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numalinx.txt. May 13, 1992.

by Paul McJones last modified 2010-06-04 15:09

numalinx.txt only contains a new, trimmed introduction and the systematic index -- for the history of the NUMAL development, and the KWIC index, see: P.W. Hemker (ed.)[1981]: NUMAL. Numerical Procedures in ALGOL 60. 7 volumes. MC Syllabus 47, Mathematical Centre, Amsterdam.

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1
 
                          INDEX TO THE LIBRARY
 
                                 NUMAL
 
            OF ALGOL 60 PROCEDURES IN NUMERICAL MATHEMATICS
 
 
 ***********************************************************************
 
 ON  REQUEST  OF  THE   ACADEMIC  COMPUTING  CENTRE  AMSTERDAM  ( SARA )
 THE  LIBRARY   NUMAL   IS DEVELOPED   AND  SUPPORTED  BY THE  NUMERICAL
 MATHEMATICS  DEPARTMENT  OF THE  MATHEMATICAL  CENTRE  (AMSTERDAM).
 THE PRESENT DOCUMENT  CONTAINS A  SURVEY OF THE PROCEDURES AVAILABLE IN
 NUMAL FOR ALGOL 60 VERSION 5.  MOREOVER, IT  DESCRIBES THE WAY IN WHICH
 ONE CAN OBTAIN FULL DOCUMENTATION OF ALL AVAILABLE PROCEDURES.
 
 
 TO GET ENTRANCE TO THE DOCUMENTATION.
 
 CLASSIFIED  ACCORDING  TO  SUBJECT,  THE  PRESENT  INDEX  CONTAINS  THE
 NAMES  OF  THE  PROCEDURES, THE  CORRESPONDING  CODE NUMBERS  IN NUMAL5
 AND  A  REFERENCE  TO  THE  DOCUMENTATION.  THIS  REFERENCE  GIVES  THE
 RECORD NUMBER IN THE DOCUMENTATION FILE.
 IN ORDER TO CONSULT A SPECIFIED RECORD OF DOCUMENTATION,  ALL PRECEDING
 RECORDS  HAVE  TO BE  SKIPPED.
 
 
 EXAMPLE.
 
 IN  ORDER  TO OBTAIN  THE  DESCRIPTION  OF  THE  PROCEDURE  "MULTISTEP"
 (SECTION  5.2.1.1.1.1.F.; RECORD NUMBER = 211)  FROM  THE DOCUMENTATION
 FILE THE FOLLOWING CONTROL CARDS CAN BE USED
                  ......
                  LABEL,TAPE,R,L=NUMAL5TAPE,ID=NUMAL5MC,D=GE,VSN=T1886H.
                  SKIPF,TAPE,1,17. TO POSITION THE TAPE
                  COMMENT.         AT THE DOCUMENT FILE
                  SKIPF,TAPE,210.
                  COPYBR,TAPE,OUTPUT.
                  ......
 IN ORDER TO OBTAIN THE SOURCE TEXT, ONE MORE RECORD HAD TO BE SKIPPED.
 
 
 STATUS OF NUMAL AND RESPONSIBILITY.
 
 IN 1979  THE INTEREST OF THE  MATHEMATICAL CENTRE  IN THE  CREATION  OF
 GENERAL  NUMERICAL  SOFTWARE  IN  ALGOL60  DECREASED  AND THE  DRAFTING
 COMMITTEE DECIDED  TO CONCLUDE THE  NUMAL PROJECT WITH THE  PUBLICATION
 OF A FINAL REVISION OF THE LIBRARY IN BOOK FORM.
 THIS RESULTED IN THE MATHEMATICAL CENTRE PUBLICATION: 
     P.W. HEMKER (ED.)[1981]: 
     NUMAL. NUMERICAL PROCEDURES IN ALGOL 60. 7 VOLUMES.
     MC SYLLABUS, MATHEMATICAL CENTRE, AMSTERDAM.
 IN  THIS FORM THE  LIBRARY  NUMAL  CAN BE SEEN AS A DESCRIPTION  OF THE
 STATE-OF-THE-ART  OF NUMERICAL ALGOL60 PROGRAMMING AT THE  MATHEMATICAL
 CENTRE AT THE END OF THE 1970-S.
 IT IS  CLAIMED  TO  CONTAIN  A VALUABLE  COLLECTION  OF  ROUTINES  IN A
 LANGUAGE THAT STILL CAN DESCRIBE NUMERICAL PROCEDURES  BETTER THAN MANY
 OTHER PROGRAMMING LANGUAGES CURRENTLY IN USE.
1
 IN 1981 A NEW TRANSLATED LIBRARY WAS PRODUCED  FOR THE USE OF THE NUMAL
 PROCEDURES UNDER  ALGOL 60 VERSION 5  ( AT THE ACADEMIC COMPUTER CENTRE
 OF AMSTERDAM ).  THIS REQUIRED  AN ADAPTATION  OF ALL  SOURCE TEXTS  TO
 ALGOL 60.1 STANDARD.   TO THIS END A COMPLETELY  NEW DOCUMENTATION FILE
 WAS MADE  DIFFERING  FROM THE FORMER ONE  (AND FROM THE ABOVE MENTIONED
 PUBLICATION) IN THE FOLLOWING WAYS: 
  1) A FEW CHANGES WERE NECESSARY WHEN THE SOURCE TEXT WAS NOT ACCORDING
     TO ALGOL 60.1 STANDARD (ESPECIALLY WHEN "INTEGER" AND "REAL"  TYPES
     TYPES WERE INTERCHANGED IN PARAMETER SUBSTITUTION).
  2) IN  ORDER  TO  PROCESS  THE  DOCUMENT FILE  AUTOMATICALLY,  SEVERAL
     IRREGULARITIES CONCERNING LAY-OUT WERE CORRECTED.
  3) A  COUPLE OF ERRORS  IN THE SPECIFICATIONS  OF THE PROCEDURES  WERE
     DETECTED AND DULY CORRECTED.
 A RECORD IS KEPT OF ALL CHANGES EXCEPT THOSE OF THE LAY-OUT.
 
 ALTHOUGH  MUCH EFFORT HAS BEEN SPENT TO KEEP THE NUMBER OF ERRORS  TO A
 MINIMUM, IT IS POSSIBLE THAT SOME MINOR ERRORS STILL REMAIN.
 THE  MATHEMATICAL  CENTRE  CANNOT  BE  HELD LIABLE  FOR  DAMAGE  DUE TO
 ILL-FUNCTIONING OF ANY PROGRAM USING THE NUMAL LIBRARY.
 THE  NUMERICAL MATHEMATICS  DEPARTMENT  WILL KEEP  A LIST OF ALL ERRORS
 IN THE DOCUMENTATION AND/OR THE PROCEDURES THAT BECOME KNOWN,  AND THIS
 LIST CAN BE OBTAINED ON REQUEST.
 
 FOR REMARKS AND REQUESTS: 
 
    MATHEMATISCH CENTRUM
    AFDELING NUMERIEKE WISKUNDE
    KRUISLAAN 413               (ADDRESS FOR VISITS)
    1098 SJ  AMSTERDAM
    POSTBUS 4079                (MAILING ADDRESS)
    1009 AB  AMSTERDAM
    TEL.: 020 - 592 9333.
 
 
 
                                                   J. KOK
                                                  (MATHEMATICAL CENTRE)
 
 
 
 
 
 
 
 
 
 
 
 
 NO  PART  OF THE  LIBRARY   NUMAL   MAY  BE  REPRODUCED,  STORED  IN  A
 RETRIEVAL  SYSTEM   OR  TRANSMITTED,  IN  ANY  FORM  OR  BY ANY  MEANS,
 ELECTRONIC,   PHOTOCOPYING,  RECORDING,  OR  OTHERWISE,   WITHOUT   THE
 PRIOR  WRITTEN  PERMISSION  OF THE  ACADEMIC COMPUTING CENTRE AMSTERDAM
 (SARA) OR  THE  MATHEMATICAL CENTRE  (AMSTERDAM).
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    1.ELEMENTARY PROCEDURES
         1.REAL VECT AND MAT OPERATIONS
              1.INITIALIZATION
                                       INIVEC                       *   31010    NOV/81       1
                                       INIMAT                       *   31011    NOV/81       1
                                       INIMATD                      *   31012    NOV/81       1
                                       INISYMD                      *   31013    NOV/81       1
                                       INISYMROW                    *   31014    NOV/81       1
              2.DUPLICATION
                                       DUPVEC                       *   31030    NOV/81       3
                                       DUPVECROW                    *   31031    NOV/81       3
                                       DUPROWVEC                    *   31032    NOV/81       3
                                       DUPVECCOL                    *   31033    NOV/81       3
                                       DUPCOLVEC                    *   31034    NOV/81       3
                                       DUPMAT                       *   31035    NOV/81       3
              3.MULTIPLICATION
                                       MULVEC                       *   31020    NOV/81       5
                                       MULROW                       *   31021    NOV/81       5
                                       MULCOL                       *   31022    NOV/81       5
                                       COLCST                       *   31131    NOV/81       5
                                       ROWCST                       *   31132    NOV/81       5
              4.SCALAR PRODUCTS
                   1.VECTOR VECTOR PRODUCTS
                                       VECVEC                       *   34010    NOV/81       7
                                       MATVEC                       *   34011    NOV/81       7
                                       TAMVEC                       *   34012    NOV/81       7
                                       MATMAT                       *   34013    NOV/81       7
                                       TAMMAT                       *   34014    NOV/81       7
                                       MATTAM                       *   34015    NOV/81       7
                                       SEQVEC                       *   34016    NOV/81       7
                                       SCAPRD1                      *   34017    NOV/81       7
                                       SYMMATVEC                    *   34018    NOV/81       7
                   2.MATRIX VECTOR PRODUCTS
                                       FULMATVEC                    *   31500    NOV/81       9
                                       FULTAMVEC                    *   31501    NOV/81       9
                                       FULSYMMATVEC                     31502    NOV/81       9
                                       RESVEC                       *   31503    NOV/81       9
                                       SYMRESVEC                        31504    NOV/81       9
                   3.MATRIX MATRIX PRODUCTS
                                       HSHVECMAT                    *   31070    NOV/81      11
                                       HSHCOLMAT                    *   31071    NOV/81      11
                                       HSHROWMAT                    *   31072    NOV/81      11
                                       HSHVECTAM                    *   31073    NOV/81      11
                                       HSHCOLTAM                    *   31074    NOV/81      11
                                       HSHROWTAM                    *   31075    NOV/81      11
              5.ELIMINATION
                                       ELMVEC                       *   34020    NOV/81      13
                                       ELMCOL                       *   34023    NOV/81      13
                                       ELMROW                       *   34024    NOV/81      13
                                       ELMVECCOL                    *   34021    NOV/81      13
                                       ELMCOLVEC                    *   34022    NOV/81      13
                                       ELMVECROW                    *   34026    NOV/81      13
                                       ELMROWVEC                    *   34027    NOV/81      13
                                       ELMCOLROW                    *   34029    NOV/81      13
    1.   1.   5.                                                       ELMROWCOL                    *   34028    NOV/81      13
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    1.   1.   5.                                                       MAXELMROW                    *   34025    NOV/81      13
              6.INTERCHANGING
                                       ICHVEC                       *   34030    NOV/81      15
                                       ICHCOL                       *   34031    NOV/81      15
                                       ICHROW                       *   34032    NOV/81      15
                                       ICHROWCOL                    *   34033    NOV/81      15
                                       ICHSEQVEC                    *   34034    NOV/81      15
                                       ICHSEQ                       *   34035    NOV/81      15
              7.ROTATION
                                       ROTCOL                       *   34040    NOV/81      17
                                       ROTROW                       *   34041    NOV/81      17
              8.NORMS
                                       INFNRMVEC                    *   31061    NOV/81      19
                                       INFNRMROW                    *   31062    NOV/81      19
                                       INFNRMCOL                    *   31063    NOV/81      19
                                       INFNRMMAT                    *   31064    NOV/81      19
                                       ONENRMVEC                    *   31065    NOV/81      19
                                       ONENRMROW                    *   31066    NOV/81      19
                                       ONENRMCOL                    *   31067    NOV/81      19
                                       ONENRMMAT                    *   31068    NOV/81      19
                                       ABSMAXMAT                    *   31069    NOV/81      19
              9.SCALING
                                       REASCL                           34183    NOV/81      21
         2.COMPL VECT AND MAT OPERATIONS
              1.
              2.
              3.MULTIPLICATION
                                       COMCOLCST                        34352    NOV/81      23
                                       COMROWCST                        34353    NOV/81      23
              4.SCALAR PRODUCTS
                                       COMMATVEC                        34354    NOV/81      25
                                       HSHCOMCOL                        34355    NOV/81      25
                                       HSHCOMPRD                        34356    NOV/81      25
              5.ELIMINATION
                                       ELMCOMVECCOL                     34376    NOV/81      27
                                       ELMCOMCOL                        34377    NOV/81      27
                                       ELMCOMROWVEC                     34378    NOV/81      27
              6.INTERCHANGING
              7.ROTATION
                                       ROTCOMCOL                        34357    NOV/81      29
                                       ROTCOMROW                        34358    NOV/81      29
                                       CHSH2                            34611    NOV/81      29
              8.NORMS
                                       COMEUCNRM                        34359    NOV/81      31
              9.SCALING
                                       COMSCL                           34193    NOV/81      33
                                       SCLCOM                           34360    NOV/81      33
         3.COMPLEX ARITHMETIC
              1.MONADIC OPERATIONS
                                       COMABS                           34340    NOV/81      35
                                       COMSQRT                          34343    NOV/81      35
                                       CARPOL                           34344    NOV/81      35
              2.DYADIC OPERATIONS
                                       COMMUL                       *   34341    NOV/81      37
    1.   3.   2.                                                       COMDIV                       *   34342    NOV/81      37
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    1.   4.LONG INTEGER ARITHMETIC
                                       LNGINTADD                        31200    NOV/81      39
                                       LNGINTSUBTRACT                   31201    NOV/81      39
                                       LNGINTMULT                       31202    NOV/81      39
                                       LNGINTDIVIDE                     31203    NOV/81      39
                                       LNGINTPOWER                      31204    NOV/81      39
         5.LONG REAL ARITHMETIC
              1.ELEM. ARITHMETIC OPERATIONS
                                       DP ADD                       *   31101    NOV/81      41
                                       DP SUB                       *   31102    NOV/81      41
                                       DP MUL                       *   31103    NOV/81      41
                                       DP DIV                       *   31104    NOV/81      41
                                       DP POW                           31109    NOV/81      41
                                       LNG ADD                      *   31105    NOV/81      41
                                       LNG SUB                      *   31106    NOV/81      41
                                       LNG MUL                      *   31107    NOV/81      41
                                       LNG DIV                      *   31108    NOV/81      41
                                       LNG POW                          31110    NOV/81      41
              2.SCALAR PRODUCTS
                                       LNGVECVEC                    *   34410    NOV/81      43
                                       LNGMATVEC                    *   34411    NOV/81      43
                                       LNGTAMVEC                    *   34412    NOV/81      43
                                       LNGMATMAT                    *   34413    NOV/81      43
                                       LNGTAMMAT                    *   34414    NOV/81      43
                                       LNGMATTAM                    *   34415    NOV/81      43
                                       LNGSEQVEC                    *   34416    NOV/81      43
                                       LNGSCAPRD1                   *   34417    NOV/81      43
                                       LNGSYMMATVEC                 *   34418    NOV/81      43
                                       LNGFULMATVEC                 *   31505    NOV/81      43
                                       LNGFULTAMVEC                 *   31506    NOV/81      43
                                       LNGFULSYMMATVEC                  31507    NOV/81      43
                                       LNGRESVEC                    *   31508    NOV/81      43
                                       LNGSYMRESVEC                     31509    NOV/81      43
              3.CONVERSION
                                       LNGREATODECI                     31100    NOV/81      45
    2.ALGEBRAIC EVALUATIONS
         1.EVAL. OF A FINITE SERIES
         2.EVAL. OF POLYNOMIALS
              1.EVAL. OF GENERAL POLYNOMIALS
                   1.POLYNOMIALS IN GRUNERT FORM
                                       POL                              31040    NOV/81      47
                                       TAYPOL                           31241    NOV/81      47
                                       NORDERPOL                        31242    NOV/81      47
                                       DERPOL                           31243    NOV/81      47
                   2.POLYNOMIALS IN NEWTON FORM
              2.EVAL. OF ORTHOGON. POLYNOMIALS
                   1.GENERAL ORTHOGON. POLYNOMIALS
                                       ORTPOL                           31044    NOV/81      49
                                       ORTPOLSYM                        31048    NOV/81      49
                                       ALLORTPOL                        31045    NOV/81      49
                                       ALLORTPOLSYM                     31049    NOV/81      49
                                       SUMORTPOL                        31047    NOV/81      49
                                       SUMORTPOLSYM                     31058    NOV/81      49
                   2.CHEBYCHEV POLYNOMIALS
    2.   2.   2.   2.                                                  CHEPOLSUM                        31046    NOV/81      51
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    2.   2.   2.   2.                                                  ODDCHEPOLSUM                     31059    NOV/81      51
                                       CHEPOL                           31042    NOV/81      51
                                       ALLCHEPOL                        31043    NOV/81      51
              3.EVAL. OF TRIGONOM. POLYNOMIALS
                   1.EVAL. OF FOURIER SERIES
                                       SINSER                           31090    NOV/81      53
                                       COSSER                           31091    NOV/81      53
                                       FOUSER                           31092    NOV/81      53
                                       FOUSER1                          31093    NOV/81      53
                                       FOUSER2                          31094    NOV/81      53
                                       COMFOUSER                        31095    NOV/81      53
                                       COMFOUSER1                       31096    NOV/81      53
                                       COMFOUSER2                       31097    NOV/81      53
         3.EVAL. OF CONTINUED FRACTIONS
                                       JFRAC                            35083    NOV/81      55
         4.OPERATIONS ON POLYNOMIALS
              1.TRANSF. OF REPRESENTATION
                                       POLCHS                           31051    NOV/81      57
                                       CHSPOL                           31052    NOV/81      57
                                       POLSHTCHS                        31053    NOV/81      57
                                       SHTCHSPOL                        31054    NOV/81      57
                                       GRNNEW                           31055    NOV/81      57
                                       NEWGRN                           31050    NOV/81      57
                                       LINTFMPOL                        31250    NOV/81      57
              2.OP. ON GENERAL POLYNOMIALS
              3.OP. ON ORTHOGONAL POLYNOMIALS
                                       INTCHS                           31248    NOV/81      59
         5.FAST FOURIER TRANSFORM
    3.LINEAR ALGEBRA
         1.LINEAR SYSTEMS
              1.FULL MATRICES
                   1.SQUARE NON-SINGULAR MATRICES
                        1.REAL MATRICES
                             1.GENERAL MATRICES
  1.PREPARATORY PROCEDURES
                                       DEC                              34300    NOV/81      61
                                       GSSELM                           34231    NOV/81      61
                                       ONENRMINV                        34240    NOV/81      61
                                       ERBELM                           34241    NOV/81      61
                                       GSSERB                           34242    NOV/81      61
                                       GSSNRI                           34252    NOV/81      61
  2.CALCULATION OF DETERMINANT
                                       DETERM                           34303    NOV/81      63
  3.SOLUTION OF LINEAR EQUATIONS
                                       SOL                              34051    NOV/81      65
                                       DECSOL                           34301    NOV/81      65
                                       SOLELM                           34061    NOV/81      65
                                       GSSSOL                           34232    NOV/81      65
                                       GSSSOLERB                        34243    NOV/81      65
  4.MATRIX INVERSION
                                       INV                              34053    NOV/81      67
                                       DECINV                           34302    NOV/81      67
                                       INV1                             34235    NOV/81      67
                                       GSSINV                           34236    NOV/81      67
    3.   1.   1.   1.   1.   1.   4.                                   GSSINVERB                        34244    NOV/81      67
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    3.   1.   1.   1.   1.   1.   5.ITERATIVELY IMPROVED SOLUTION
                                       ITISOL                           34250    NOV/81      69
                                       GSSITISOL                        34251    NOV/81      69
                                       ITISOLERB                        34253    NOV/81      69
                                       GSSITISOLERB                     34254    NOV/81      69
                             2.SYMMETRIC POS DEF MATRICES
  1.PREPARATORY PROCEDURES
                                       CHLDEC2                          34310    NOV/81      71
                                       CHLDEC1                          34311    NOV/81      71
  2.CALCULATION OF DETERMINANT
                                       CHLDETERM2                       34312    NOV/81      73
                                       CHLDETERM1                       34313    NOV/81      73
  3.SOLUTION OF LINEAR EQUATIONS
                                       CHLSOL2                          34390    NOV/81      75
                                       CHLSOL1                          34391    NOV/81      75
                                       CHLDECSOL2                       34392    NOV/81      75
                                       CHLDECSOL1                       34393    NOV/81      75
  4.MATRIX INVERSION
                                       CHLINV2                          34400    NOV/81      77
                                       CHLINV1                          34401    NOV/81      77
                                       CHLDECINV2                       34402    NOV/81      77
                                       CHLDECINV1                       34403    NOV/81      77
                             3.GENERAL SYMMETRIC MATRICES
  1.PREPARATORY PROCEDURES
                                       DECSYM2                          34291    NOV/81      79
  2.CALCULATION OF DETERMINANT
                                       DETERMSYM2                       34294    NOV/81      81
  3.SOLUTION OF LINEAR EQUATIONS
                                       SOLSYM2                          34292    NOV/81      83
                                       DECSOLSYM2                       34293    NOV/81      83
                        2.COMPLEX MATRICES
                   2.FULL RANK OVERDETERM SYSTEMS
                        1.REAL MATRICES
                             1.PREPARATORY PROCEDURES
                                       LSQORTDEC                        34134    NOV/81      85
                                       LSQDGLINV                        34132    NOV/81      85
                             2.LEAST SQUARES SOLUTION
                                       LSQSOL                           34131    NOV/81      87
                                       LSQORTDECSOL                     34135    NOV/81      87
                             3.INVERSE MATRIX OF NORMAL EQN.
                                       LSQINV                           34136    NOV/81      89
                             4.LEAST SQRS WITH LIN. CONSTR.
                                       LSQDECOMP                        34137    NOV/81      91
                                       LSQREFSOL                        34138    NOV/81      91
                        2.COMPLEX MATRICES
                   3.OTHER PROBLEMS
                        1.REAL MATRICES
                             1.SOLUTION OVERDETERMINED SYST
                                       SOLSVDOVR                        34280    NOV/81      93
                                       SOLOVR                           34281    NOV/81      93
                             2.SOLUTION UNDERDETERM SYSTEMS
                                       SOLSVDUND                        34282    NOV/81      95
                                       SOLUND                           34283    NOV/81      95
                             3.SOLUTION HOMOGENEOUS EQUATION
    3.   1.   1.   3.   1.   3.                                        HOMSOLSVD                        34284    NOV/81      97
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    3.   1.   1.   3.   1.   3.                                        HOMSOL                           34285    NOV/81      97
                             4.PSEUDO-INVERSION
                                       PSDINVSVD                        34286    NOV/81      99
                                       PSDINV                           34287    NOV/81      99
                        2.COMPLEX MATRICES
              2.SPARSE MATRICES
                   1.DIRECT METHODS
                        1.REAL MATRICES
                             1.NON-SYMMETRIC MATRICES
  1.BAND MATRICES
       1.PREPARATORY PROCEDURES
                                       DECBND                           34320    NOV/81     101
       2.CALCULATION OF DETERMINANT
                                       DETERMBND                        34321    NOV/81     103
       3.SOLUTION OF LINEAR EQUATIONS
                                       SOLBND                           34071    NOV/81     105
                                       DECSOLBND                        34322    NOV/81     105
  2.TRIDIAGONAL MATRICES
       1.PREPARATORY PROCEDURES
                                       DECTRI                           34423    NOV/81     107
                                       DECTRIPIV                        34426    NOV/81     107
       2.CALCULATION OF DETERMINANT
       3.SOLUTION OF LINEAR EQUATIONS
                                       SOLTRI                           34424    NOV/81     109
                                       DECSOLTRI                        34425    NOV/81     109
                                       SOLTRIPIV                        34427    NOV/81     109
                                       DECSOLTRIPIV                     34428    NOV/81     109
  3.BLOC-TRIDIAGONAL MATRICES
                             2.SYMMETRIC POS DEF MATRICES
  1.BAND MATRICES
       1.PREPARATORY PROCEDURES
                                       CHLDECBND                        34330    NOV/81     111
       2.CALCULATION OF DETERMINANT
                                       CHLDETERMBND                     34331    NOV/81     113
       3.SOLUTION OF LINEAR EQUATIONS
                                       CHLSOLBND                        34332    NOV/81     115
                                       CHLDECSOLBND                     34333    NOV/81     115
  2.TRIDIAGONAL MATRICES
       1.PREPARATORY PROCEDURES
                                       DECSYMTRI                        34420    NOV/81     117
       2.CALCULATION OF DETERMINANT
       3.SOLUTION OF LINEAR EQUATIONS
                                       SOLSYMTRI                        34421    NOV/81     119
                                       DECSOLSYMTRI                     34422    NOV/81     119
  3.BLOC-TRIDIAGONAL MATRICES
                        2.COMPLEX MATRICES
                   2.ITERATIVE METHODS
                        1.REAL MATRICES
                                       CONJ GRAD                        34220    NOV/81     121
                        2.COMPLEX MATRICES
         2.TRANSFORMATION TO SPECIAL FORM
              1.SIMILARITY TRANSFORMATIONS
                   1.EQUILIBRATION
                        1.REAL MATRICES
    3.   2.   1.   1.   1.                                             EQILBR                           34173    NOV/81     123
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    3.   2.   1.   1.   1.                                             BAKLBR                           34174    NOV/81     123
                        2.COMPLEX MATRICES
                                       EQILBRCOM                        34361    NOV/81     125
                                       BAKLBRCOM                        34362    NOV/81     125
                   2.TRANSF TO HESSENBERG FORM
                        1.REAL MATRICES
                             1.SYMMETRIC MATRICES
                                       TFMSYMTRI2                       34140    NOV/81     127
                                       BAKSYMTRI2                       34141    NOV/81     127
                                       TFMPREVEC                        34142    NOV/81     127
                                       TFMSYMTRI1                       34143    NOV/81     127
                                       BAKSYMTRI1                       34144    NOV/81     127
                             2.ASYMMETRIC MATRICES
                                       TFMREAHES                        34170    NOV/81     129
                                       BAKREAHES1                       34171    NOV/81     129
                                       BAKREAHES2                       34172    NOV/81     129
                        2.COMPLEX MATRICES
                             1.HERMITIAN MATRICES
                                       HSHHRMTRI                        34363    NOV/81     131
                                       HSHHRMTRIVAL                     34364    NOV/81     131
                                       BAKHRMTRI                        34365    NOV/81     131
                             2.NON-HERMITIAN MATRICES
                                       HSHCOMHES                        34366    NOV/81     133
                                       BAKCOMHES                        34367    NOV/81     133
              2.OTHER TRANSFORMATIONS
                   1.TRANSF TO BIDIAGONAL FORM
                        1.REAL MATRICES
                                       HSHREABID                        34260    NOV/81     135
                                       PSTTFMMAT                        34261    NOV/81     135
                                       PRETFMMAT                        34262    NOV/81     135
                        2.COMPLEX MATRICES
         3.THE (ORDINARY) EIGENV PROBLEM
              1.REAL MATRICES
                   1.SYMMETRIC MATRICES
                        1.TRIDIAGONAL MATRICES
                                       VALSYMTRI                        34151    NOV/81     137
                                       VECSYMTRI                        34152    NOV/81     137
                                       QRIVALSYMTRI                     34160    NOV/81     137
                                       QRISYMTRI                        34161    NOV/81     137
                        2.FULL MATRICES
                                       EIGVALSYM2                       34153    NOV/81     139
                                       EIGSYM2                          34154    NOV/81     139
                                       EIGVALSYM1                       34155    NOV/81     139
                                       EIGSYM1                          34156    NOV/81     139
                                       QRIVALSYM2                       34162    NOV/81     139
                                       QRISYM                           34163    NOV/81     139
                                       QRIVALSYM1                       34164    NOV/81     139
                        3.ITERATIVE IMPROVEMENT
                             1.AUXILIARY PROCEDURES
                                       MERGESORT                        36405    NOV/81     141
                                       VECPERM                          36404    NOV/81     141
                                       ROWPERM                          36403    NOV/81     141
                             2.ORTHOGONALISATION
                                       ORTHOG                           36402    NOV/81     143
    3.   3.   1.   1.   3.   3.IMPROVEMENT AND ERRORBOUNDS
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    3.   3.   1.   1.   3.   3.                                        SYMEIGINP                        36401    NOV/81     145
                   2.ASYMMETRIC MATRICES
                        1.MATRICES IN HESSENBERG FORM
                                       REAVALQRI                        34180    NOV/81     147
                                       REAVECHES                        34181    NOV/81     147
                                       REAQRI                           34186    NOV/81     147
                                       COMVALQRI                        34190    NOV/81     147
                                       COMVECHES                        34191    NOV/81     147
                        2.FULL MATRICES
                                       REAEIGVAL                        34182    NOV/81     149
                                       REAEIG1                          34184    NOV/81     149
                                       REAEIG3                          34187    NOV/81     149
                                       COMEIGVAL                        34192    NOV/81     149
                                       COMEIG1                          34194    NOV/81     149
              2.COMPLEX MATRICES
                   1.HERMITIAN MATRICES
                                       EIGVALHRM                        34368    NOV/81     151
                                       EIGHRM                           34369    NOV/81     151
                                       QRIVALHRM                        34370    NOV/81     151
                                       QRIHRM                           34371    NOV/81     151
                   2.NON-HERMITIAN MATRICES
                        1.MATRICES IN HESSENBERG FORM
                                       VALQRICOM                        34372    NOV/81     153
                                       QRICOM                           34373    NOV/81     153
                        2.FULL MATRICES
                                       EIGVALCOM                        34374    NOV/81     155
                                       EIGCOM                           34375    NOV/81     155
         4.THE GENERALIZED EIGENV PROBLEM
              1.REAL MATRICES
                   1.SYMMETRIC MATRICES
                   2.ASYMMETRIC MATRICES
                                       QZIVAL                           34600    NOV/81     157
                                       QZI                              34601    NOV/81     157
                                       HSHDECMUL                        34602    NOV/81     157
                                       HESTGL3                          34603    NOV/81     157
                                       HESTGL2                          34604    NOV/81     157
                                       HSH2COL                          34605    NOV/81     157
                                       HSH3COL                          34606    NOV/81     157
                                       HSH2ROW3                         34607    NOV/81     157
                                       HSH2ROW2                         34608    NOV/81     157
                                       HSH3ROW3                         34609    NOV/81     157
                                       HSH3ROW2                         34610    NOV/81     157
         5.SINGULAR VALUES
              1.REAL MATRICES
                   1.BIDIAGONAL MATRICES
                                       QRISNGVALBID                     34270    NOV/81     159
                                       QRISNGVALDECBID                  34271    NOV/81     159
                   2.FULL MATRICES
                                       QRISNGVAL                        34272    NOV/81     161
                                       QRISNGVALDEC                     34273    NOV/81     161
              2.COMPLEX MATRICES
         6.ZEROS OF POLYNOMIALS
              1.ZEROS OF GENERAL REAL POLYNOM.
                                       ZERPOL                           34501    NOV/81     163
    3.   6.   1.                                                       BOUNDS                           34502    NOV/81     163
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    3.   6.   2.ZEROS OF ORTHOGONAL POLYNOM.
                                       ALLZERORTPOL                     31362    NOV/81     165
                                       LUPZERORTPOL                     31363    NOV/81     165
                                       SELZERORTPOL                     31364    NOV/81     165
                                       ALLJACZER                        31370    NOV/81     165
                                       ALLLAGZER                        31371    NOV/81     165
              3.ZEROS OF COMPLEX POLYNOMIALS
                                       COMKWD                           34345    NOV/81     167
    4.ANALYTIC EVALUATIONS
         1.EVAL. OF AN INFINITE SERIES
                                       EULER                            32010    NOV/81     169
                                       SUMPOSSERIES                     32020    NOV/81     169
         2.QUADRATURE
              1.ONE-DIMENSIONAL QUADRATURE
                                       QADRAT                           32070    NOV/81     171
                                       INTEGRAL                         32051    NOV/81     173
              2.MULTIDIMENSIONAL QUADRATURE
                                       TRICUB                           32075    NOV/81     175
              3.GAUSSIAN QUADRATURE
                   1.GENERAL WEIGHTS
                                       RECCOF                           31254    NOV/81     177
                                       GSSWTS                           31253    NOV/81     177
                                       GSSWTSSYM                        31252    NOV/81     177
                   2.SPECIAL WEIGHTS
                                       GSSJACWGHTS                      31425    NOV/81     179
                                       GSSLAGWGHTS                      31427    NOV/81     179
         3.NUMERICAL DIFFERENTIATION
              1.FUNCTIONS OF ONE VARIABLE
              2.FUNCTIONS OF MORE VARIABLES
                   1.CALC. WITH DIFFERENCE FORMULAS
                                       JACOBNNF                         34437    NOV/81     181
                                       JACOBNMF                         34438    NOV/81     181
                                       JACOBNBNDF                       34439    NOV/81     181
    5.ANALYTICAL PROBLEMS
         1.ANALYTICAL EQUATIONS
              1.NON-LINEAR EQUATIONS
                   1.A SINGLE EQUATION
                        1.NO DERIVATIVE AVAILABLE
                                       ZEROIN                           34150    NOV/81     183
                                       ZEROINRAT                        34436    NOV/81     183
                        2.DERIVATIVE AVAILABLE
                                       ZEROINDER                        34453    NOV/81     185
                   2.A SYSTEM OF EQUATIONS
                        1.AUXILIARY PROCEDURES
                        2.JACOBIAN MATRIX NOT AVAILABLE
                                       QUANEWBND                        34430    NOV/81     187
                                       QUANEWBND1                       34431    NOV/81     187
                        3.JACOBIAN MATRIX AVAILABLE
                   3.POLYNOMIAL EQUATIONS
                     SEE ALSO SECTION 3.6
              2.UNCONSTRAINED OPTIMIZATION
                   1.FUNCTIONS OF ONE VARIABLE
                        1.DERIVATIVE NOT AVAILABLE
                                       MININ                            34433    NOV/81     189
    5.   1.   2.   1.   2.DERIVATIVE AVAILABLE
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    5.   1.   2.   1.   2.                                             MININDER                         34435    NOV/81     191
                   2.FUNCTIONS OF MORE VARIABLES
                        1.AUXILIARY PROCEDURES
                                       LINEMIN                          34210    NOV/81     193
                                       RNK1UPD                          34211    NOV/81     193
                                       DAVUPD                           34212    NOV/81     193
                                       FLEUPD                           34213    NOV/81     193
                        2.NO DERIVATIVES AVAILABLE
                                       PRAXIS                           34432    NOV/81     195
                        3.GRADIENT AVAILABLE
                                       RNK1MIN                          34214    NOV/81     197
                                       FLEMIN                           34215    NOV/81     197
                        4.GRADIENT & JACOBIAN AVAILABLE
              3.OVERDETERMINED NONLINEAR SYST.
                   1.LEAST SQUARES SOLUTIONS
                     SEE ALSO SECTION 7.
                        1.AUXILIARY PROCEDURES
                        2.JACOBIAN MATRIX NOT AVAILABLE
                          SEE ALSO SECTION 5.1.2.2.2.
                        3.JACOBIAN MATRIX AVAILABLE
                                       MARQUARDT                        34440    NOV/81     199
                                       GSSNEWTON                        34441    NOV/81     199
         2.FUNCTIONAL EQUATIONS
              1.DIFFERENTIAL EQUATIONS
                   1.INITIAL VALUE PROBLEMS
                        1.FIRST ORDER ORDINARY D.E.
                             1.NO DERIVATIVES RHS AVAILABLE
                                       RK1                              33010    NOV/81     201
                                       RKE                              33033    NOV/81     203
                                       RK4A                             33016    NOV/81     205
                                       RK4NA                            33017    NOV/81     207
                                       RK5NA                            33018    NOV/81     209
                                       MULTISTEP                        33080    NOV/81     211
                                       DIFFSYS                          33180    NOV/81     213
                                       ARK                              33061    NOV/81     215
                                       EFRK                             33070    NOV/81     217
                             2.JACOBIAN MATRIX AVAILABLE
                                       EFSIRK                           33160    NOV/81     219
                                       EFERK                            33120    NOV/81     221
                                       LINIGER1VS                       33132    NOV/81     223
                                       LINIGER2                         33131    NOV/81     225
                                       GMS                              33191    NOV/81     227
                                       IMPEX                            33135    NOV/81     229
                               SEE ALSO PROC. MULTISTEP (5.2.1.1.1.1)
                             3.SEVERAL DERIVATIVES AVAILABLE
                                       MODIFIED TAYLOR                  33040    NOV/81     231
                                       EXPONENTIALLY FITTED TAYLOR      33050    NOV/81     233
                        2.SECOND ORDER ORDINARY D.E.
                             1.NO DERIVATIVES RHS AVAILABLE
                                       RK2                              33012    NOV/81     235
                                       RK2N                             33013    NOV/81     237
                                       RK3                              33014    NOV/81     239
                                       RK3N                             33015    NOV/81     241
                             2.SEVERAL DERIV. RHS AVAILABLE
    5.   2.   1.   1.   3.INITIAL BOUNDARY VALUE PROBLEM
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    5.   2.   1.   1.   3.                                             ARKMAT                           33066    NOV/81     243
                   2.BOUNDARY VALUE PROBLEMS
                        1.TWO POINT B.V.P.
                             1.SHOOTING METHODS
                               SEE ALSO SECTION 5.2.1.3.1
                             2.LINEAR GLOBAL METHODS
  1.SECOND ORDER TPBVP
       1.SELF ADJOINT TPBVP
                                       FEM LAG SYM                      33300    NOV/81     245
                                       FEM LAG                          33301    NOV/81     245
                                       FEM LAG SPHER                    33308    NOV/81     245
       2.SKEW ADJOINT TPBVP
                                       FEM LAG SKEW                     33302    NOV/81     247
  2.FOURTH ORDER TPBVP
       1.SELF ADJOINT TPBVP
                                       FEM HERM SYM                     33303    NOV/81     249
       2.SKEW ADJOINT TPBVP
                             3.NON-LINEAR GLOBAL METHODS
                                       NON LIN FEM LAG SKEW             33314    NOV/81     251
                        2.TWO-DIMENSIONAL B.V.P.
                             1.ELLIPTIC B.V.P.S
  1.DISCRETIZATION PROCEDURES
  2.SPECIAL LINEAR SYSTEMS
                                       RICHARDSON                       33170    NOV/81     253
                                       ELIMINATION                      33171    NOV/81     253
    SEE ALSO SECTION 3.1.2
  3.SPECIAL NON-LINEAR SYSTEMS
                             2.PARABOLIC & HYPERBOLIC B.V.P.S
                        3.MULTI-DIMENSIONAL B.V.P.
                        4.OVER-DETERMINED PROBLEMS
                   3.PARAMETER ESTIMATION IN D.E.
                        1.P.E. IN INITIAL VALUE PROBLEMS
                                       PEIDE                            34444    NOV/81     255
              2.INTEGRAL EQUATIONS
              3.INTEGRO- DIFFERENTIAL EQS
              4.DIFFERENCE EQUATIONS
              5.CONVOLUTION EQUATIONS
    6.SPECIAL FUNCTIONS & CONSTANTS
         1.MATHEMATICAL CONSTANTS
                                       PI                           *   30006    NOV/81     257
                                       E                            *   30007    NOV/81     257
         2.MACHINE CONSTANTS
                                       MBASE                        *   30001    NOV/81     259
                                       ARREB                        *   30002    NOV/81     259
                                       DWARF                        *   30003    NOV/81     259
                                       GIANT                        *   30004    NOV/81     259
                                       INTCAP                       *   30005    NOV/81     259
                                       OVERFLOW                     *   30008    NOV/81     259
                                       UNDERFLOW                    *   30009    NOV/81     259
         3.RANDOM NUMBERS
                                       RANDOM                           30010    NOT YET AVAILABLE
                                       SETRANDOM                        30011    NOT YET AVAILABLE
         4.ELEMENTARY FUNCTIONS
              1.CIRCULAR FUNCTIONS
    6.   4.   1.                                                       TAN                              35120    NOV/81     261
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    6.   4.   1.                                                       ARCSIN                           35121    NOV/81     261
                                       ARCCOS                           35122    NOV/81     261
              2.HYPERBOLIC FUNCTIONS
                                       SINH                             35111    NOV/81     263
                                       COSH                             35112    NOV/81     263
                                       TANH                             35113    NOV/81     263
                                       ARCSINH                          35114    NOV/81     263
                                       ARCCOSH                          35115    NOV/81     263
                                       ARCTANH                          35116    NOV/81     263
              3.LOGARITHMIC FUNCTION
                                       LOG ONE PLUS X                   35130    NOV/81     265
         5.EXPONENTIAL INTEGRAL, ETC.
              1.EXPONENTIAL INTEGRAL
                                       EI                               35080    NOV/81     267
                                       EI ALPHA                         35081    NOV/81     267
                                       ENX                              35086    NOV/81     267
                                       NONEXP ENX                       35087    NOV/81     267
              2.SINE AND COSINE INTEGRAL
                                       SINCOSINT                        35084    NOV/81     269
                                       SINCOSFG                         35085    NOV/81     269
         6.GAMMA FUNCTION, ETC.
                                       GAMMA                            35061    NOV/81     271
                                       RECIP GAMMA                      35060    NOV/81     271
                                       LOG GAMMA                        35062    NOV/81     271
                                       INCOMGAM                         35030    NOV/81     271
                                       INCBETA                          35050    NOV/81     271
                                       IBPPLUSN                         35051    NOV/81     271
                                       IBQPLUSN                         35052    NOV/81     271
                                       IXQFIX                           35053    NOV/81     271
                                       IXPFIX                           35054    NOV/81     271
                                       FORWARD                          35055    NOV/81     271
                                       BACKWARD                         35056    NOV/81     271
         7.ERROR FUNCTION, ETC.
                                       ERRORFUNCTION                    35021    NOV/81     273
                                       NONEXPERFC                       35022    NOV/81     273
                                       INVERSE ERROR FUNCTION           35023    NOV/81     273
                                       FRESNEL                          35027    NOV/81     273
                                       FG                               35028    NOV/81     273
         8.LEGENDRE FUNCTIONS
         9.BESSEL FUNCTIONS OF INT. ORDER
              1.BESSEL FUNCTIONS J AND Y
                                       BESS J0                          35160    NOV/81     275
                                       BESS J1                          35161    NOV/81     275
                                       BESS J                           35162    NOV/81     275
                                       BESS Y01                         35163    NOV/81     275
                                       BESS Y                           35164    NOV/81     275
                                       BESS PQ0                         35165    NOV/81     275
                                       BESS PQ1                         35166    NOV/81     275
              2.BESSEL FUNCTIONS I AND K
                                       BESS I0                          35170    NOV/81     277
                                       BESS I1                          35171    NOV/81     277
                                       BESS I                           35172    NOV/81     277
                                       BESS K01                         35173    NOV/81     277
                                       BESS K                           35174    NOV/81     277
    6.   9.   2.                                                       NONEXP BESS I0                   35175    NOV/81     277
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    6.   9.   2.                                                       NONEXP BESS I1                   35176    NOV/81     277
                                       NONEXP BESS I                    35177    NOV/81     277
                                       NONEXP BESS K01                  35178    NOV/81     277
                                       NONEXP BESS K                    35179    NOV/81     277
              3.KELVIN FUNCTIONS
        10.BESSEL FUNCTIONS OF REAL ORDER
              1.BESSEL FUNCTIONS J AND Y
                                       BESS JAPLUSN                     35180    NOV/81     279
                                       BESS YA01                        35181    NOV/81     279
                                       BESS YAPLUSN                     35182    NOV/81     279
                                       BESS PQA01                       35183    NOV/81     279
                                       BESS ZEROS                       35184    NOV/81     279
                                       START                            35185    NOV/81     279
              2.BESSEL FUNCTIONS I AND K
                                       BESS IAPLUSN                     35190    NOV/81     281
                                       BESS KA01                        35191    NOV/81     281
                                       BESS KAPLUSN                     35192    NOV/81     281
                                       NONEXP BESS IAPLUSN              35193    NOV/81     281
                                       NONEXP BESS KA01                 35194    NOV/81     281
                                       NONEXP BESS KAPLUSN              35195    NOV/81     281
              3.SPHERICAL BESSEL FUNCTIONS
                                       SPHER BESS J                     35150    NOV/81     283
                                       SPHER BESS Y                     35151    NOV/81     283
                                       SPHER BESS I                     35152    NOV/81     283
                                       SPHER BESS K                     35153    NOV/81     283
                                       NONEXP SPHER BESS I              35154    NOV/81     283
                                       NONEXP SPHER BESS K              35155    NOV/81     283
              4.AIRY FUNCTIONS
                                       AIRY                             35140    NOV/81     285
                                       AIRYZEROS                        35145    NOV/81     285
    7.INTERPOLATION & APPROXIMATION
         1.REAL DATA IN ONE DIMENSION
              1.INTERPOLATION, WITH
                   1.POLYNOMIALS
                        1.GENERAL POLYNOMIALS
                                       NEWTON                           36010    NOV/81     287
                        2.ORTHOGONAL POLYNOMIALS
                   2.SPLINES
                        1.GENERAL SPLINES
                        2.NATURAL SPLINES
                   3.TRIGONOMETRIC SERIES
                        1.FOURIER SERIES
                        2.SINE SERIES
                        3.COSINE SERIES
                   4.RATIONAL FUNCTIONS
                   5.EXPONENTIAL FUNCTIONS
              2.APPROXIMATION IN 2-NORM, WITH
                   1.GENERAL FUNCTIONS
                     SEE ALSO SECTION 5.1.3.1
                   2.POLYNOMIALS
                        1.GENERAL POLYNOMIALS
                        2.ORTHOGONAL POLYNOMIALS
                   3.SPLINES
                        1.GENERAL SPLINES
    7.   1.   2.   3.   2.NATURAL SPLINES
1                    INDEX                                             PROCEDURE                        CODE     MNT/YR    RECORD
                                                                                           NUMBER
 
 
 
    7.   1.   2.   4.TRIGONOMETRIC SERIES
                   5.RATIONAL FUNCTIONS
                   6.EXPONENTIAL FUNCTIONS
              3.APPROXIMATION IN INF-NORM,WITH
                   1.GENERAL FUNCTIONS
                   2.POLYNOMIALS
                        1.GENERAL POLYNOMIALS
                                       INI                              36020    NOV/81     289
                                       SNDREMEZ                         36021    NOV/81     289
                                       MINMAXPOL                        36022    NOV/81     289
                        2.ORTHOGONAL POLYNOMIALS
                   3.TRIGONOMETRIC SERIES
                   4.RATIONAL FUNCTIONS
              4.APPROXIMATION IN 1-NORM, WITH
                   1.GENERAL FUNCTIONS
                   2.POLYNOMIALS
         2.REAL DATA IN MORE DIMENSIONS
         3.REAL FUNCTIONS IN 1 DIMENSION
              1.POLYNOMIALS
    8.NUMBER THEORY
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VERSION: 811214
1
 
 OBSOLETE PROCEDURES
 
 PROCEDURE                     CODE       WITHDRAWAL   EXPIRATION      REPLACED BY
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 VERSION: 811214
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