF THE PARAMETERS , FOR EXAMPLE X, Y, INFO. 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 3 DATA AND RESULTS: SEE EXAMPLE OF USE, AND REF[2]. PROCEDURES USED: INIVEC= CP31010, MULVEC= CP31020, MULROW= CP31021, DUPVEC= CP31030, MATVEC= CP34011, ELMVEC= CP34020, VECVEC= CP34010, DEC = CP34300, SOL = CP34051. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH : CIRCA 20 + M * (5 + M). RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO SOLVE. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: LINIGER1VS: INTEGRATES THE SYSTEM OF DIFFERENTIAL EQUATIONS FROM X0 UNTIL XE, BY MEANS OF A FIRST ORDER FORMULA. THE INTEGRATION METHOD IS BASED ON THE F(1) FORMULA DESCRIBED BY LINIGER AND WILLOUGHBY (SEE REF[1]). ERROR ESTIMATES AND A STEPSIZE STRATEGY FOR THIS METHOD ARE DESCRIBED IN [2] , AND A VARIABLE STEP METHOD IS CONSTRUCTED FOR THE CONVENIENCE OF THE USER. HOWEVER, THE STEPSIZE STRATEGY REQUIRES MANY EXTRA ARRAY OPERATIONS. THE USER MAY AVOID THIS EXTRA WORK BY GIVING AETA AND RETA A NEGATIVE VALUE AND PRESCRIBING A STEPSIZE (HMAX) HIMSELF. REFERENCES: [1]. W.LINIGER AND R.A.WILLOUGHBY. EFFICIENT INTEGRATION METHODS FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. SIAM J. NUM. ANAL. 7 (1970) PAGE 47. [2]. K.DEKKER. ERROR ESTIMATES AND STEPSIZE STRATEGIES FOR THE LINIGER- WILLOUGHBY FORMULAE. (TO APPEAR IN 1974). 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 4 EXAMPLE OF USE: CONSIDER THE SYSTEM OF DIFFERENTIAL EQUATIONS: DY[1]/DX = -Y[1] + Y[1] * Y[2] + .99 * Y[2] DY[2]/DX = -1000 * ( -Y[1] + Y[1] * Y[2] + Y[2] ) WITH THE INITIAL CONDITIONS AT X = 0: Y[1] = 1 AND Y[2] = 0. THE SOLUTION AT X = 50 IS APPROXIMATELY: Y[1] = .765 878 320 2487 AND Y[2] = .433 710 353 5768. THE FOLLOWING PROGRAM SHOWS INTEGRATION OF THIS PROBLEM WITH VARIABLE AND CONSTANT STEPSIZES: "BEGIN" "COMMENT" TEST LINIGER1VS; "INTEGER" ITMAX; "REAL" X,SIGMA,RETA,TIME; "REAL" "ARRAY" Y[1:2],J[1:2,1:2],INFO[1:9]; "PROCEDURE" F(A); "ARRAY" A; "BEGIN" "REAL" A1,A2; A1:=A[1]; A2:=A[2]; A[1]:=(A1+.99)*(A2-1)+.99; A[2]:=1000*((1+A1)*(1-A2)-1); "END"; "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; "BEGIN" J[1,1]:=Y[2]-1; J[1,2]:=.99+Y[1]; J[2,1]:=1000*(1-Y[2]); J[2,2]:=-1000*(1+Y[1]); SIGMA:=ABS(J[2,2]+J[1,1]-SQRT((J[2,2]-J[1,1])**2+ 4*J[2,1]*J[1,2]))/2; "END" JACOBIAN; "PROCEDURE" OUT; "IF" X=50 "THEN" OUTPUT(61,"("6(3ZDB),2BD"-ZD,2(2B+.3DB3D),-3ZD.3D,/")", INFO[1],INFO[2],INFO[3],INFO[4],INFO[5],INFO[6],INFO[9],Y[1], Y[2],CLOCK-TIME); "FOR" RETA:="-2,"-4,"-6 "DO" "BEGIN" X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; LINIGER1VS(X,50.0,2,Y,SIGMA,F,J,JACOBIAN,10,.1,50.0,RETA, RETA,INFO,OUT); "END"; OUTPUT(61,"("//")"); "FOR" RETA:=-"-2,-"-4,-"-6 "DO" "BEGIN" X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; LINIGER1VS(X,50.0,2,Y,SIGMA,F,J,JACOBIAN,10,.1,1.0,RETA, RETA,INFO,OUT); "END"; "END" 17 21 8 2 0 2 2" -2 +.772 017 +.435 672 0.525 13 25 23 2 0 2 2" -2 +.767 414 +.434 202 0.717 105 210 105 2 0 2 2" -2 +.766 027 +.433 758 4.741 50 52 1 0 50 2 0" 0 +.766 670 +.433 081 0.549 50 104 3 0 50 3 0" 0 +.766 183 +.433 811 1.158 50 152 12 0 50 4 0" 0 +.766 185 +.433 809 1.653 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 5 SOURCE TEXT(S): "CODE" 33132; "PROCEDURE" LINIGER1VS(X,XE,M,Y,SIGMA,DERIVATIVE,J, JACOBIAN,ITMAX,HMIN,HMAX,AETA,RETA,INFO,OUTPUT); "VALUE" M; "INTEGER" M,ITMAX; "REAL" X,XE,SIGMA,AETA,RETA,HMIN,HMAX; "ARRAY" Y,J,INFO; "PROCEDURE" DERIVATIVE,JACOBIAN,OUTPUT; "BEGIN" "INTEGER" I,ST,LASTJAC; "REAL" H,HNEW,MU,MU1,BETA,P,E,E1,ETA,ETA1,DISCR; "BOOLEAN" LAST,FIRST,EVALJAC,EVALCOEF; "INTEGER" "ARRAY" PI[1:M]; "REAL" "ARRAY" DY,YL,YR,F[1:M],A[1:M,1:M],AUX[1:3]; "REAL" "PROCEDURE" NORM(A); "ARRAY" A; NORM:=SQRT(VECVEC(1,M,0,A,A)); "PROCEDURE" COEFFICIENT; "BEGIN" "REAL" B,E; B:=ABS(H*SIGMA); "IF" B>40 "THEN" "BEGIN" MU:=1/B; BETA:=1; P:=2+2/(B-2) "END" "ELSE" "IF" B<.04 "THEN" "BEGIN" E:=B*B/30; P:=3-E; MU:=.5-B/12*(1-E/2); BETA:=.5+B/6*(1-E) "END" "ELSE" "BEGIN" E:=EXP(B)-1; MU:=1/B-1/E; BETA:=(1-B/E)*(1+1/E); P:=(BETA-MU)/(.5-MU) "END"; MU1:=H*(1-MU); LUDECOMP "END" COEFFICIENT; "COMMENT" 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 6 ; "PROCEDURE" LUDECOMP; "BEGIN" "INTEGER" I; "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" MULROW(1,M,I,I,A,J,-MU1); A[I,I]:=A[I,I]+1 "END"; AUX[2]:=0; DEC(A,M,AUX,PI) "END" LUDECOMP; "PROCEDURE" STEPSIZE; "IF" ETA<0 "THEN" "BEGIN" "REAL" HL; HL:=H; H:=HNEW:=HMAX; INFO[5]:=INFO[5]+1; "IF" 1.1*HNEW>XE-X "THEN" "BEGIN" LAST:="TRUE"; H:=HNEW:=XE-X "END"; EVALCOEF:=H^=HL; "END" "ELSE" "IF" FIRST "THEN" "BEGIN" H:=HNEW:=HMIN; FIRST:="FALSE"; INFO[4]:=INFO[4]+1 "END" "ELSE" "BEGIN" "REAL" B,HL; B:=DISCR/ETA; HL:=H; "IF" B<.01 "THEN" B:=.01; HNEW:= "IF" B>0 "THEN" H*B**(-1/P) "ELSE" HMAX; "IF" HNEWHMAX "THEN" "BEGIN" HNEW:=HMAX; INFO[5]:=INFO[5]+1 "END"; "IF" 1.1*HNEW>=XE-X "THEN" "BEGIN" LAST:="TRUE"; H:=HNEW:=XE-X "END" "ELSE" "IF" ABS(H/HNEW-1)>.1 "THEN" H:=HNEW; EVALCOEF:=H^=HL "END" STEPSIZE; "PROCEDURE" LINEARITY(ERROR); "REAL" ERROR; "BEGIN" "INTEGER" K; "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" DY[K]:=Y[K]-MU1*F[K]; SOL(A,M,PI,DY); ELMVEC(1,M,0,DY,Y,-1); ERROR:=NORM(DY) "END" LINEARITY; "COMMENT" 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 7 ; "PROCEDURE" ITERATION(I); "INTEGER" I; "IF" RETA<0 "THEN" "BEGIN" "INTEGER" K; "IF" I=1 "THEN" "BEGIN" MULVEC(1,M,0,DY,F,H); "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" YL[K]:=Y[K]+MU*DY[K]; SOL(A,M,PI,DY); E:=1; "END" "ELSE" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" DY[K]:=YL[K]-Y[K]+MU1*F[K]; "IF" E*NORM(Y)>E1*E1 "THEN" "BEGIN" EVALJAC:=I>=3; "IF" I>3 "THEN" "BEGIN" INFO[3]:=INFO[3]+1; JACOBIAN(J,Y); LUDECOMP "END"; "END"; SOL(A,M,PI,DY) "END"; E1:=E; E:=NORM(DY); ELMVEC(1,M,0,Y,DY,1); ETA:=NORM(Y)*RETA+AETA; DISCR:=0; DUPVEC(1,M,0,F,Y); DERIVATIVE(F); INFO[2]:=INFO[2]+1; "END" "ELSE" "BEGIN" "INTEGER" K; "IF" I=1 "THEN" "BEGIN" LINEARITY(E); "IF" E*(ST-LASTJAC)>ETA "THEN" "BEGIN" JACOBIAN(J,Y); LASTJAC:=ST; INFO[3]:=INFO[3]+1; H:=HNEW; COEFFICIENT; LINEARITY(E) "END"; EVALJAC:= E*(ST+1-LASTJAC)>ETA; MULVEC(1,M,0,DY,F,H); "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" YL[K]:=Y[K]+MU*DY[K]; SOL(A,M,PI,DY); "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" YR[K]:=H*BETA*MATVEC(1,M,K,J,DY); SOL(A,M,PI,YR); ELMVEC(1,M,0,YR,DY,1); "COMMENT" 1SECTION : 5.2.1.1.1.2.C (OCTOBER 1974) PAGE 8 ; "END" "ELSE" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" DY[K]:=YL[K]-Y[K]+MU1*F[K]; "IF" E>ETA1 "AND" DISCR>ETA1 "THEN" "BEGIN" INFO[3]:=INFO[3]+1; JACOBIAN(J,Y); LUDECOMP "END"; SOL(A,M,PI,DY); E:=NORM(DY) "END"; ELMVEC(1,M,0,Y,DY,1); ETA:=NORM(Y)*RETA+AETA; ETA1:=ETA/SQRT(RETA); DUPVEC(1,M,0,F,Y); DERIVATIVE(F); INFO[2]:=INFO[2]+1; "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" DY[K]:=YR[K]-H*F[K]; DISCR:=NORM(DY)/2 "END" ITERATION; FIRST:=EVALJAC:="TRUE"; LAST:=EVALCOEF:="FALSE"; INIVEC(1,9,INFO,0); ETA:=RETA*NORM(Y)+AETA; ETA1:=ETA/SQRT(ABS(RETA)); DUPVEC(1,M,0,F,Y); DERIVATIVE(F); INFO[2]:=1; "FOR" ST:=1,ST+1 "WHILE" ^LAST "DO" "BEGIN" STEPSIZE; INFO[1]:=INFO[1]+1; "IF" EVALJAC "THEN" "BEGIN" JACOBIAN(J,Y); INFO[3]:=INFO[3]+1; H:=HNEW; COEFFICIENT; EVALJAC:="FALSE"; LASTJAC:=ST "END" "ELSE" "IF" EVALCOEF "THEN" COEFFICIENT; "FOR" I:=1,I+1 "WHILE" E>ABS(ETA) "AND" DISCR>1.3*ETA "AND" I<=ITMAX "DO" "BEGIN" ITERATION(I); "IF" I>INFO[6] "THEN" INFO[6]:=I; "END"; INFO[7]:=ETA; INFO[8]:=DISCR; X:=X+H; "IF" DISCR>INFO[9] "THEN" INFO[9]:=DISCR; OUTPUT; "END"; "END" LINIGER1VS; "EOP" 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 1 AUTHOR: K.DEKKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 1973/07/16. BRIEF DESCRIPTION: LINIGER2 SOLVES INITIAL VALUE PROBLEMS , GIVEN AS AN AUTONOMOUS SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS , BY MEANS OF AN EXPONENTIALLY FITTED ONESTEP METHOD. NO AUTOMATIC STEPSIZE CONTROL IS PROVIDED. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS. KEYWORDS: DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEMS, STIFF EQUATIONS, EXPONENTIAL FITTING, IMPLICIT ONESTEP METHODS. 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 2 CALLING SEQUENCE : THE HEADING OF THE PROCEDURE LINIGER2 READS: "PROCEDURE" LINIGER2(X,XE,M,Y,SIGMA1,SIGMA2,F,EVALUATE,J, JACOBIAN,K,ITMAX,STEP,AETA,RETA,OUTPUT); "INTEGER" M,K,ITMAX; "REAL" X,XE,SIGMA1,SIGMA2,STEP,AETA,RETA; "ARRAY" Y,J; "BOOLEAN" "PROCEDURE" EVALUATE; "REAL" "PROCEDURE" F; "PROCEDURE" JACOBIAN,OUTPUT; "CODE" 33131; THE MEANING OF THE FORMAL PARAMETERS IS: X: ; THE INDEPENDENT VARIABLE X; CAN BE USED IN F, JACOBIAN, OUTPUT, ETC.; ENTRY: THE INITIAL VALUE X0; EXIT : THE FINAL VALUE XE; XE: ; THE FINAL VALUE OF X (XE>=X); M: ; THE NUMBER OF EQUATIONS; Y: ; "ARRAY" Y[1:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SYSTEM OF DIFFERENTIAL EQUATIONS: Y[I] AT X=X0; EXIT : THE FINAL VALUES OF THE SOLUTION: Y[I] AT X=XE; SIGMA1: ; THE MODULUS OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED; THIS POINT MAY BE COMPLEX OR REAL AND NEGATIVE; SIGMA2: ; SIGMA2 MAY DEFINE THREE DIFFERENT TYPES OF EXPONENTIAL FITTING: FITTING IN TWO COMPLEX CONJUGATED POINTS , FITTING IN TWO REAL NEGATIVE POINTS , OR FITTING IN ONE POINT COMBINED WITH THIRD ORDER ACCURACY; IF THIRD ORDER ACCURACY IS DESIRED , SIGMA2 SHOULD HAVE THE VALUE 0; IF FITTING IN A SECOND NEGATIVE POINT IS DESIRED , SIGMA2 SHOULD HAVE THE VALUE OF THE MODULUS OF THIS POINT; IF FITTING IN TWO COMPLEX CONJUGATED POINTS IS DESIRED , THEN SIGMA2 SHOULD BE MINUS THE VALUE OF THE ARGUMENT OF THE POINT IN THE SECOND QUADRANT (THUS A VALUE BETWEEN - PI AND - PI/2); F: ; THE HEADING OF THIS PROCEDURE READS: "REAL" "PROCEDURE" F(I); "INTEGER" I; THIS PROCEDURE SHOULD DELIVER THE RIGHT HAND SIDE OF THE I-TH DIFFERENTIAL EQUATION AS F; 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 3 EVALUATE: ; THE HEADING OF THIS PROCEDURE READS: "BOOLEAN" "PROCEDURE" EVALUATE(ITNUM); "INTEGER" ITNUM; EVALUATE SHOULD HAVE THE VALUE "TRUE", IF IT IS DESIRED THAT THE JACOBIAN OF THE SYSTEM IS UPDATED IN THE ITNUM-TH ITERATION STEP OF THE NEWTON PROCESS; J: ; "ARRAY" J[1:M,1:M]; THE JACOBIAN MATRIX OF THE SYSTEM; THE ARRAY J SHOULD BE UPDATED IN THE PROCEDURE JACOBIAN; JACOBIAN: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; IN THIS PROCEDURE THE JACOBIAN HAS TO BE ASSIGNED TO THE THE ARRAY J , OR AN APPROXIMATION OF THE JACOBIAN , IF ONLY SECOND ORDER ACCURACY IS REQUIRED; K: ; COUNTS THE NUMBER OF INTEGRATION STEPS TAKEN; FOR EXAMPLE, CAN BE USED IN EVALUATE; ITMAX: ; AN UPPERBOUND FOR THE NUMBER OF ITERATIONS IN NEWTON'S PROCESS, USED TO SOLVE THE IMPLICIT EQUATIONS; STEP: ; THE LENGTH OF THE INTEGRATION STEP, TO BE PRESCRIBED BY THE THE USER; AETA: ; REQUIRED ABSOLUTE PRECISION IN THE NEWTON PROCESS, USED TO SOLVE THE IMPLICIT EQUATIONS; RETA: ; REQUIRED RELATIVE PRECISION IN THE NEWTON PROCESS, USED TO SOLVE THE IMPLICIT EQUATIONS; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS; "PROCEDURE" OUTPUT; THIS PROCEDURE IS CALLED AT THE END OF EACH INTEGRATION STEP ; THE USER CAN ASK FOR OUTPUT OF THE PARAMETERS , FOR EXAMPLE X, K, Y. DATA AND RESULTS: SEE EXAMPLE OF USE, AND REF[3]. PROCEDURES USED: VECVEC= CP34010, MATVEC= CP34011, MATMAT= CP34013, DEC = CP34300, SOL = CP34051. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 20 + M * (4+M). 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 4 RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO SOLVE. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: LINIGER2: INTEGRATES THE SYSTEM OF DIFFERENTIAL EQUATIONS FROM X0 UNTIL XE, BY MEANS OF A SECOND ORDER FORMULA (IF SIGMA2=0 AND EVALUATE="TRUE" EVEN THIRD ORDER). SEE ALSO REF[1] AND REF[2]. REFERENCES: [1]. W.LINIGER AND R.A.WILLOUGHBY. EFFICIENT INTEGRATION METHODS FOR STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. SIAM J. NUM. ANAL. 7 (1970) PAGE 47. [2]. T.J.DEKKER, P.W.HEMKER AND P.W.VAN DER HOUWEN. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 1 (DUTCH). MC SYLLABUS 15.1, (1972) MATHEMATICAL CENTRE. [3]. P.A.BEENTJES, K.DEKKER, H.C.HEMKER, S.P.N.VAN KAMPEN AND G.M.WILLEMS. COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 2 (DUTCH). MC SYLLABUS 15.2, (1973) MATHEMATICAL CENTRE. EXAMPLE OF USE: CONSIDER THE SYSTEM OF DIFFERENTIAL EQUATIONS: DY[1]/DX = -Y[1] + Y[1] * Y[2] + .99 * Y[2] DY[2]/DX = -1000 * ( -Y[1] + Y[1] * Y[2] + Y[2] ) WITH THE INITIAL CONDITIONS AT X = 0: Y[1] = 1 AND Y[2] = 0. (SEE REF[2], PAGE 11). THE SOLUTION AT X = 50 IS APPROXIMATELY: Y[1] = .765 878 320 2487 AND Y[2] = .433 710 353 5768. THE FOLLOWING PROGRAM SHOWS SOME DIFFERENT CALLS OF THE PROCEDURE LINIGER2, AND ILLUSTRATES THE ACCURACY WHICH MAY BE OBTAINED BY IT: "BEGIN" "INTEGER" K,ITMAX,PASSES,PASJAC; "REAL" X,SIGMA,STEP,TIME; "REAL" "ARRAY" Y[1:2],J[1:2,1:2]; "REAL" "PROCEDURE" F(I); "INTEGER" I; "IF" I=1 "THEN" F:=(Y[1]+.99)*(Y[2]-1)+.99 "ELSE" "BEGIN" PASSES:=PASSES+1; F:=1000*((1+Y[1])*(1-Y[2])-1) "END"; 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 5 "PROCEDURE" JACOBIAN(J,Y); "ARRAY" J,Y; "BEGIN" J[1,1]:=Y[2]-1; J[1,2]:=.99+Y[1]; J[2,1]:=1000*(1-Y[2]); J[2,2]:=-1000*(1+Y[1]); SIGMA:=ABS(J[2,2]+J[1,1]-SQRT((J[2,2]-J[1,1])**2+ 4*J[2,1]*J[1,2]))/2; PASJAC:=PASJAC+1 "END" JACOBIAN; "BOOLEAN" "PROCEDURE" EVALUATE1(I); "INTEGER" I; EVALUATE1:= I=1; "BOOLEAN" "PROCEDURE" EVALUATE2(I); "INTEGER" I; EVALUATE2:= "TRUE"; "PROCEDURE" OUT; "IF" X=50 "THEN" OUTPUT(61,"("3(-4ZDB),2(4B+.3DB3DB3D),-5ZD.3D,/")",K,PASSES, PASJAC,Y[1],Y[2],CLOCK-TIME); OUTPUT(61,"(""(" THIS LINE AND THE FOLLOWING TEXT IS ")" "("PRINTED BY THIS PROGRAM")",//, "(" THE RESULTS WITH LINIGER2 -SECOND ORDER- ARE:")",/, "(" K DER.EV. JAC.EV. Y[1] Y[2]")" "(" TIME")",/")"); "FOR" STEP:=10,1 "DO" "FOR" ITMAX:=1,3 "DO" "BEGIN" PASSES:=PASJAC:=0; X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; LINIGER2(X,50.0,2,Y,SIGMA,0.0,F,EVALUATE1,J,JACOBIAN,K,ITMAX, STEP,"-4,"-4,OUT); "END"; OUTPUT(61,"("//, "(" THE RESULTS WITH LINIGER2 -THIRD ORDER- ARE:")",/, "(" K DER.EV. JAC.EV. Y[1] Y[2]")" "(" TIME")",/")"); "FOR" STEP:=10,1 "DO" "FOR" ITMAX:=1,3 "DO" "BEGIN" PASSES:=PASJAC:=0; X:=Y[2]:=0; Y[1]:=1; TIME:=CLOCK; LINIGER2(X,50.0,2,Y,SIGMA,0.0,F,EVALUATE2,J,JACOBIAN,K,ITMAX, STEP,"-4,"-4,OUT); "END"; "END" THIS LINE AND THE FOLLOWING TEXT IS PRINTED BY THIS PROGRAM: THE RESULTS WITH LINIGER2 -SECOND ORDER- ARE: K DER.EV. JAC.EV. Y[1] Y[2] TIME 5 5 5 +.766 392 210 +.434 218 863 0.092 5 15 5 +.765 755 853 +.433 671 223 0.175 50 50 50 +.765 884 310 +.433 715 687 0.949 50 101 50 +.765 877 388 +.433 710 059 1.494 THE RESULTS WITH LINIGER2 -THIRD ORDER- ARE: K DER.EV. JAC.EV. Y[1] Y[2] TIME 5 5 5 +.766 392 210 +.434 218 863 0.092 5 15 15 +.765 882 250 +.433 711 614 0.300 50 50 50 +.765 884 310 +.433 715 687 0.949 50 101 101 +.765 878 873 +.433 710 531 2.080 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 6 SOURCE TEXT(S): 0"CODE" 33131; "PROCEDURE" LINIGER2(X,XE,M,Y,SIGMA1,SIGMA2,F,EVALUATE,J, JACOBIAN,K,ITMAX,STEP,AETA,RETA,OUTPUT); "INTEGER" M,K,ITMAX; "REAL" X,XE,SIGMA1,SIGMA2,STEP,AETA,RETA; "ARRAY" Y,J; "BOOLEAN" "PROCEDURE" EVALUATE; "REAL" "PROCEDURE" F; "PROCEDURE" JACOBIAN,OUTPUT; "BEGIN" "INTEGER" I; "REAL" H,HL,B1,B2,P,Q,C0,C1,C2,C3,C4; "BOOLEAN" LAST; "INTEGER" "ARRAY" PI[1:M]; "REAL" "ARRAY" DY,YL,FL[1:M],A[1:M,1:M],AUX[1:3]; "PROCEDURE" STEPSIZE; "BEGIN" H:=STEP; "IF" 1.1*H>=XE-X "THEN" "BEGIN" LAST:="TRUE"; H:=XE-X; X:=XE "END" "ELSE" X:=X+H "END" STEPSIZE; "PROCEDURE" COEFFICIENT; "BEGIN" "REAL" R1,R2,EX,ZETA,ETA,SINL,COSL,SINH,COSH,D; "REAL" "PROCEDURE" R(X); "VALUE" X; "REAL" X; "IF" X>40 "THEN" R:=X/(X-2) "ELSE" "BEGIN" EX:=EXP(-X); R:=X*(1-EX)/(X-2+(X+2)*EX) "END"; B1:=H*SIGMA1; B2:=H*SIGMA2; "IF" B1<.1 "THEN" "BEGIN" P:=0; Q:=1/3; "GOTO" OUT "END"; "IF" B2<0 "THEN" "GOTO" COMPLEX; "IF" B1<1 "OR" B2<.1 "THEN" "GOTO" THIRDORDER; "IF" ABS(B1-B2)40 "THEN" EX:=0; R2:=B1/(1-EX); R2:=1-EX*R2*R2; Q:=1/(R1*R1*R2); P:=R1*Q-2/B1; "GOTO" OUT; COMPLEX: ETA:=ABS(B1*SIN(SIGMA2)); ZETA:=ABS(B1*COS(SIGMA2)); "IF" ETA40 "THEN" "BEGIN" P:=1-4*ZETA/B1/B1; Q:=4*(1-ZETA)/B1/B1+1 "END" "ELSE" "BEGIN" EX:=EXP(ZETA); SINL:=SIN(ETA); COSL:=COS(ETA); SINH:=.5*(EX-1/EX); COSH:=.5*(EX+1/EX); D:=ETA*(COSH-COSL)-.5*B1*B1*SINL; P:=(ZETA*SINL+ETA*SINH-4*ZETA*ETA/B1/B1*(COSH-COSL))/D; Q:=ETA*((COSH-COSL-ZETA*SINH-ETA*SINL)*4/B1/B1+COSH+COSL)/D "END"; OUT: C0:=.25*H*H*(P+Q); C1:=.5*H*(1+P); C2:=H-C1; C3:=.25*H*H*(Q-P); C4:=.5*H*P; ELEMENTS OF MATRIX "END" COEFFICIENT; "PROCEDURE" ELEMENTS OF MATRIX; "BEGIN" "INTEGER" K; "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" "FOR" K:=1 "STEP" 1 "UNTIL" M "DO" A[I,K]:=C0*MATMAT(1,M,I,K,J,J)-C1*J[I,K]; A[I,I]:=A[I,I]+1 "END"; AUX[2]:=0; DEC(A,M,AUX,PI) "END" ELOFMAT; "COMMENT" 1SECTION : 5.2.1.1.1.2.D (AUGUST 1974) PAGE 8 ; "PROCEDURE" NEWTON ITERATION; "BEGIN" "INTEGER" ITNUM; "REAL" JFL,ETA,DISCR; ITNUM:=0; NEXT: ITNUM:=ITNUM+1; "IF" EVALUATE(ITNUM) "THEN" "BEGIN" JACOBIAN(J,Y); COEFFICIENT "END" "ELSE" "IF" ITNUM=1 "AND" H^=HL "THEN" COEFFICIENT; "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" FL[I]:=F(I); "IF" ITNUM=1 "THEN" "BEGIN" "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" "BEGIN" JFL:=MATVEC(1,M,I,J,FL); DY[I]:=H*(FL[I]-C4*JFL); YL[I]:=Y[I]+C2*FL[I]+C3*JFL "END" "END" "ELSE" "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" DY[I]:=YL[I]-Y[I]+C1*FL[I]-C0*MATVEC(1,M,I,J,FL); SOL(A,M,PI,DY); "FOR" I:=1 "STEP" 1 "UNTIL" M "DO" Y[I]:=Y[I]+DY[I]; "IF" ITNUM; THE INDEPENDENT VARIABLE X; ENTRY: THE INITIAL VALUE OF X; EXIT : THE END VALUE OF X; XE: ; ENTRY: THE END VALUE OF X; R: ; ENTRY: THE NUMBER OF DIFFERENTIAL EQUATIONS; 1SECTION : 5.2.1.1.1.2.E (MARCH 1977) PAGE 2 Y: ; "ARRAY" Y[1:R]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUE OF Y; EXIT : THE SOLUTION Y AT THE POINT X AFTER EACH INTEGRATION STEP; H: ; ENTRY: THE STEPLENGTH WHEN THE INTEGRATION HAS TO BE PERFORMED WITHOUT THE STEPSIZE MECHANISM, OTHER- WISE THE INITIAL STEPLENGTH (SEE THE PARAMETERS HMIN AND HMAX); HMIN, HMAX: ; ENTRY: MINIMAL AND MAXIMAL STEPLENGTH BY WHICH THE INTE- GRATION IS ALLOWED TO BE PERFORMED; BY PUTTING HMIN = HMAX THE STEPSIZE MECHANISM IS ELIMINATED; IN THIS CASE THE GIVEN VALUES FOR HMIN AND HMAX ARE IRRELEVANT, WHILE THE INTEGRATION IS PERFORMED WITH THE STEPLENGTH GIVEN BY H; DELTA: ; ENTRY: THE REAL PART OF THE POINT AT WHICH EXPONENTIAL FITTING IS DESIRED; (SEE METHOD AND PERFORMANCE); ALTERNATIVES: DELTA = (AN ESTIMATE OF) THE REAL PART OF THE LARGEST EIGENVALUE IN MODULUS OF THE JACOBIAN MATRIX OF THE SYSTEM; DELTA <= -10**15, IN ORDER TO OBTAIN ASYMPTOTIC STABILITY; DELTA = 0, IN ORDER TO OBTAIN A HIGHER ORDER OF ACCURACY IN CASE OF LINEAR EQUATIONS; DERIVATIVE: ; "PROCEDURE" DERIVATIVE(Y); "ARRAY" Y; ; JACOBIAN: ; "PROCEDURE" JACOBIAN(J, Y); "ARRAY" J, Y; WHEN IN GMS JACOBIAN IS CALLED THE ARRAY Y CONTAINS THE VALUES OF THE DEPENDENT VARIABLE; UPON COMPLETION OF A CALL OF JACOBIAN THE ARRAY J SHOULD CONTAIN THE VALUES OF THE JACOBIAN MATRIX OF F(Y); AETA, RETA: ; ENTRY: MEASURE OF THE ABSOLUTE AND RELATIVE LOCAL ACCURACY REQUIRED; THESE VALUES ARE IRRELEVANT WHEN THE INTEGRATION IS PER- FORMED WITHOUT THE STEPSIZE MECHANISM; N: ; EXIT : THE NUMBER OF INTEGRATION STEPS; JEV: ; EXIT: THE NUMBER OF JACOBIAN EVALUATIONS; LU: ; EXIT: THE NUMBER OF LU-DECOMPOSITIONS; 1SECTION : 5.2.1.1.1.2.E (OCTOBER 1974) PAGE 3 NSJEV: ; ENTRY: NUMBER OF INTEGRATION STEPS PER JACOBIAN EVALUATION; THE VALUE OF NSJEV IS RELEVANT ONLY WHEN THE INTEGRATION IS PERFORMED WITHOUT THE STEPSIZE MECHANISM AND THE SYSTEM TO BE SOLVED IS NON-LINEAR; LINEAR: ; ENTRY: TRUE WHEN THE SYSTEM TO BE INTEGRATED IS LINEAR, OTHERWISE FALSE; IF LINEAR IS TRUE THE STEPSIZE MECHANISM IS AUTOMATICALLY ELIMINATED; OUT: ; "PROCEDURE" OUT; ; DATA AND RESULTS: SEE REF[2]. PROCEDURES USED: VECVEC = CP34010, MATVEC = CP34011, MATMAT = CP34013, ELMROW = CP34024, ELMVEC = CP34020, DUPVEC = CP31030, GSSELM = CP34231, SOLELM = CP34061, COLCST = CP31131, MULVEC = CP31020. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: 8 * R + 3 * R * R; RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO BE SOLVED. LANGUAGE: ALGOL 60. 1SECTION : 5.2.1.1.1.2.E (MARCH 1977) PAGE 4 METHOD AND PERFORMANCE: THE PROCEDURE GMS DESCRIBES AN IMPLEMENTATION OF A THIRD ORDER THREE-STEP GENERALIZED LINEAR MULTISTEP METHOD WITH QUASI-ZERO PARASITIC ROOTS AND QUASI-ADAPTIVE STABILITY FUNCTION. IN PARTI- CULAR THE ALGORITHM IS DEVELOPED FOR THE INTEGRATION OF STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS. THE PROCEDURE SUPPLIES THE ADDITIONAL STARTING VALUES AND PERFORMS A STEPSIZE CONTROL WHICH IS BASED ON THE NON-LINEARITY OF THE DIFFERENTIAL EQUATION. BY THIS CONTROL THE JACOBIAN MATRIX IS INCIDENTALLY EVALUATED. IT IS POSSIBLE TO ELIMINATE THE STEPSIZE CONTROL. THEN, ONE HAS TO GIVE THE NUMBER OF INTEGRATION STEPS PER JACOBIAN EVALUATION. FOR LINEAR EQUATIONS THE STEPSIZE CONTROL IS AUTOMATICALLY ELIMIN- ATED, WHILE THE PROCEDURE PERFORMS ONE EVALUATION OF THE JACOBIAN. MOREOVER, IN THIS CASE THE THREE-STEP SCHEME IS REDUCED TO A ONE- STEP SCHEME. THE PROCEDURE USES ONE FUNCTION EVALUATION PER INTE- GRATION STEP AND IT DOES NOT REJECT INTEGRATION STEPS. EACH CHANGE IN THE STEPLENGTH OR EACH REEVALUATION OF THE JACOBIAN COSTS ONE LU-DECOMPOSITION. IT IS POSSIBLE TO FIT EXPONENTIALLY, THIS FIT- TING IS EQUIVALENT TO FITTING IN THE SENSE OF LINIGER AND WILLOUGHBY, ONLY WHEN THE JACOBIAN MATRIX IS EVALUATED AT EACH IN- TEGRATION STEP. WHEN THE SYSTEM TO BE INTEGRATED IS NON-LINEAR AND THE JACOBIAN MATRIX IS NOT EVALUATED AT EACH INTEGRATION STEP, IT IS RECOMMENDED TO FIT AT INFINITY (DELTA <= -10**15). DETAILS ARE GIVEN IN REFERENCE 2. REFERENCES: [1] HOUWEN, P. J. VAN DER AND VERWER, J. G., GENERALIZED LINEAR MULTISTEP METHODS 1, DEVELOPMENT OF ALGO- RITHMS WITH ZERO-PARASITIC ROOTS, REPORT NW 10/74, MATHEMATISCH CENTRUM, AMSTERDAM 1974. [2] VERWER, J. G., GENERALIZED LINEAR MULTISTEP METHODS 2, NUMERICAL APPLICATIONS, REPORT NW 12/74, MATHEMATISCH CENTRUM, AMSTERDAM, 1974. EXAMPLE OF USE: WE CONSIDER THE DIFFERENTIAL EQUATION DY1/DX = -1000 * Y1 * (Y1 + Y2 -1.999987), DY2/DX = -2500 * Y2 * (Y1 + Y2 - 2), ON THE INTERVAL [0,50], WITH INITIAL VALUE Y1(0) = Y2(0) = 1. THE REFERENCE SOLUTION AT X = 50 IS GIVEN BY: Y1(50) = .5976546988, Y2(50) = 1.4023434075. 1SECTION : 5.2.1.1.1.2.E (MARCH 1977) PAGE 5 "BEGIN" "PROCEDURE" DER(Y); "ARRAY" Y; "BEGIN" "REAL" Y1, Y2; Y1:= Y[1]; Y2:= Y[2]; Y[1]:= -1000 * Y1 * (Y1 + Y2 - 1.999987); Y[2]:= -2500 * Y2 * (Y1 + Y2 - 2) "END" DER; "PROCEDURE" JAC(J, Y); "ARRAY" J, Y; "BEGIN" "REAL" Y1, Y2; Y1:= Y[1]; Y2:= Y[2]; J[1,1]:= 1999.987 - 1000 * (2 * Y1 + Y2); J[1,2]:= -1000 * Y1; J[2,1]:= -2500 * Y2; J[2,2]:= 2500 * (2 - Y1 - 2 * Y2) "END" JAC; "PROCEDURE" OUTP; "IF" X = 50 "THEN" "BEGIN" "REAL" YE1, YE2; YE1:= .5976546988; YE2:= 1.4023434075; OUTPUT(61, "(" "("X = ")", 2D2B, "("N = ")", 3ZD2B, "("JEV = ")", 3ZD2B, "("LU = ")", 3ZD, 2/, "("Y1 = ")", +.13D"+2D2B, "("REL. ERR. = ")", .2D"+2D, /, "("Y2 = ")", +.13D"+2D2B, "("REL. ERR. = ")", .2D"+2D")", X, N, JEV, LU, Y[1], ABS((Y[1] - YE1) / YE1), Y[2], ABS((Y[2] - YE2) / YE2)) "END" OUTP; "INTEGER" N, JEV, LU; "REAL" X; "ARRAY" Y[1:2]; X:= 0.0; Y[1]:= Y[2]:= 1.0; GMS(X, 50.0, 2, Y, .01, .001, .5, -"15, DER, JAC, "-5, "-5, N, JEV, LU, 0, "FALSE", OUTP) "END" THIS PROGRAM DELIVERS: X = 50 N = 109 JEV = 3 LU = 12 Y1 = +.5976547958004"+00 REL. ERR. = .16"-06 Y2 = +.1402343310813"+01 REL. ERR. = .69"-07 1SECTION : 5.2.1.1.1.2.E (MARCH 1977) PAGE 6 SOURCE TEXT: "CODE" 33191; "PROCEDURE" GMS(X, XE, R, Y, H, HMIN, HMAX, DELTA, DERIVATIVE, JACOBIAN, AETA, RETA, N, JEV, LU, NSJEV, LINEAR, OUT); "VALUE" R; "REAL" X, XE, H, HMIN, HMAX, DELTA, AETA, RETA; "INTEGER" R, N, JEV, NSJEV, LU; "BOOLEAN" LINEAR; "ARRAY" Y; "PROCEDURE" DERIVATIVE, JACOBIAN, OUT; "BEGIN" "INTEGER" I, J, K, L, NSJEV1, COUNT, COUNT1, KCHANGE; "REAL" A, A1, ALFA, E, S1, S2, Z1, X0, XL0, XL1, XL2, ETA, H0, H1, Q, Q1, Q2, Q12, Q22, Q1Q2, DISCR; "BOOLEAN" UPDATE, CHANGE, REEVAL, STRATEGY; "INTEGER" "ARRAY" RI, CI[1:R]; "ARRAY" AUX[1:9], BD1, BD2[1:3,1:3], Y1, Y0[1:R], HJAC, H2JAC2, RQZ[1:R,1:R], YL, FL[1:3 * R]; "PROCEDURE" INITIALIZATION; "BEGIN" LU:= JEV:= N:= NSJEV1:= KCHANGE:= 0; X0:= X; DISCR:= 0; K:=1; H1:= H0:= H; COUNT:= -2; AUX[2]:= "-14; AUX[4]:= 8; DUPVEC(1, R, 0, YL, Y); REEVAL:= CHANGE:= "TRUE"; STRATEGY:= HMIN ^= HMAX "AND" ^LINEAR; Q1:= -1; Q2:= -2; COUNT1:= 0; XL0:= XL1:= XL2:= 0 "END" INITIALIZATION; "COMMENT" 1SECTION : 5.2.1.1.1.2.E (OCTOBER 1974) PAGE 7 ; "PROCEDURE" COEFFICIENT; "BEGIN" XL2:= XL1; XL1:= XL0; XL0:= X0; "IF" CHANGE "THEN" "BEGIN" "IF" N > 2 "THEN" "BEGIN" Q1:= (XL1 - XL0) / H1; Q2:= (XL2 - XL0) / H1 "END"; Q12:= Q1 * Q1; Q22:= Q2 * Q2; Q1Q2:= Q1 * Q2; A:= -(3 * ALFA + 1) / 12; BD1[1,3]:= 1 + (1 / 3 - (Q1 + Q2) * .5) / Q1Q2; BD1[2,3]:= (1 / 3 - Q2 * .5) / (Q12 - Q1Q2); BD1[3,3]:= (1 / 3 - Q1 * .5) / (Q22 - Q1Q2); BD2[1,3]:= -ALFA * .5 + A * (1 - Q1 - Q2) / Q1Q2; BD2[2,3]:= A * (1 - Q2) / (Q12 - Q1Q2); BD2[3,3]:= A * (1 - Q1) / (Q22 - Q1Q2); "IF" STRATEGY "OR" N <= 2 "THEN" "BEGIN" BD1[2,2]:= 1 / (2 * Q1); BD1[1,2]:= 1 - BD1[2,2]; BD2[2,2]:= -(3 * ALFA + 1) / (12 * Q1); BD2[1,2]:= -BD2[2,2] - ALFA * .5 "END" "END" "END" COEFFICIENT; "PROCEDURE" OPERATOR CONSTRUCTION; "BEGIN" "IF" REEVAL "THEN" "BEGIN" JACOBIAN(HJAC, Y); JEV:= JEV + 1; NSJEV1:= 0; "IF" DELTA <= -"15 "THEN" ALFA:= 1 / 3 "ELSE" "BEGIN" Z1:= H1 * DELTA; A:= Z1 * Z1 + 12; A1:= 6 * Z1; "IF" ABS(Z1) < .1 "THEN" ALFA:= (Z1 * Z1 / 140 - 1) * Z1 / 30 "ELSE" "IF" Z1 < -33 "THEN" ALFA:= (A + A1) / (3 * Z1 * (2 + Z1)) "ELSE" "BEGIN" E:= EXP(Z1); ALFA:= ((A - A1) * E - A - A1) / (((2 - Z1) * E - 2 - Z1) * Z1 * 3) "END" "END"; S1:= -(1 + ALFA) * .5; S2:= (ALFA * 3 + 1) / 12 "END"; "COMMENT" 1SECTION : 5.2.1.1.1.2.E (OCTOBER 1974) PAGE 8 ; A:= H1 / H0; A1:= A * A; "IF" REEVAL "THEN" A:= H1; "IF" A ^= 1 "THEN" "FOR" J:= 1 "STEP" 1 "UNTIL" R "DO" COLCST(1, R, J, HJAC, A); "FOR" I:= 1 "STEP" 1 "UNTIL" R "DO" "BEGIN" "FOR" J:= 1 "STEP" 1 "UNTIL" R "DO" "BEGIN" Q:= H2JAC2[I,J]:= "IF" REEVAL "THEN" MATMAT(1, R, I, J, HJAC, HJAC) "ELSE" H2JAC2[I,J] * A1; RQZ[I,J]:= S2 * Q "END"; RQZ[I,I]:= RQZ[I,I] + 1; ELMROW(1, R, I, I, RQZ, HJAC, S1) "END"; GSSELM(RQZ, R, AUX, RI, CI); LU:= LU + 1; REEVAL:= UPDATE:= "FALSE" "END" OPERATOR CONSTRUCTION; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" "IF" COUNT ^= 1 "THEN" "BEGIN" DUPVEC(1, R, 0, FL, YL); DERIVATIVE(FL); N:= N + 1; NSJEV1:= NSJEV1 + 1 "END"; MULVEC(1, R, 0, Y0, YL, (1 - ALFA) / 2 - BD1[1,K]); "FOR" L:= 2 "STEP" 1 "UNTIL" K "DO" ELMVEC(1, R, R * (L - 1), Y0, YL, -BD1[L,K]); "FOR" L:= 1 "STEP" 1 "UNTIL" K "DO" ELMVEC(1, R, R * (L - 1), Y0, FL, H1 * BD2[L,K]); "FOR" I:= 1 "STEP" 1 "UNTIL" R "DO" Y[I]:= MATVEC(1, R, I, HJAC, Y0); MULVEC(1, R, 0, Y0, YL, (1 - 3 * ALFA) / 12 - BD2[1,K]); "FOR" L:= 2 "STEP" 1 "UNTIL" K "DO" ELMVEC(1, R, R * (L - 1), Y0, YL, -BD2[L,K]); "FOR" I:= 1 "STEP" 1 "UNTIL" R "DO" Y[I]:= Y[I] + MATVEC(1, R, I, H2JAC2, Y0); DUPVEC(1, R, 0, Y0, YL); "FOR" L:= 1 "STEP" 1 "UNTIL" K "DO" ELMVEC(1, R, R * (L - 1), Y0, FL, H1 * BD1[L,K]); ELMVEC(1, R, 0, Y, Y0, 1); SOLELM(RQZ, R, RI, CI, Y) "END" DIFFERENCE SCHEME; "PROCEDURE" NEXT INTEGRATION STEP; "BEGIN" "FOR" L:= 2, 1 "DO" "BEGIN" DUPVEC(L * R + 1, (L + 1) * R, -R, YL, YL); DUPVEC(L * R + 1, (L + 1) * R, -R, FL, FL) "END"; DUPVEC(1, R, 0, YL, Y) "END" NEXT INTEGRATION STEP; "COMMENT" 1SECTION : 5.2.1.1.1.2.E (OCTOBER 1974) PAGE 9 ; "PROCEDURE" TEST ACCURACY; "BEGIN" K:= 2; DUPVEC(1, R, 0, Y1, Y); DIFFERENCE SCHEME; K:= 3; ETA:= AETA + RETA * SQRT(VECVEC(1, R, 0, Y1, Y1)); ELMVEC(1, R, 0, Y, Y1, -1); DISCR:= SQRT(VECVEC(1, R, 0, Y, Y)); DUPVEC(1, R, 0, Y, Y1) "END" TEST ACCURACY; "PROCEDURE" STEPSIZE; "BEGIN" X0:= X; H0:= H1; "IF" N <= 2 "AND" ^LINEAR "THEN" K:= K + 1; "IF" COUNT = 1 "THEN" "BEGIN" A:= ETA / (.75 * (ETA + DISCR)) + .33; H1:= "IF" A <= .9 "OR" A >= 1.1 "THEN" A * H0 "ELSE" H0; COUNT:= 0; REEVAL:= A <= .9 "AND" NSJEV1 ^= 1; COUNT1:= "IF" A >= 1 "OR" REEVAL "THEN" 0 "ELSE" COUNT1 + 1; "IF" COUNT1 = 10 "THEN" "BEGIN" COUNT1:= 0; REEVAL:= "TRUE"; H1:= A * H0 "END" "END" "ELSE" "BEGIN" H1:= H; REEVAL:= NSJEV = NSJEV1 "AND" ^STRATEGY "AND" ^LINEAR "END"; "IF" STRATEGY "THEN" H1:= "IF" H1 > HMAX "THEN" HMAX "ELSE" "IF" H1 < HMIN "THEN" HMIN "ELSE" H1; X:= X + H1; "IF" X >= XE "THEN" "BEGIN" H1:= XE - X0; X:= XE "END"; "IF" N <= 2 "AND" ^LINEAR "THEN" REEVAL:= "TRUE"; "IF" H1 ^= H0 "THEN" "BEGIN" UPDATE:= "TRUE"; KCHANGE:= 3 "END"; "IF" REEVAL "THEN" UPDATE:= "TRUE"; CHANGE:= KCHANGE > 0 "AND" ^LINEAR; KCHANGE:= KCHANGE - 1 "END" STEPSIZE; INITIALIZATION; OUT; X:= X + H1; OPERATOR CONSTRUCTION; BD1[1,1]:= 1; BD2[1,1]:= -ALFA * .5; "IF" ^LINEAR "THEN" COEFFICIENT; NEXT STEP: DIFFERENCE SCHEME; "IF" STRATEGY "THEN" COUNT:= COUNT + 1; "IF" COUNT = 1 "THEN" TEST ACCURACY; OUT; "IF" X >= XE "THEN" "GOTO" END; STEPSIZE; "IF" UPDATE "THEN" OPERATOR CONSTRUCTION; "IF" ^LINEAR "THEN" COEFFICIENT; NEXT INTEGRATION STEP; "GOTO" NEXT STEP; END: "END" GMS; "EOP" 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 1 AUTHOR: B.LINDBERG. CONTRIBUTOR: K.DEKKER. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 741101. BRIEF DESCRIPTION: IMPEX SOLVES AN INITIAL VALUE PROBLEM,GIVEN AS AN AUTONOMOUS SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS , BY MEANS OF THE IMPLICIT MIDPOINT RULE WITH SMOOTHING AND EXTRAPOLATION. AUTOMATIC STEPSIZE CONTROL IS PROVIDED. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS. KEYWORDS: DIFFERENTIAL EQUATIONS, INITIAL VALUE PROBLEMS, STIFF EQUATIONS, IMPLICIT MIDPOINT RULE, SMOOTHING, EXTRAPOLATION. CALLING SEQUENCE: THE HEADING OF THE PROCEDURE IMPEX READS: "PROCEDURE" IMPEX (N, T0, TEND, Y0, DERIV, AVAILABLE, H0, HMAX, PRESCH, EPS, WEIGHTS, UPDATE, FAIL, CONTROL); "VALUE" N; "INTEGER" N; "REAL" T0,TEND,H0,HMAX,EPS; "BOOLEAN" PRESCH,FAIL; "ARRAY" Y0,WEIGHTS; "BOOLEAN" "PROCEDURE" AVAILABLE; "PROCEDURE" DERIV,UPDATE,CONTROL; "CODE" 33135; IMPEX: INTEGRATES THE SYSTEM OF DIFFERENTIAL EQUATIONS , GIVEN AS DY/DT=F(T,Y), FROM T0 UNTIL TEND. THE MEANING OF THE FORMAL PARAMETERS IS: N: ; THE NUMBER OF EQUATIONS; T0: ; THE INITIAL VALUE OF THE INDEPENDENT VARIABLE; 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 2 TEND: ; THE FINAL VALUE OF THE INDEPENDENT VARIABLE; Y0: ; "REAL" "ARRAY" Y0[1:N]; ENTRY: THE INITIAL VALUES OF THE SYSTEM OF DIFFERENTIAL EQUATIONS: Y0[I] AT T=T0; DERIV: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIV(T,Y,F,N); "INTEGER" N; "REAL" T; "ARRAY" Y,F; THIS PROCEDURE SHOULD DELIVER THE VALUE OF F(T,Y) IN THE ARRAY F[1:N]; AVAILABLE: ; THE HEADING OF THIS PROCEDURE READS: "BOOLEAN" "PROCEDURE" AVAILABLE(T,Y,A,N); "INTEGER" N; "REAL" T; "ARRAY" Y,A; IF AN ANALYTIC EXPRESSION OF THE JACOBIAN MATRIX AT THE POINT (T,Y) IS NOT AVAILABLE, THIS PROCEDURE SHOULD DELIVER THE VALUE "FALSE"; OTHERWISE THE PROCEDURE SHOULD DELIVER THE VALUE "TRUE", AND THE JACOBIAN MATRIX SHOULD BE ASSIGNED TO THE ARRAY A[1:N,1:N]; H0: ; THE INITIAL STEPSIZE; HMAX: ; MAXIMAL STEPSIZE BY WHICH THE INTEGRATION IS PERFORMED; PRESCH: ; INDICATOR FOR CHOICE OF STEPSIZE; THE STEPSIZE IS AUTOMATICALLY CONTROLLED IF PRESCH="FALSE"; OTHERWISE THE STEPSIZE HAS TO BE PRESCRIBED , EITHER ONLY INITIALLY OR ALSO BY THE PROCEDURE CONTROL; EPS: ; BOUND FOR THE ESTIMATE OF THE LOCAL ERROR; WEIGHTS: ; "REAL" "ARRAY" WEIGHTS[1:N]; WEIGHTS FOR THE COMPUTATION OF THE WEIGHTED EUCLIDEAN NORM OF THE ERRORS; ENTRY: INITIAL WEIGHTS; NOTE THAT THE CHOICE WEIGHTS[I] = 1 IMPLIES AN ESTIMATION OF THE ABSOLUTE ERROR , WHEREAS WEIGTHS[I] = Y[I] DEFINES A RELATIVE ERROR; UPDATE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" UPDATE(WEIGHTS,Y2,N); "INTEGER" N; "ARRAY" WEIGHTS,Y2; THIS PROCEDURE MAY CHANGE THE ARRAY WEIGHTS , ACCORDING TO THE VALUE OF AN APPROXIMATION FOR Y(T) , GIVEN IN THE ARRAY Y2[1:N]; 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 3 FAIL: ; EXIT : IF THE PROCEDURE FAILS TO SOLVE THE SYSTEM OF EQUATIONS, FAIL WILL HAVE THE VALUE "TRUE" UPON EXIT; THIS MAY OCCUR UPON DIVERGENCE OF THE ITERATION PROCESS, USED IN THE MIDPOINT RULE , WHILE INTEGRATION IS PERFORMED WITH A USER DEFINED PRESCRIBED STEPSIZE; CONTROL: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" CONTROL(TPRINT,T,H,HNEW,Y,ERROR,N); "INTEGER" N; "REAL" TPRINT,T,H,HNEW; "ARRAY" Y,ERROR; CONTROL IS CALLED ON ENTRY OF IMPEX, AND FURTHER AS SOON AS THE INEQUALITY TPRINT <= T HOLDS; DURING A CALL OF CONTROL PRINTING OF RESULTS AND CHANGE OF STEPSIZE (IF PRESCH = "TRUE") IS THEN POSSIBLE; THE MEANING OF THE FORMAL PARAMETERS IS: TPRINT: ; ENTRY: THE VALUE OF THE INDEPENDENT VARIABLE AT WHICH A CALL OF CONTROL WAS DESIRED; EXIT: A NEW VALUE (TPRINT>T) AT WHICH A CALL OF CONTROL IS DESIRED; T: ; THE ACTUAL VALUE OF THE INDEPENDENT VARIABLE, UP TO WHICH INTEGRATION HAS BEEN PERFORMED; H: ; HALVE THE ACTUAL STEPSIZE; HNEW: ; THE NEW STEPSIZE; IF PRESCH="TRUE", THEN THE USER MAY PRESCRIBE A NEW STEPSIZE BY CHANGING HNEW; Y: ; "REAL" "ARRAY" Y[1:5,1:N]; THE VALUE OF THE DEPENDENT VARIABLE AND ITS FIRST FOUR DIVIDED DIFFERENCES AT THE POINT T ARE GIVEN IN THIS ARRAY; ERROR: ; "REAL" "ARRAY" ERROR[1:3]; THE ELEMENTS OF THIS ARRAY CONTAIN THE FOLLOWING ERRORS: ERROR[1]: THE LOCAL ERROR; ERROR[2]: THE GLOBAL ERROR OF SECOND ORDER IN H; ERROR[3]: THE GLOBAL ERROR OF FOURTH ORDER IN H; N: ; THE NUMBER OF EQUATIONS; EXAMPLE OF USE: SEE EXAMPLE OF USE OF THE PROCEDURE IMPEX; DATA AND RESULTS: FOR DATA, SEE REF[1]. THE RESULTS OF THE INTEGRATION ARE ATTAINABLE THROUGH THE PROCEDURE CONTROL , WHICH IS CALLED AT SPECIFIED, USER DEFINED, VALUES OF THE INDEPENDENT VARIABLE . IN PARTICULAR , THE VALUES OF THE DEPENDENT VARIABLE AT THE ENDPOINT OF INTEGRATION ARE OBTAINED BY A CALL OF CONTROL WITH TPRINT=TEND. 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 4 PROCEDURES USED: INIVEC = CP31010, INIMAT = CP31011, MULVEC = CP31020, MULROW = CP31021, DUPVEC = CP31030, DUPROWVEC= CP31032, DUPMAT = CP31035, VECVEC = CP34010, MATVEC = CP34011, MATMAT = CP34013, ELMVEC = CP34020, ELMROW = CP34024, DEC = CP34300, SOL = CP34051. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA N * ( 23 + 2 * N ) (DECIMAL). RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO SOLVE. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: THE INTEGRATION METHOD (REF[1]) IS BASED ON THE COMPUTATION OF TWO INDEPENDENT SOLUTIONS Y(T,H) AND Y(T,H/2) BY THE IMPLICIT MIDPOINT RULE. PASSIVE SMOOTHING AND PASSIVE EXTRAPOLATION IS PERFORMED TO OBTAIN STABILITY AND HIGH ACCURACY . THE ALGORITHM USES FOR EACH STEP AT LEAST THREE FUNCTION EVALUATIONS, AND ON CHANGE OF STEPSIZE OR AT SLOW CONVERGENCE IN THE ITERATION PROCESS AN APPROXIMATION OF THE JACOBIAN MATRIX (COMPUTED BY DIVIDED DIFFERENCES OR EXPLICITLY SPECIFIED BY THE USER) . IF THE COMPUTED LOCAL ERROR EXCEEDS THE TOLERANCE , THE LAST STEP IS REJECTED. MOREOVER , TWO GLOBAL ERRORS ARE COMPUTED. REFERENCES: [1]. B.LINDBERG. IMPEX 2 , A PROCEDURE FOR THE SOLUTION OF SYSTEMS OF STIFF DIFFERENTIAL EQUATIONS. ROYAL INSTITUTE OF TECHNOLOGY,STOCKHOLM. TRITA-NA-7303 (1973). [2]. (TO APPEAR) COLLOQUIUM STIFF DIFFERENTIAL EQUATIONS 3 (DUTCH). M.C. SYLLABUS 15.3 (1974), MATHEMATICAL CENTRE. 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 5 EXAMPLE OF USE: CONSIDER THE AUTONOMOUS SYSTEM OF DIFFERENTIAL EQUATIONS: DY[1]/DX = .2 * ( Y[2] - Y[1] ), DY[2]/DX = 10 * Y[1] - ( 60 - Y[3]/8 ) * Y[2] + Y[3]/8, DY[3]/DX = 1, WITH INITIAL CONDITIONS AT X=0 : Y[1]=Y[2]=Y[3]=0 (SEE REF[2]). THE SOLUTION AT SEVERAL POINTS IN THE INTERVAL [0, 400] MAY BE OBTAINED BY THE FOLLOWING PROGRAM: (THE SOLUTION AT X=400 IS: Y[1]=22.24222011, Y[2]=27.11071335) "BEGIN" "INTEGER" N,NFE,NJE,POINT; "REAL" T,TEND,EPS,HMAX,L,H2,TIME; "ARRAY" Y,SW[1:3],PRINT[1:5]; "BOOLEAN" FAIL; "PROCEDURE" LIPEST(L,Y,EPS,T,F,N); "REAL" T,L,EPS; "ARRAY" Y; "INTEGER" N; "PROCEDURE" F; "BEGIN" "REAL" N1,N2; "INTEGER" I,IT; "ARRAY" F1,F2,Z,X[1:N]; "REAL" "PROCEDURE" NORM(Y); "ARRAY" Y; NORM:=SQRT(VECVEC(1,N,0,Y,Y)); DUPVEC(1,N,0,Z,Y); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" X[I]:="IF" Y[I]=0 "THEN" EPS "ELSE" (1+EPS)*Y[I]; N1:=NORM(X)*EPS; F(T,X,F1,N); "FOR" IT:=1 "STEP" 1 "UNTIL" 5 "DO" "BEGIN" F(T,Z,F2,N); ELMVEC(1,N,0,F2,F1,-1); N2:=N1/NORM(F2); DUPVEC(1,N,0,Z,X); ELMVEC(1,N,0,Z,F2,N2) "END"; F(T,Z,F2,N); ELMVEC(1,N,0,F2,F1,-1); L:=NORM(F2)/N1 "END" LIPEST; "PROCEDURE" F(T,Y,F1,N); "VALUE" T; "REAL" T; "ARRAY" Y,F1; "INTEGER" N; "BEGIN" NFE:=NFE+1; F1[1]:=0.2*(Y[2]-Y[1]); F1[2]:=10*Y[1]-(60-.125*Y[3])*Y[2]+.125*Y[3]; F1[3]:=1 "END"; "BOOLEAN" "PROCEDURE" AVAILABLE(T,Y,A,N); "INTEGER" N; "REAL" T; "ARRAY" Y,A; "BEGIN" NJE:=NJE+1; AVAILABLE:="TRUE"; A[1,1]:=-.2; A[1,2]:=.2; A[1,3]:=A[3,1]:=A[3,2]:=A[3,3]:=0; A[2,1]:=10; A[2,2]:=.125*Y[3]-60; A[2,3]:=.125*(1+Y[2]) "END" 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 6 "PROCEDURE" UPDATE(SW,R1,N); "INTEGER" N; "ARRAY" SW,R1; "BEGIN" "REAL" S1,S2; "INTEGER" I; "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" S1:=1/SW[I]; S2:=ABS(R1[I]); "IF" S1.2 "THEN" "BEGIN" "IF" LIT=0 "THEN" "GOTO" FAIL; LIT:=0; CONV:=1; IT:=1; RECOMP(A,H,T,Z,FAIL,PS); "END"; ASS: MAX1:=MAX; "GOTO" ATER; OUT: "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" Z[I]:=2*Z[I]-Y[I]; "END" ITERATE; "PROCEDURE" RECOMP(A,H,T,Y,FAIL,PS); "REAL" H,T; "ARRAY" A,Y; "LABEL" FAIL; "INTEGER" "ARRAY" PS; "BEGIN" "REAL" SL; "ARRAY" AUX[1:3]; SL:=H/2; "IF" "NOT" AVAILABLE(T,Y,A,N) "THEN" DFDY(T,Y,A); "FOR" I:=1 "STEP" 1 "UNTIL" N "DO" "BEGIN" MULROW(1,N,I,I,A,A,-SL); A[I,I]:=1+A[I,I] "END"; AUX[2]:="-14; DEC(A,N,AUX,PS); "IF" AUX[3]1 "THEN" SN:=1; "IF" START "THEN" "BEGIN" SN:=SN**4; LR:=LR*4 "END"; EHR[I]:=B3:=SN*EHR[I]+LR; C3:=C3+B3*B3*W; "END" I; B0:=ERR[1]; ERR[1]:=B1:=SQRT(C0); ERR[2]:=SQRT(C1); ERR[3]:=SQRT(C3)+SQRT(C2)/2; LQ:=EPS/("IF" B0=80 "THEN" LQ:=10; "END"; "PROCEDURE" REJECT; "IF" START "THEN" "BEGIN" HNEW:=LQ**(1/5)*H/2; "GOTO" INIT "END" "ELSE" "BEGIN" "FOR" K:=1,2,3,4,1,2,3 "DO" ELMROW(1,N,K,K+1,R,R,-1); "FOR" K:=1,2,3,4 "DO" ELMROW(1,N,K,K+1,RF,RF,-1); T:=T-H2; HALV:="TRUE"; HNEW:=H; "GOTO" MSTP "END" 1SECTION : 5.2.1.1.1.2.F (OCTOBER 1975) PAGE 12 ; "PROCEDURE" STEPSIZE; "IF" LQ<2 "THEN" "BEGIN" HALV:="TRUE"; HNEW:=H "END" "ELSE" "BEGIN" "IF" LQ>80 "THEN" HNEW:=("IF" LQ>5120 "THEN" (LQ/5)**(1/5) "ELSE" 2)*H2; "IF" HNEW>HMAX "THEN" HNEW:=HMAX; "IF" TEND>T "AND" TEND-THMAX "THEN" H:=HMAX "ELSE" H:=H0; "IF" H>(TEND-T0)/4 "THEN" H:=(TEND-T0)/4; "END"; HNEW:=H; ALF:=0; T:=TP:=T0; INIVEC(1,3,ERR,0); INIVEC(1,N,EHR,0); DUPROWVEC(1,N,1,R,Y0); INIMAT(2, 5, 1, N, R, 0.0); CONTROL(TP,T,H,HNEW,R,ERR,N); INIT: INITIALIZATION; START:="TRUE"; "FOR" ECI:=0,1,2,3 "DO" "BEGIN" ONE LARGE STEP; T:=T+H2; "IF" ECI>0 "THEN" "BEGIN" BACKWARD DIFFERENCES; UPDATE(WEIGHTS,S2,N) "END" "END"; ECI:=4; MSTP: "IF" HNEW^=H2 "THEN" "BEGIN" ECI:=1; CHANGE OF INFORMATION; ONE LARGE STEP; T:=T+H2; ECI:=2; "END"; ONE LARGE STEP; BACKWARD DIFFERENCES; UPDATE(WEIGHTS,S2,N); ERROR ESTIMATES; "IF" ECI<4 "AND" LQ>80 "THEN" LQ:=20; HALV:=TWO:="FALSE"; "IF" PRESCH "THEN" "GOTO" TRYCK; "IF" LQ<1 "THEN" REJECT "ELSE" STEPSIZE; TRYCK: "IF" TP<=T "THEN" CONTROL(TP,T,H,HNEW,R,ERR,N); "IF" START "THEN" START:="FALSE"; "IF" HNEW=H2 "THEN" T:=T+H2; ECI:=ECI+1; "IF" T1 "THEN" T:=T-H2; HALV:=TWO:="FALSE"; HNEW:=H2/2; "IF" START "THEN" "GOTO" INIT "ELSE" "GOTO" TRYCK "END"; END: "END" IMPEX; "EOP" 1SECTION : 5.2.1.1.1.3 (AUGUST 1974) PAGE 1 SECTION 5.2.1.1.1.3 CONTAINS TWO ALTERNATIVE PROCEDURES FOR SOLVING FIRST-ORDER INITIAL VALUE PROBLEMS WITH SEVERAL DERIVATIVES AVAILABLE. A. MODIFIED TAYLOR SOLVES AN INITIAL ( BOUNDARY ) VALUE PROBLEM, GIVEN AS A SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS , BY MEANS OF A ONE-STEP TAYLOR-METHOD. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF LARGE SYSTEMS ARISING FROM PARTIAL DIFFERENTIAL EQUATIONS, PROVIDED THAT HIGHER ORDER DERIVATIVES CAN BE EASILY OBTAINED. B. EXPONENTIALLY FITTED TAYLOR SOLVES AN INITIAL VALUE PROBLEM, GIVEN AS A SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS , BY MEANS OF A ONE-STEP TAYLOR-METHOD . AUTOMATIC STEPSIZE CONTROL IS PROVIDED. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF STIFF DIFFERENTIAL EQUATIONS , PROVIDED THAT HIGHER ORDER DERIVATIVES CAN BE EASILY OBTAINED. 1SECTION : 5.2.1.1.1.3.A (AUGUST 1974) PAGE 1 AUTHORS: P.J. VAN DER HOUWEN AND P.A.BEENTJES. INSTITUTE: MATHEMATICAL CENTRE. RECEIVED: 730616. BRIEF DESCRIPTION: MODIFIED TAYLOR SOLVES AN INITIAL ( BOUNDARY ) VALUE PROBLEM, GIVEN AS A SYSTEM OF FIRST ORDER DIFFERENTIAL EQUATIONS , BY MEANS OF A ONE-STEP TAYLOR-METHOD. IN PARTICULAR THIS METHOD IS SUITABLE FOR THE INTEGRATION OF LARGE SYSTEMS ARISING FROM PARTIAL DIFFERENTIAL EQUATIONS , PROVIDED THAT HIGHER ORDER DERIVATIVES CAN BE EASILY OBTAINED. KEYWORDS: DIFFERENTIAL EQUATIONS, INITIAL (BOUNDARY) VALUE PROBLEMS, ONE-STEP TAYLOR-METHOD. 1SECTION : 5.2.1.1.1.3.A (DECEMBER 1975) PAGE 2 CALLING SEQUENCE: THE HEADING OF THE PROCEDURE READS: "PROCEDURE" MODIFIED TAYLOR (T,TE,M0,M,U,SIGMA,TAUMIN,I,DERIVATIVE, K,DATA,ALFA,NORM,AETA,RETA,ETA,RHO,OUT); "INTEGER" M0,M,I,K,NORM; "REAL" T,TE,SIGMA,TAUMIN,ALFA,AETA,RETA,ETA,RHO; "ARRAY" U,DATA; "PROCEDURE" DERIVATIVE,OUT; "CODE" 33040; THE MEANING OF THE FORMAL PARAMETERS IS: T: ; THE INDEPENDENT VARIABLE T; MAY BE USED IN DERIVATIVE, SIGMA ETC.; ENTRY: THE INITIAL VALUE T0; EXIT : THE FINAL VALUE TE; TE: ; THE FINAL VALUE OF T (TE >= T); M0,M: ; INDICES OF THE FIRST AND LAST EQUATION OF THE SYSTEM TO BE SOLVED; U: ; "ARRAY" U[M0:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT T = T0; EXIT : THE VALUES OF THE SOLUTION AT T = TE; SIGMA: ; THE SPECTRAL RADIUS OF THE JACOBIAN MATRIX WITH RESPECT TO THOSE EIGENVALUES WHICH ARE LOCATED IN THE LEFT HALFPLANE; IF SIGMA TENDS TO INFINITY , PROCEDURE MODIFIED TAYLOR TERMINATES; TAUMIN: ; MINIMAL STEP LENGTH BY WHICH THE INTEGRATION IS PERFORMED; HOWEVER,ACTUAL STEPSIZES WILL ALWAYS BE WITHIN THE INTERVAL [MIN(HMIN,HSTAB),HSTAB],WHERE HSTAB(= DATA[0]/SIGMA) IS THE STEPLENGTH PRESCRIBED BY STABILITY CONSIDERATIONS; I: ; A JENSEN PARAMETER FOR PROCEDURE DERIVATIVE; MAY BE USED IN M0 AND M; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(I,A); "INTEGER" I; "ARRAY" A; WHEN THIS PROCEDURE IS CALLED, ARRAY A[M0 : M] CONTAINS THE COMPONENTS OF THE (I-1)-ST DERIVATIVE OF U AT THE POINT T; UPON COMPLETION OF DERIVATIVE, ARRAY A SHOULD CONTAIN THE COMPONENTS OF THE I-TH DERIVATIVE OF U AT THE POINT T; K: ; INDICATES THE NUMBER OF INTEGRATION STEPS PERFORMED; ENTRY: K = 0; 1SECTION : 5.2.1.1.1.3.A (AUGUST 1974) PAGE 3 DATA: ; "ARRAY" DATA[-2 : DATA[-2]]; ENTRY: DATA[-2]: THE ORDER OF THE HIGHEST DERIVATIVE UPON WHICH THE TAYLOR METHOD IS BASED; DATA[-1]: ORDER OF ACCURACY OF THE METHOD; DATA[0] : STABILITY PARAMETER; DATA[1] , ... , DATA[DATA[-2]] : POLYNOMIAL COEFFICIENTS; FOR FURTHER EXPLANATION AND POSSIBLE VALUES OF THE ELEMENTS OF ARRAY DATA SEE REFERENCES [2] AND [3]; ALFA: ; GROWTH FACTOR FOR THE INTEGRATION STEP LENGTH; NORM: ; IF NORM = 1 DISCREPANCY AND TOLERANCE ARE ESTIMATED IN THE MAXIMUM NORM, OTHERWISE IN THE EUCLIDIAN NORM; AETA,RETA: ; DESIRED ABSOLUTE AND RELATIVE ACCURACY; IF BOTH AETA AND RETA ARE NEGATIVE , ACCURACY CONDITIONS WILL BE IGNORED; ETA,RHO: ; COMPUTED TOLERANCE AND DISCREPANCY; OUT: ; THE HEADING OF THIS PROCEDURE READS : "PROCEDURE" OUT; THROUGH THIS PROCEDURE THE VALUES AFTER EACH INTEGRATION STEP OF FOR INSTANCE T, U, ETA AND RHO ARE ACCESSIBLE. DATA AND RESULTS: FOR FURTHER EXPLANATION OF THE PARAMETERS AETA, RETA, ETA, RHO, M0, M AND THE ARRAY DATA SEE REFERENCES [2] AND [3]. AS FOR THE INDICES M0 AND M THE FOLLOWING MAY BE REMARKED: WHEN THE METHOD OF LINES IS APPLIED TO HYPERBOLIC DIFFERENTIAL EQUATIONS THE NUMBER OF RELEVANT ORDINARY DIFFERENTIAL EQUATIONS DECREASES DURING THE INTEGRATION PROCESS. IN PROCEDURE MODIFIED TAYLOR , THIS MAY BE REALIZED BY INTEGER PROCEDURES M0 AND M WHICH ARE DEFINED AS FUNCTIONS OF I, K AND DATA[-2]. PROCEDURES USED: VECVEC = CP34010. REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 75 + M - M0. RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO BE SOLVED. LANGUAGE: ALGOL 60. 1SECTION : 5.2.1.1.1.3.A (AUGUST 1974) PAGE 4 METHOD AND PERFORMANCE: SEE REFERENCES. REFERENCES: [1].P.J. VAN DER HOUWEN. ONE-STEP METHODS FOR LINEAR INITIAL VALUE PROBLEMS I, POLYNOMIAL METHODS,TW REPORT 119, MATHEMATICAL CENTRE, AMSTERDAM (1970). [2].P.J. VAN DER HOUWEN, P.BEENTJES, K.DEKKER AND E.SLAGT ONE-STEP METHODS FOR LINEAR INITIAL VALUE PROBLEMS III, NUMERICAL EXAMPLES, TW REPORT 130/71, MATHEMATICAL CENTRE, AMSTERDAM (1971). [3].P.J. VAN DER HOUWEN, J.KOK. NUMERICAL SOLUTION OF A MINIMAX PROBLEM, TW REPORT 123/71, MATHEMATICAL CENTRE, AMSTERDAM (1971). EXAMPLE OF USE: THE SOLUTION AT T=EXP(1) AND T=EXP(2) OF THE DIFFERENTIAL EQUATION DU/DT=-EXP(T)*(U-LN(T)) + 1/T WITH INITIAL CONDITION U(.01)=LN(.01) AND ANALYTICAL SOLUTION U(T) = LN(T), MAY BE OBTAINED AS FOLLOWS: "BEGIN" "INTEGER" I,K;"REAL" T,TE,ETA,RHO,EXPT,LNT,C0,C1,C2,C3; "ARRAY" U[0:0],DATA[-2:4]; "PROCEDURE" OP;"IF" T=TE "THEN" OUTPUT(61,"(""("NUMBER OF STEPS:")",3ZD,/, "("SOLUTION: T= ")",+D.5D, "(" U(T) = ")",+D.7D,//")",K,T,U[0]); "PROCEDURE" DER(I,A);"INTEGER" I;"ARRAY" A; "BEGIN" "IF" I=1 "THEN" "BEGIN" EXPT:=EXP(T);LNT:=LN(T);C0:=A[0]; C1:=A[0]:=-EXPT*C0+1/T+EXPT*LNT "END"; "IF" I=2 "THEN" C2:=A[0]:=EXPT*(LNT+1/T-C0-C1)-1/T/T; "IF" I=3 "THEN" C3:=A[0]:= EXPT*(LNT+2/T-C0-2*C1-C2-1/T/T)+2/T/T/T; "IF" I=4 "THEN" A[0]:=C3-2*(1+3/T)/T/T/T+ EXPT*((1-(2-2/T)/T)/T-C1-C2*2-C3) "END"; I:=-2;"FOR" T:=4,3,6.025,1,.5,1/6,.018455702 "DO" "BEGIN" DATA[I]:=T;I:=I+1 "END"; T:=U[0]:="-2;K:=0;"FOR" TE:=EXP(1),TE*TE "DO" MODIFIED TAYLOR(T,TE,0,0,U,EXP(T),"-4,I,DER,K,DATA,1.5,1,"-5, "-4,ETA,RHO,OP) "END" 1SECTION : 5.2.1.1.1.3.A (AUGUST 1974) PAGE 5 THIS PROGRAM DELIVERS: NUMBER OF STEPS: 46 SOLUTION: T= +2.71828 U(T) = +1.0000285 NUMBER OF STEPS: 424 SOLUTION: T= +7.38906 U(T) = +1.9999967 SOURCE TEXT(S): 0"CODE" 33040; "PROCEDURE" MODIFIED TAYLOR(T,TE,M0,M,U,SIGMA,TAUMIN,I,DERIVATIVE,K, DATA,ALFA,NORM,AETA,RETA,ETA,RHO,OUT); "INTEGER" M0,M,I,K,NORM; "REAL" T,TE,SIGMA,TAUMIN,ALFA,AETA,RETA,ETA,RHO; "ARRAY" U,DATA; "PROCEDURE" DERIVATIVE,OUT; "BEGIN" I:=0; "BEGIN" "INTEGER" N,P,Q; "OWN" "REAL" EC0,EC1,EC2,TAU0,TAU1,TAU2,TAUS,T2; "REAL" T0,TAU,TAUI,TAUEC,ECL,BETAN,GAMMA; "REAL" "ARRAY" C[M0:M],BETA,BETHA[1:DATA[-2]]; "BOOLEAN" START,STEP1,LAST; "PROCEDURE" COEFFICIENT; "BEGIN" "INTEGER" J;"REAL" IFAC; IFAC:=1; GAMMA:=.5; N:=DATA[-2]; P:=DATA[-1]; BETAN:=DATA[0]; Q:= "IF" PS "THEN" S:=X "END" "END" "ELSE" S:=SQRT(VECVEC(M0,M,0,W,W)); NORMFUNCTION:=S "END"; "PROCEDURE" LOCAL ERROR BOUND; ETA:=AETA+RETA * NORMFUNCTION(NORM,U); "PROCEDURE" LOCAL ERROR CONSTRUCTION(I);"INTEGER" I; "BEGIN" "IF" I=P "THEN" "BEGIN" ECL:=0;TAUEC:=1 "END"; "IF" I>P+1 "THEN" TAUEC:=TAUEC*TAU; ECL:=ECL+ABS(BETHA[I])*TAUEC*NORMFUNCTION(NORM,C); "IF" I=N "THEN" "BEGIN" EC0:=EC1;EC1:=EC2;EC2:=ECL; RHO:=ECL*TAU**Q "END" "END"; "PROCEDURE" STEPSIZE; "BEGIN" "REAL" TAUACC,TAUSTAB,AA,BB,CC,EC; LOCAL ERROR BOUND; "IF" ETA>0 "THEN" "BEGIN" "IF" START "THEN" "BEGIN" "IF" K=0 "THEN" "BEGIN" "INTEGER" J; "FOR" J:=M0 "STEP" 1 "UNTIL" M "DO" C[J]:=U[J]; I:=1; DERIVATIVE(I,C); TAUACC:=ETA/NORMFUNCTION(NORM,C); STEP1:="TRUE" "END" "ELSE" "IF" STEP1 "THEN" "BEGIN" TAUACC:=(ETA/RHO)**(1/Q)*TAU2; "IF" TAUACC>10*TAU2 "THEN" TAUACC:=10*TAU2 "ELSE" STEP1:="FALSE" "END" "ELSE" "BEGIN" BB:=(EC2-EC1)/TAU1; CC:=EC2-BB*T2; EC:=BB*T+CC; TAUACC:="IF" EC<0 "THEN" TAU2 "ELSE" (ETA/EC)**(1/Q); START:="FALSE" "END" 1SECTION : 5.2.1.1.1.3.A (AUGUST 1974) PAGE 7 "END" "ELSE" "BEGIN" AA:=((EC0-EC1)/TAU0+(EC2-EC1)/TAU1)/ (TAU1+TAU0); BB:=(EC2-EC1)/TAU1-AA*(2*T2-TAU1); CC:=EC2-T2*(BB+AA*T2); EC:=CC+T*(BB+T*AA); TAUACC:="IF" EC<0 "THEN" TAUS "ELSE" (ETA/EC)**(1/Q); "IF" TAUACC>ALFA*TAUS "THEN" TAUACC:=ALFA*TAUS; "IF" TAUACCTAUSTAB "THEN" TAUSTAB "ELSE" TAUACC; TAUS:=TAU; "IF" TAU>=TE-T "THEN" "BEGIN" TAU:=TE-T;LAST:= "TRUE" "END"; TAU0:=TAU1;TAU1:=TAU2;TAU2:=TAU "END"; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" "INTEGER" J; "REAL" B; "FOR" J:=M0 "STEP" 1 "UNTIL" M "DO" C[J]:=U[J]; TAUI:=1; NEXT TERM: I:=I+1; DERIVATIVE(I,C); TAUI:=TAUI*TAU; B:=BETA[I]*TAUI; "IF" ETA>0 "AND" I>=P "THEN" LOCAL ERROR CONSTRUCTION(I); "FOR" J:=M0 "STEP" 1 "UNTIL" M "DO" U[J]:=U[J]+B*C[J]; "IF" I; THE INDEPENDENT VARIABLE T; MAY BE USED IN DERIVATIVE, SIGMA ETC.; ENTRY: THE INITIAL VALUE T0; EXIT : THE FINAL VALUE TE; TE: ; THE FINAL VALUE OF T (TE >= T); M0: ; INDEX OF THE FIRST EQUATION OF THE SYSTEM TO BE SOLVED; M: ; INDEX OF THE LAST EQUATION OF THE SYSTEM TO BE SOLVED; U: ; "ARRAY" U[M0:M]; THE DEPENDENT VARIABLE; ENTRY: THE INITIAL VALUES OF THE SOLUTION OF THE SYSTEM OF DIFFERENTIAL EQUATIONS AT T = T0; EXIT : THE VALUES OF THE SOLUTION AT T = TE; SIGMA: ; THE MODULUS OF THE (COMPLEX) POINT AT WHICH EXPONENTIAL FITTING IS DESIRED , FOR EXAMPLE AN APPROXIMATION OF THE MODULUS OF THE CENTRE OF THE LEFT HAND CLUSTER; PHI: ; THE ARGUMENT OF THE (COMPLEX) POINT AT WHICH EXPONENTIAL FITTING IS DESIRED; PHI SHOULD HAVE A VALUE FROM THE RANGE [PI/2,PI]; DIAMETER: ; THE DIAMETER OF THE LEFT HAND CLUSTER; DERIVATIVE: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" DERIVATIVE(I,A); "INTEGER" I; "ARRAY" A; I ASSUMES THE VALUES 1,2,3 AND A IS A ONE-DIMENSIONAL ARRAY A[M0:M]; WHEN THIS PROCEDURE IS CALLED , ARRAY A CONTAINS THE COMPONENTS OF THE (I-1)-ST DERIVATIVE OF U AT THE POINT T; UPON COMPLETION OF DERIVATIVE, ARRAY A SHOULD CONTAIN THE COMPONENTS OF THE I-TH DERIVATIVE OF U AT THE POINT T; I: ; A JENSEN PARAMETER FOR PROCEDURE DERIVATIVE; MAY BE USED IN M0 AND M; 1SECTION : 5.2.1.1.1.3.B (AUGUST 1974) PAGE 3 K: ; INDICATES THE NUMBER OF INTEGRATION STEPS PERFORMED; ENTRY: K = 0; EXIT : THE NUMBER OF INTEGRATION STEPS PERFORMED; ALFA: ; MAXIMAL GROWTH FACTOR FOR THE INTEGRATION STEP LENGTH; NORM: ; IF NORM = 1 DISCREPANCY AND TOLERANCE ARE ESTIMATED IN THE MAXIMUM NORM, OTHERWISE IN THE EUCLIDIAN NORM; AETA: ; DESIRED ABSOLUTE LOCAL ACCURACY ; AETA SHOULD BE POSITIVE; RETA: ; DESIRED RELATIVE LOCAL ACCURACY ; RETA SHOULD BE POSITIVE; ETA: ; COMPUTED TOLERANCE; RHO: ; COMPUTED DISCREPANCY; HMIN: ; MINIMAL STEPSIZE BY WHICH THE INTEGRATION IS PERFORMED; HOWEVER, A SMALLER STEP WILL BE TAKEN IF HMIN EXCEEDS THE STEPSIZE HSTAB , PRESCRIBED BY THE STABILITY CONDITIONS (SEE REF[2], FORMULA 6.12); IF HSTAB TENDS TO ZERO, THE PROCEDURE TERMINATES; HSTART: ; ENTRY: THE INTITIAL STEPSIZE ; HOWEVER, IF K = 0 ON ENTRY, THE VALUE OF HSTART IS NOT TAKEN INTO CONSIDERATION; EXIT: A SUGGESTION FOR THE STEPSIZE , IF THE INTEGRATION SHOULD BE CONTINUED FOR T>TE; HSTART MAY BE USED IN SUCCESSIVE CALLS OF THE PROCEDURE, IN ORDER TO OBTAIN THE SOLUTION IN SEVERAL POINTS TE1,TE2,ETC; OUTPUT: ; THE HEADING OF THIS PROCEDURE READS: "PROCEDURE" OUTPUT; THROUGH THIS PROCEDURE THE VALUES AFTER EACH INTEGRATION STEP OF FOR INSTANCE T, U, ETA AND RHO ARE ACCESSIBLE; DATA AND RESULTS: FOR FURTHER EXPLANATION OF THE PARAMETERS SIGMA,PHI,DIAMETER,AETA, RETA,ETA,RHO,M0,M SEE REF[2]; FOR RESULTS: SEE EXAMPLE OF USE AND REF[2]; PROCEDURES USED: INIVEC = CP 31010; DUPVEC = CP 31030; VECVEC = CP 34010; ELMVEC = CP 34020; ZEROIN = CP 34150. 1SECTION : 5.2.1.1.1.3.B (AUGUST 1974) PAGE 4 REQUIRED CENTRAL MEMORY: EXECUTION FIELD LENGTH: CIRCA 40 + 2 * (M - M0). RUNNING TIME: DEPENDS STRONGLY ON THE DIFFERENTIAL EQUATION TO BE SOLVED. LANGUAGE: ALGOL 60. METHOD AND PERFORMANCE: SEE REFERENCES. REFERENCES: [1]. P.J. VAN DER HOUWEN. ONE-STEP METHODS FOR LINEAR INITIAL VALUE PROBLEMS II, POLYNOMIAL METHODS. TW REPORT 122, (1970) MATHEMATICAL CENTRE. [2]. P.J. VAN DER HOUWEN, P.BEENTJES, K.DEKKER AND E.SLAGT. ONE-STEP METHODS FOR LINEAR INITIAL VALUE PROBLEMS III, NUMERICAL EXAMPLES. TW REPORT 130, (1971) MATHEMATICAL CENTRE. EXAMPLE OF USE: THE SOLUTION AT T=EXP(1) AND T=EXP(2) OF THE DIFFERENTIAL EQUATION DU/DT=-EXP(T)*(U-LN(T)) + 1/T WITH INITIAL CONDITION U(.01)=LN(.01) AND ANALYTICAL SOLUTION U(T) = LN(T), MAY BE OBTAINED AS FOLLOWS: "BEGIN" "INTEGER" I,K; "REAL" T,TE,TE1,TE2,RETA,ETA,RHO,PI,HS,EXPT,LNT,TIME,U0,U1,U2; "REAL" "ARRAY" U[0:0]; "PROCEDURE" DERIVATIVE(I,U); "INTEGER" I; "ARRAY" U; "IF" I=1 "THEN" "BEGIN" EXPT:=EXP(T); LNT:=LN(T); U0:=U[0]; U1:=U[0]:=EXPT*(LNT-U0)+1/T "END" "ELSE" "IF" I=2 "THEN" U2:=U[0]:=EXPT*(LNT-U0-U1+1/T)-1/T/T "ELSE" U[0]:=EXPT*(LNT-U0-2*U1-U2+2/T-1/T/T)+2/T/T/T; "PROCEDURE" OUT; "IF" T=TE "THEN" OUTPUT(61,"("6ZD,+3ZD.3DB3DB3D")",K,U[0]); 1SECTION : 5.2.1.1.1.3.B (AUGUST 1974) PAGE 5 "PROCEDURE" OUT1; OUTPUT(61,"("4BD"-D,3Z.3D,/")",RETA,CLOCK-TIME); OUTPUT(61,"(""(" THIS LINE AND THE FOLLOWING TEXT IS ")" "("PRINTED BY THIS PROGRAM")",//, "(" THE RESULTS WITH EFT ARE -CONFER REF[2]- :")",/, "(" K U(TE1) K U(TE2)")" "(" RETA TIME")",/")"); PI:=4*ARCTAN(1); TE1:=EXP(1); TE2:=EXP(2); "FOR" RETA:="-1,"-2,"-3,"-4 "DO" "BEGIN" T:=.01; U[0]:=LN(T); K:=0; HS:=0; TIME:=CLOCK; "FOR" TE:=TE1,TE2 "DO" EXPONENTIALLY FITTED TAYLOR (T,TE,0,0,U,EXP(T),PI,2*EXP(2*T/3),DERIVATIVE,I,K,1.5,2, RETA/10,RETA,ETA,RHO,"-4,HS,OUT); OUT1 "END"; OUTPUT(61,"("//,"(" WITH RELAXED ACCURACY CONDITIONS FOR ")" "("T>3:")",/,"(" K U(TE1) K U(TE2)")" "(" RETA TIME")",/")"); "FOR" RETA:="-1,"-2,"-3,"-4 "DO" "BEGIN" T:=.01; U[0]:=LN(T); K:=0; HS:=0; TIME:=CLOCK; "FOR" TE:=TE1,TE2 "DO" EXPONENTIALLY FITTED TAYLOR (T,TE,0,0,U,EXP(T),PI,2*EXP(2*T/3),DERIVATIVE,I,K,1.5,2, RETA/10*("IF" T<3 "THEN" 1 "ELSE" EXP(2*(T-3))), RETA*("IF" T<3 "THEN" 1 "ELSE" EXP(2*(T-3))), ETA,RHO,"-4,HS,OUT); OUT1 "END" "END" THIS LINE AND THE FOLLOWING TEXT IS PRINTED BY THIS PROGRAM THE RESULTS WITH EFT ARE -CONFER REF[2]- : K U(TE1) K U(TE2) RETA TIME 15 +1.003 845 001 42 +2.000 076 417 1"-1 .938 22 +1.001 211 286 52 +2.000 066 067 1"-2 1.121 36 +1.000 108 738 92 +2.000 020 495 1"-3 1.872 56 +1.000 045 271 171 +2.000 000 925 1"-4 3.493 WITH RELAXED ACCURACY CONDITIONS FOR T>3: K U(TE1) K U(TE2) RETA TIME 15 +1.003 845 001 42 +2.000 076 417 1"-1 1.037 22 +1.001 211 286 50 +2.000 049 978 1"-2 1.154 36 +1.000 108 738 68 +2.000 023 330 1"-3 1.419 56 +1.000 045 271 98 +2.000 065 056 1"-4 2.008 1SECTION : 5.2.1.1.1.3.B (AUGUST 1974) PAGE 6 SOURCE TEXT(S): 0"CODE" 33050; "PROCEDURE" EXPONENTIALLY FITTED TAYLOR(T,TE,M0,M,U,SIGMA,PHI,DIAMETER, DERIVATIVE,I,K,ALFA,NORM,AETA,RETA,ETA,RHO,HMIN,HSTART,OUTPUT); "INTEGER" M0,M,I,K,NORM; "REAL" T,TE,SIGMA,PHI,DIAMETER,ALFA,AETA,RETA,ETA,RHO,HMIN,HSTART; "ARRAY" U; "PROCEDURE" DERIVATIVE,OUTPUT; "BEGIN" "INTEGER" KL; "REAL" Q,EC0,EC1,EC2,H,HI,H0,H1,H2,BETAN,T2,SIGMAL,PHIL; "REAL" "ARRAY" C,RO[M0:M],BETA,BETHA[1:3]; "BOOLEAN" LAST,START; "PROCEDURE" COEFFICIENT; "BEGIN" "REAL" B,B1,B2,BB,E,BETA2,BETA3; B:=H*SIGMAL; B1:=B*COS(PHIL); BB:=B*B; "IF" ABS(B)<"-3 "THEN" "BEGIN" BETA2:=.5-BB/24; BETA3:=1/6+B1/12; BETHA[3]:=.5+B1/3 "END" "ELSE" "IF" B1<-40 "THEN" "BEGIN" BETA2:=(-2*B1-4*B1*B1/BB+1)/BB; BETA3:=(1+2*B1/BB)/BB; BETHA[3]:=1/BB "END" "ELSE" "BEGIN" E:=EXP(B1)/BB; B2:=B*SIN(PHIL); BETA2:=(-2*B1-4*B1*B1/BB+1)/BB; BETA3:=(1+2*B1/BB)/BB; "IF" ABS(B2/B)<"-5 "THEN" "BEGIN" BETA2:=BETA2-E*(B1-3); BETA3:=BETA3+E*(B1-2)/B1; BETHA[3]:=1/BB+E*(B1-1) "END" "ELSE" "BEGIN" BETA2:=BETA2-E*SIN(B2-3*PHIL)/B2*B; BETA3:=BETA3+E*SIN(B2-2*PHIL)/B2; BETHA[3]:=1/BB+E*SIN(B2-PHIL)/B2*B; "END" "END"; BETA[1]:=BETHA[1]:=1; BETA[2]:=BETA2; BETA[3]:=BETA3; BETHA[2]:=1-BB*BETA3; B:=ABS(B); Q:="IF" B<1.5 "THEN" 4-2*B/3 "ELSE" "IF" B<6 "THEN" (30-2*B)/9 "ELSE" 2; "END"; "COMMENT" 1SECTION : 5.2.1.1.1.3.B (AUGUST 1974) PAGE 7 ; "REAL" "PROCEDURE" NORMFUNCTION(NORM,W); "INTEGER" NORM; "ARRAY" W; "BEGIN" "INTEGER" J; "REAL" S,X; S:=0; "IF" NORM=1 "THEN" "BEGIN" "FOR" J:=M0 "STEP" 1 "UNTIL" M "DO" "BEGIN" X:=ABS(W[J]); "IF" X>S "THEN" S:=X "END" "END" "ELSE" S:=SQRT(VECVEC(M0,M,0,W,W)); NORMFUNCTION:=S; "END"; "PROCEDURE" LOCAL ERROR BOUND; ETA:=AETA+RETA * NORMFUNCTION(NORM,U); "PROCEDURE" LOCAL ERROR CONSTRUCTION(I); "INTEGER" I; "BEGIN" "IF" I=1 "THEN" INIVEC(M0,M,RO,0); "IF" I<4 "THEN" ELMVEC(M0,M,0,RO,C,BETHA[I]*HI); "IF" I=4 "THEN" "BEGIN" ELMVEC(M0,M,0,RO,C,-H); RHO:=NORMFUNCTION(NORM,RO); EC0:=EC1;EC1:=EC2;EC2:=RHO/H**Q; "END" "END"; "PROCEDURE" STEPSIZE; "BEGIN" "REAL" HACC,HSTAB,HCR,HMAX,A,B,C; "IF" "NOT" START "THEN" LOCAL ERROR BOUND; "IF" START "THEN" "BEGIN" H1:=H2:=HACC:=HSTART; EC2:=EC1:=1; KL:=1; START:="FALSE" "END" "ELSE" "IF" KL<3 "THEN" "BEGIN" HACC:=(ETA/RHO)**(1/Q)*H2; "IF" HACC>10*H2 "THEN" HACC:=10*H2 "ELSE" KL:=KL+1 "END" "ELSE" "BEGIN" A:=(H0*(EC2-EC1)-H1*(EC1-EC0))/(H2*H0-H1*H1); H:=H2*("IF" ETA0 "THEN" "BEGIN" B:=(EC2-EC1-A*(H2-H1))/H1; C:=EC2-A*H2-B*T2; HACC:=0; HMAX:=H; "IF" ^ZEROIN(HACC,H,HACC**Q*(A*HACC+B*T+C)-ETA, "-3*H2) "THEN" HACC:=HMAX "END" "ELSE" HACC:=H; "IF" HACC<.5*H2 "THEN" HACC:=.5*H2; "END"; "IF" HACC1 "THEN" "BEGIN" A:=ABS(DIAMETER/SIGMAL+"-14)/2; B:=2*ABS(SIN(PHIL)); BETAN:=("IF" A>B "THEN" 1/A "ELSE" 1/B)/A; HSTAB:=ABS(BETAN/SIGMAL); "IF" HSTAB<"-14*T "THEN" "GOTO" ENDOFEFT; "IF" H>HSTAB "THEN" H:=HSTAB "END"; HCR:=H2*H2/H1; "IF" KL>2 "AND" ABS(H-HCR)<"-6*HCR "THEN" H:="IF" HTE "THEN" "BEGIN" LAST:="TRUE"; HSTART:=H; H:=TE-T "END"; H0:=H1;H1:=H2;H2:=H; "END"; "PROCEDURE" DIFFERENCE SCHEME; "BEGIN" HI:=1; SIGMAL:=SIGMA; PHIL:=PHI; STEPSIZE; COEFFICIENT; "FOR" I:=1,2,3 "DO" "BEGIN" HI:=HI*H; "IF" I>1 "THEN" DERIVATIVE(I,C); LOCALERRORCONSTRUCTION(I); ELMVEC(M0,M,0,U,C,BETA[I]*HI) "END"; T2:=T; K:=K+1; "IF" LAST "THEN" "BEGIN" LAST:="FALSE"; T:=TE; START:="TRUE" "END" "ELSE" T:=T+H; DUPVEC(M0,M,0,C,U); DERIVATIVE(1,C); LOCALERRORCONSTRUCTION(4); OUTPUT; "END"; START:="TRUE"; LAST:="FALSE"; DUPVEC(M0,M,0,C,U); DERIVATIVE(1,C); "IF" K=0 "THEN" "BEGIN" LOCAL ERROR BOUND; HSTART:=ETA/NORMFUNCTION(NORM,C) "END"; NEXT LEVEL: DIFFERENCE SCHEME; "IF" T^=TE "THEN" "GOTO" NEXT LEVEL; ENDOFEFT: "END" EXPONENTIAL FITTED TAYLOR; "EOP"