SUBROUTINE DINT1 (A,B,ANSWER,WORK,IOPT) c Copyright (c) 1996 California Institute of Technology, Pasadena, CA. c ALL RIGHTS RESERVED. c Based on Government Sponsored Research NAS7-03001. C>> 1996-03-31 DINT1 Krogh Removed unused variable in common. c>> 1995-11-20 DINT1 Krogh Converted from SFTRAN to Fortran 77. c>> 1994-10-19 DINT1 Krogh Changes to use M77CON c>> 1994-07-07 DINT1 Snyder set up for CHGTYP. C>> 1993-05-18 DINT1 Krogh -- Changed "END" to "END PROGRAM" C>> 1991-09-20 DINT1 Krogh converted '(1)' dimensioning to '(*)'. C>> 1987-11-19 DINT1 Snyder Initial code. C c--D replaces "?": ?INT1, ?INTA, ?intc, ?intec, ?INTF, ?INTOP, ?INTNC C ****************************************************************** C C THIS SUBROUTINE ATTEMPTS TO CALCULATE THE INTEGRAL OF A FUNCTION C OVER THE INTERVAL A TO B TO WITHIN A TOLERANCE LEVEL SPECIFIED C BY THE USER. THE FUNCTION IS PROVIDED BY THE USER VIA A C SUBPROGRAM REFERENCED BY CALL DINTF (F,X,IFLAG), OR BY REVERSE C COMMUNICATION. IN THE CASE OF ONE-DIMENSIONAL QUADRATURE, IFLAG C IS ALWAYS ZERO. ALL ABSCISSAE ARE WITHIN THE INTERVAL OF C INTEGRATION. C C THE RESULT IS OBTAINED USING QUADRATURE FORMULAE DUE TO C T. N. L. PATTERSON, MATHEMATICS OF COMPUTATION, VOLUME 22, C PAGES 847-856, 1968. C C ***** WARNING *********************************************** C C THE RELIABILITY AND EFFICIENCY OF THIS PROGRAM ARE STRONGLY C INFLUENCED BY DISCONTINUITIES IN THE FUNCTION OR IT'S C DERIVATIVES, INTEGRABLE SINGULARITIES ON THE INTERVAL OF C INTEGRATION, AND NON-INTEGRABLE SINGULARITIES NEAR THIS INTERVAL C (INCLUDING COMPLEX VALUES). THE EFFICIENCY AND RELIABILITY C OF INTEGRATING SUCH FUNCTIONS MAY BE GREATLY IMPROVED BY MANUALLY C SUBDIVIDING THE INTERVAL OF INTEGRATION AT THE DISCONTINUITY, C SINGULARITY OR CLOSEST POINT TO A POLE, AND SUMMING THE ANSWERS. C A CHANGE OF VARIABLE TO ELIMINATE OR REDUCE THE STRENGTH OF THE C SINGULARITY WILL SIGNIFICANTLY IMPROVE PERFORMANCE. C C ***** FORMAL ARGUMENTS ************************************* C C A LOWER LIMIT OF INTERVAL OF INTEGRATION. DOUBLE PRECISION A C B UPPER LIMIT OF INTERVAL OF INTEGRATION. DOUBLE PRECISION B C ANSWER IS THE ESTIMATE OF THE INTEGRAL. WHEN REVERSE COMMUNICATION C IS SPECIFIED IT IS USED TO PASS FUNCTION VALUES FROM C THE USER PROGRAM TO DINTA. DOUBLE PRECISION ANSWER C WORK IS A WORKING STORAGE VECTOR USED BY OPTIONS (SEE IOPT). C WHEN THE INTEGRATION IS COMPLETE, WORK(1) CONTAINS THE C ESTIMATED ABSOLUTE ERROR. WHEN REVERSE COMMUNICATION C IS SPECIFIED, WORK(1) IS USED TO PASS ABSCISSAE FROM C DINTA TO THE USER PROGRAM. DOUBLE PRECISION WORK(*) C IOPT IS A VECTOR OF INTEGERS USED TO RETURN THE STATUS AND TO C SPECIFY OPTIONS. INTEGER IOPT(*) C IOPT(1) IS USED TO RETURN A STATUS INDICATOR TO THE USER. C VALUES OF THE STATUS INDICATOR ARE C -1 - NORMAL TERMINATION, EITHER THE ABSOLUTE OR THE RELATIVE C ERROR CRITERIA ARE SATISFIED. C -2 - NORMAL TERMINATION, NEITHER THE ABSOLUTE NOR THE C RELATIVE ERROR CRITERIA ARE SATISFIED, BUT THE RELATIVE C TO THE OBTAINABLE ERROR CRITERION IS SATISFIED. C -3 - NORMAL TERMINATION, BUT NONE OF THE ERROR CRITERIA ARE C SATISFIED. C +4 - BAD IOPT VALUE. C +5 - TOO MANY FUNCTION VALUES NEEDED. C +6 - APPARENT NON-INTEGRABLE SINGULARITY NEAR ANSWER. C ENTRIES IN IOPT STARTING AT IOPT(2) ARE DESCRIBED BELOW. C IOPT(I) ENTRY IN IOPT(I) MEANS C 0 NO MORE OPTIONS, BEGIN INTEGRATION. C 1 IOPT(I+1) CONTAINS THE UNIT NUMBER FOR OUTPUT. C 2 IOPT(I+1) IS DIAGNOSTIC PRINT LEVEL C 0 - NO PRINTING C 1 - MINIMUM PRINTING - ERROR MESSAGES (DEFAULT) C 2 - PANEL BOUNDARIES AND ANSWERS C 3 - ERROR ESTIMATES FOR EACH QUADRATURE FORMULA C 4 - DETAILED OUTPUT (DIFFERENCE LINES, ETC). C 3 WORK(IOPT(I+1)) CONTAINS THE ABSOLUTE TOLERANCE, C WORK(IOPT(I+1)+1) CONTAINS TOLERANCE RELATIVE TO C THE LOCALLY OBTAINABLE PRECISION, AND C WORK(IOPT(I+1)+2) CONTAINS TOLERANCE RELATIVE TO C THE VALUE OF THE INTEGRAL. THE TOLERANCE RELATIVE C TO THE LOCALLY OBTAINABLE PRECISION IS SPECIFIED AS C THE FRACTION OF LOCALLY OBTAINABLE DIGITS THAT ARE C PERMITTED TO BE WRONG, AND IS INTERNALLY BOUNDED C BETWEEN 0.0 AND 1.0. IF ANY OF THE ERROR CONTROL C CRITERIA ARE SATISFIED, THE ANSWER IS ACCEPTED. IF C THIS OPTION IS NOT USED, THE ABSOLUTE AND RELATIVE C TOLERANCES ARE SET TO ZERO, AND THE TOLERANCE C RELATIVE TO THE LOCALLY OBTAINABLE PRECISION IS SET C TO 0.25. C 4 WORK(IOPT(I+1)) CONTAINS ABSOLUTE ERROR COMMITTED C COMPUTING F. WORK(IOPT(I+1)) MAY BE CHANGED DURING C THE INTEGRATION. C 5 WORK(IOPT(I+1)) CONTAINS RELATIVE ERROR EXPECTED TO C BE COMMITTED COMPUTING F. CHANGES TO THIS VALUE C DURING INTEGRATION WILL NOT BE DETECTED. C 6 USE REVERSE COMMUNICATION - C CALL DINT1 (A,B,ANSWER,WORK,IOPT) C 1 CALL DINTA (ANSWER,WORK,IFLAG) C IF (IFLAG.NE.0) GO TO 2 C ANSWER=F(WORK(1)) C GO TO 1 C 2 CONTINUE C IFLAG AT THIS POINT CONTAINS THE SAME VALUE AS C THE STATUS FLAG WOULD (IF SELECTED) FOR THE C FORWARD COMMUNICATION METHOD. C 7 SET MINIMUM INDEX OF QUADRATURE FORMULA TO USE C BEFORE SUBDIVISION TO IOPT(I+1). C 8 NO EFFECT. MAY BE USED TO PASS INFORMATION INTO C DINTF. INCREMENT I BY 2. C 9 IOPT(I+1) IS THE MAXIMUM NUMBER OF FUNCTION VALUES C TO USE TO INTEGRATE THE ENTIRE PROBLEM. IF C IOPT(I+1) .LE. 0, THE NUMBER OF FUNCTION VALUES C IS NOT CONTROLLED. C 10 IOPT(I+1) IS USED TO RETURN THE NUMBER OF FUNCTION C VALUES USED TO INTEGRATE THE ENTIRE PROBLEM. C 11 WORK(IABS(IOPT(I+1))) IS THE LOCATION OF A C SINGULARITY OR DISCONTINUITY. IF THE LOCATION IS C INSIDE THE INTERVAL, THE INTERVAL WILL BE C IMMEDIATELY SUBDIVIDED. IF IOPT(I+1).GT.0, A C T**2 SUBSTITUTION BASED AT WORK(...) WILL BE USED. C IF IOPT(I+1) .LT. 0 A T**4 SUBSTITUTION WILL BE C USED. IF IOPT(I+1) .EQ. 0, NO SUBSTITUTION WILL BE C USED. C 12 WORK(IOPT(I+1)) CONTAINS THE ABSOLUTE ERROR IN THE C LOWER LIMIT OF THE CURRENT DIMENSION. THE ERROR IN C THE UPPER LIMIT IS IN WORK(IOPT(I+1)+1). C 13 IS USED ONLY FOR MULTIDIMENSIONAL PROBLEMS. C C ALL OPTIONS ARE SET TO THEIR NOMINAL VALUES BEFORE THE OPTION C VECTOR IS PROCESSED. C C ***** EXTERNAL REFERENCES ********************************* C C DINTA TO PERFORM THE INTEGRATION. C DINTOP TO SET OPTIONS. C C ***** COMMON STORAGE ************************************** C C Common blocks are: DINTNC, DINTC, and DINTEC C COMMON /DINTNC/ CONTAINS VARIABLES NOT SEPARATELY SAVED FOR C EACH DIMENSION OF A MULTIPLE QUADRATURE. COMMON /DINTC/ C CONTAINS VARIABLES THAT MUST BE SAVED FOR EACH DIMENSION OF THE C QUADRATURE. THE VARIABLES IN EACH COMMON BLOCK ARE STORED IN THE C ORDER - ALWAYS DOUBLE, DOUBLE IF DOUBLE PRECISION PROGRAM, DOUBLE C IF DOUBLE PRECISION PROGRAM AND EXPONENT RANGE OF DOUBLE AND C SINGLE VERY DIFFERENT, SINGLE, INTEGER, LOGICAL. A PAD OF LOGICAL C VARIABLES IS INCLUDED AT THE END OF /DINTC/. THE DIMENSION OF C THE PAD MAY NEED TO BE VARIED SO THAT NO VARIABLES BEYOND THE END C OF THE COMMON BLOCK ARE ALTERED. C C DECLARATIONS OF COMMON /DINTNC/ VARIABLES. C DOUBLE PRECISION AINIT, BINIT, FNCVAL, S, TP DOUBLE PRECISION FER, FER1, RELOBT, TPS, XJ, XJP INTEGER FEA, FEA1, INC, INC2, IPRINT, 1 ISTOP(2,2),JPRINT, KDIM, KK, KMAXF, NDIM, 2 NFINDX, NFMAX, NFMAXM, RELTOL, REVERM, REVERS, 3 WHEREM LOGICAL NEEDH C C DECLARATIONS OF COMMON /DINTC/ VARIABLES. C c--D Next line special: S => D, X => Q, D => D, P => D DOUBLE PRECISION ACUM, PACUM, RESULT(2) C 139 $.TYPE.$ VARIABLES DOUBLE PRECISION 1 AACUM, ABSCIS, DELMIN, DELTA, DIFF, DISCX(2), 2 END(2), ERRINA, ERRINB, FAT(2), FSAVE, 3 FUNCT(24), F1, F2, LOCAL(4), PAACUM, PF1, 4 PF2, PHISUM, PHTSUM, PX, SPACE(6), 5 STEP(2), START(2), SUM, T, TA, TASAVE, 6 TB, TEND, WORRY(2), X, X1, 7 X2, XT(17), FT(17), PHI(34) c Note XT, FT, and PHI above are last, because they must be in adjacent c locations in DINTC. C 30 $DSTYP$ VARIABLES DOUBLE PRECISION 1 ABSDIF, COUNT, EDUE2A, EDUE2B, EP, EPNOIZ, 2 EPS, EPSMAX, EPSMIN, EPSO, EPSR, EPSS, 3 ERR, ERRAT(2), ERRC, ERRF, ERRI, ERRT(2), 4 ESOLD, EXTRA, PEPSMN, RE, RELEPS, REP, 5 REPROD, RNDC, TLEN, XJUMP C 29 INTEGER VARIABLES INTEGER DISCF, DISCHK, ENDPTS, I, INEW, 1 IOLD, IP, IXKDIM, J, J1, J1OLD, 2 J2, J2OLD, K, KAIMT, KMAX, KMIN, 3 L, LENDT, NFEVAL, NFJUMP, NSUB, NSUBSV, 4 NXKDIM, PART, SEARCH, TALOC, WHERE, WHERE2 C 11 TO 18 LOGICALS (7 ARE PADDING). LOGICAL DID1, FAIL, FATS(2), FSAVED, HAVDIF, 1 IEND, INIT, ROUNDF, XCDOBT(2), PAD(7) C C THE COMMON BLOCKS. C COMMON /DINTNC/ c 1 2 3 4 5 6 7 8 W AINIT, BINIT, FNCVAL, S, TP, FER, FER1, RELOBT, c 9 10 11 12 13 1 2 3 X TPS, XJ, XJP, FEA, FEA1, KDIM, INC, INC2, c 4 (2,2) 8 9 10 11 12 13 14 Y ISTOP, JPRINT, IPRINT, KK, KMAXF, NDIM, NFINDX, NFMAX, c 15 16 17 18 19 20 Z NFMAXM, RELTOL, REVERM, REVERS, WHEREM, NEEDH COMMON /DINTC/ 1 ACUM, PACUM, RESULT COMMON /DINTC/ c 1 2 (4) 6 7 8 9 10 11 (2) 1 AACUM, LOCAL, ABSCIS, TA, DELTA, DELMIN, DIFF, DISCX, c 13 (2) 15 16 17 (2) 19 20 (24) 44 2 END, ERRINA, ERRINB, FAT, FSAVE, FUNCT, F2, c 45 46 47 48 49 50 51 (6) 3 PAACUM, PF1, PF2, PHISUM, PHTSUM, PX, SPACE, c 57 (2) 59 (2) 61 62 63 64 65 4 STEP, START, SUM, T, TASAVE, TB, TEND, c 66 (2) 68 69 70 71 72 5 WORRY, X1, X2, X, F1, COUNT, c 73 (17) 90 (17) 107 (34) 6 XT, FT, PHI COMMON /DINTC/ c 141 142 143 144 145 146 1 ABSDIF, EDUE2A, EDUE2B, EP, EPNOIZ, EPSMAX, c 147 148 149 150 (2) 152 153 2 EPSO, EPSR, EPSS, ERRAT, ERRC, ERRF, c 154 (2) 156 157 158 159 160 3 ERRT, ESOLD, EXTRA, PEPSMN, RELEPS, REP, c 161 162 163 4 RNDC, TLEN, XJUMP, c 164 165 166 167 168 169 5 ERRI, ERR, EPSMIN, EPS, RE, REPROD COMMON /DINTC/ c 170 171 172 1 DISCF, DISCHK, ENDPTS, INEW, IOLD, IP, IXKDIM, 2 J, J1, J1OLD, J2, J2OLD, KMAX, KMIN, 3 L, LENDT, NFJUMP, NSUBSV, NXKDIM, TALOC, WHERE2, c 1 2 3 4 5 6 7 8 4 I, K, KAIMT, NSUB, PART, SEARCH, WHERE, NFEVAL COMMON /DINTC/ 1 DID1, FAIL, FATS, FSAVED, HAVDIF, IEND, INIT, ROUNDF, 2 XCDOBT, PAD SAVE /DINTNC/, /DINTC/ C C THE VARIABLES HERE DEFINE THE MACHINE ENVIRONMENT. ALL ARE SET C IN DINTOP. THE MEANING ATTACHED TO THESE VARIABLES CAN BE C FOUND BY LOOKING AT THE DEFINITIONS IN DINTOP. DOUBLE PRECISION 1 EMEPS, EEPSM8, EDELM2, EDELM3, ESQEPS, ERSQEP, ERSQE6, EMINF, 2 ESMALL, ENZER, EDELM1, ENINF COMMON /DINTEC/ 1 EMEPS, EEPSM8, EDELM2, EDELM3, ESQEPS, ERSQEP, ERSQE6, EMINF, 2 ESMALL, ENZER, EDELM1, ENINF SAVE /DINTEC/ C C ***** PROCEDURES ****************************************** C WHERE=0 NDIM=1 KDIM=1 NFEVAL=0 CALL DINTOP (IOPT,WORK) IF (IOPT(1).EQ.0) THEN C C CALL DINTA TO DO THE INTEGRATION. C AINIT=A BINIT=B IF (REVERS.EQ.0) CALL DINTA (ANSWER,WORK,IOPT) END IF RETURN C END .