*DECK DQAWSE SUBROUTINE DQAWSE (F, A, B, ALFA, BETA, INTEGR, EPSABS, EPSREL, + LIMIT, RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, + IORD, LAST) C***BEGIN PROLOGUE DQAWSE C***PURPOSE The routine calculates an approximation result to a given C definite integral I = Integral of F*W over (A,B), C (where W shows a singular behaviour at the end points, C see parameter INTEGR). C Hopefully satisfying following claim for accuracy C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A2A1 C***TYPE DOUBLE PRECISION (QAWSE-S, DQAWSE-D) C***KEYWORDS ALGEBRAIC-LOGARITHMIC END POINT SINGULARITIES, C AUTOMATIC INTEGRATOR, CLENSHAW-CURTIS METHOD, QUADPACK, C QUADRATURE, SPECIAL-PURPOSE C***AUTHOR Piessens, Robert C Applied Mathematics and Programming Division C K. U. Leuven C de Doncker, Elise C Applied Mathematics and Programming Division C K. U. Leuven C***DESCRIPTION C C Integration of functions having algebraico-logarithmic C end point singularities C Standard fortran subroutine C Double precision version C C PARAMETERS C ON ENTRY C F - Double precision C Function subprogram defining the integrand C function F(X). The actual name for F needs to be C declared E X T E R N A L in the driver program. C C A - Double precision C Lower limit of integration C C B - Double precision C Upper limit of integration, B.GT.A C If B.LE.A, the routine will end with IER = 6. C C ALFA - Double precision C Parameter in the WEIGHT function, ALFA.GT.(-1) C If ALFA.LE.(-1), the routine will end with C IER = 6. C C BETA - Double precision C Parameter in the WEIGHT function, BETA.GT.(-1) C If BETA.LE.(-1), the routine will end with C IER = 6. C C INTEGR - Integer C Indicates which WEIGHT function is to be used C = 1 (X-A)**ALFA*(B-X)**BETA C = 2 (X-A)**ALFA*(B-X)**BETA*LOG(X-A) C = 3 (X-A)**ALFA*(B-X)**BETA*LOG(B-X) C = 4 (X-A)**ALFA*(B-X)**BETA*LOG(X-A)*LOG(B-X) C If INTEGR.LT.1 or INTEGR.GT.4, the routine C will end with IER = 6. C C EPSABS - Double precision C Absolute accuracy requested C EPSREL - Double precision C Relative accuracy requested C If EPSABS.LE.0 C and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28), C the routine will end with IER = 6. C C LIMIT - Integer C Gives an upper bound on the number of subintervals C in the partition of (A,B), LIMIT.GE.2 C If LIMIT.LT.2, the routine will end with IER = 6. C C ON RETURN C RESULT - Double precision C Approximation to the integral C C ABSERR - Double precision C Estimate of the modulus of the absolute error, C which should equal or exceed ABS(I-RESULT) C C NEVAL - Integer C Number of integrand evaluations C C IER - Integer C IER = 0 Normal and reliable termination of the C routine. It is assumed that the requested C accuracy has been achieved. C IER.GT.0 Abnormal termination of the routine C the estimates for the integral and error C are less reliable. It is assumed that the C requested accuracy has not been achieved. C ERROR MESSAGES C = 1 Maximum number of subdivisions allowed C has been achieved. One can allow more C subdivisions by increasing the value of C LIMIT. However, if this yields no C improvement, it is advised to analyze the C integrand in order to determine the C integration difficulties which prevent the C requested tolerance from being achieved. C In case of a jump DISCONTINUITY or a local C SINGULARITY of algebraico-logarithmic type C at one or more interior points of the C integration range, one should proceed by C splitting up the interval at these C points and calling the integrator on the C subranges. C = 2 The occurrence of roundoff error is C detected, which prevents the requested C tolerance from being achieved. C = 3 Extremely bad integrand behaviour occurs C at some points of the integration C interval. C = 6 The input is invalid, because C B.LE.A or ALFA.LE.(-1) or BETA.LE.(-1), or C INTEGR.LT.1 or INTEGR.GT.4, or C (EPSABS.LE.0 and C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28), C or LIMIT.LT.2. C RESULT, ABSERR, NEVAL, RLIST(1), ELIST(1), C IORD(1) and LAST are set to zero. ALIST(1) C and BLIST(1) are set to A and B C respectively. C C ALIST - Double precision C Vector of dimension at least LIMIT, the first C LAST elements of which are the left C end points of the subintervals in the partition C of the given integration range (A,B) C C BLIST - Double precision C Vector of dimension at least LIMIT, the first C LAST elements of which are the right C end points of the subintervals in the partition C of the given integration range (A,B) C C RLIST - Double precision C Vector of dimension at least LIMIT, the first C LAST elements of which are the integral C approximations on the subintervals C C ELIST - Double precision C Vector of dimension at least LIMIT, the first C LAST elements of which are the moduli of the C absolute error estimates on the subintervals C C IORD - Integer C Vector of dimension at least LIMIT, the first K C of which are pointers to the error C estimates over the subintervals, so that C ELIST(IORD(1)), ..., ELIST(IORD(K)) with K = LAST C If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST C otherwise form a decreasing sequence C C LAST - Integer C Number of subintervals actually produced in C the subdivision process C C***REFERENCES (NONE) C***ROUTINES CALLED D1MACH, DQC25S, DQMOMO, DQPSRT C***REVISION HISTORY (YYMMDD) C 800101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C***END PROLOGUE DQAWSE C DOUBLE PRECISION A,ABSERR,ALFA,ALIST,AREA,AREA1,AREA12,AREA2,A1, 1 A2,B,BETA,BLIST,B1,B2,CENTRE,D1MACH,ELIST,EPMACH, 2 EPSABS,EPSREL,ERRBND,ERRMAX,ERROR1,ERRO12,ERROR2,ERRSUM,F, 3 RESAS1,RESAS2,RESULT,RG,RH,RI,RJ,RLIST,UFLOW INTEGER IER,INTEGR,IORD,IROFF1,IROFF2,K,LAST,LIMIT,MAXERR,NEV, 1 NEVAL,NRMAX C EXTERNAL F C DIMENSION ALIST(*),BLIST(*),RLIST(*),ELIST(*), 1 IORD(*),RI(25),RJ(25),RH(25),RG(25) C C LIST OF MAJOR VARIABLES C ----------------------- C C ALIST - LIST OF LEFT END POINTS OF ALL SUBINTERVALS C CONSIDERED UP TO NOW C BLIST - LIST OF RIGHT END POINTS OF ALL SUBINTERVALS C CONSIDERED UP TO NOW C RLIST(I) - APPROXIMATION TO THE INTEGRAL OVER C (ALIST(I),BLIST(I)) C ELIST(I) - ERROR ESTIMATE APPLYING TO RLIST(I) C MAXERR - POINTER TO THE INTERVAL WITH LARGEST C ERROR ESTIMATE C ERRMAX - ELIST(MAXERR) C AREA - SUM OF THE INTEGRALS OVER THE SUBINTERVALS C ERRSUM - SUM OF THE ERRORS OVER THE SUBINTERVALS C ERRBND - REQUESTED ACCURACY MAX(EPSABS,EPSREL* C ABS(RESULT)) C *****1 - VARIABLE FOR THE LEFT SUBINTERVAL C *****2 - VARIABLE FOR THE RIGHT SUBINTERVAL C LAST - INDEX FOR SUBDIVISION C C C MACHINE DEPENDENT CONSTANTS C --------------------------- C C EPMACH IS THE LARGEST RELATIVE SPACING. C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE. C C***FIRST EXECUTABLE STATEMENT DQAWSE EPMACH = D1MACH(4) UFLOW = D1MACH(1) C C TEST ON VALIDITY OF PARAMETERS C ------------------------------ C IER = 6 NEVAL = 0 LAST = 0 RLIST(1) = 0.0D+00 ELIST(1) = 0.0D+00 IORD(1) = 0 RESULT = 0.0D+00 ABSERR = 0.0D+00 IF (B.LE.A.OR.(EPSABS.EQ.0.0D+00.AND. 1 EPSREL.LT.MAX(0.5D+02*EPMACH,0.5D-28)).OR.ALFA.LE.(-0.1D+01) 2 .OR.BETA.LE.(-0.1D+01).OR.INTEGR.LT.1.OR.INTEGR.GT.4.OR. 3 LIMIT.LT.2) GO TO 999 IER = 0 C C COMPUTE THE MODIFIED CHEBYSHEV MOMENTS. C CALL DQMOMO(ALFA,BETA,RI,RJ,RG,RH,INTEGR) C C INTEGRATE OVER THE INTERVALS (A,(A+B)/2) AND ((A+B)/2,B). C CENTRE = 0.5D+00*(B+A) CALL DQC25S(F,A,B,A,CENTRE,ALFA,BETA,RI,RJ,RG,RH,AREA1, 1 ERROR1,RESAS1,INTEGR,NEV) NEVAL = NEV CALL DQC25S(F,A,B,CENTRE,B,ALFA,BETA,RI,RJ,RG,RH,AREA2, 1 ERROR2,RESAS2,INTEGR,NEV) LAST = 2 NEVAL = NEVAL+NEV RESULT = AREA1+AREA2 ABSERR = ERROR1+ERROR2 C C TEST ON ACCURACY. C ERRBND = MAX(EPSABS,EPSREL*ABS(RESULT)) C C INITIALIZATION C -------------- C IF(ERROR2.GT.ERROR1) GO TO 10 ALIST(1) = A ALIST(2) = CENTRE BLIST(1) = CENTRE BLIST(2) = B RLIST(1) = AREA1 RLIST(2) = AREA2 ELIST(1) = ERROR1 ELIST(2) = ERROR2 GO TO 20 10 ALIST(1) = CENTRE ALIST(2) = A BLIST(1) = B BLIST(2) = CENTRE RLIST(1) = AREA2 RLIST(2) = AREA1 ELIST(1) = ERROR2 ELIST(2) = ERROR1 20 IORD(1) = 1 IORD(2) = 2 IF(LIMIT.EQ.2) IER = 1 IF(ABSERR.LE.ERRBND.OR.IER.EQ.1) GO TO 999 ERRMAX = ELIST(1) MAXERR = 1 NRMAX = 1 AREA = RESULT ERRSUM = ABSERR IROFF1 = 0 IROFF2 = 0 C C MAIN DO-LOOP C ------------ C DO 60 LAST = 3,LIMIT C C BISECT THE SUBINTERVAL WITH LARGEST ERROR ESTIMATE. C A1 = ALIST(MAXERR) B1 = 0.5D+00*(ALIST(MAXERR)+BLIST(MAXERR)) A2 = B1 B2 = BLIST(MAXERR) C CALL DQC25S(F,A,B,A1,B1,ALFA,BETA,RI,RJ,RG,RH,AREA1, 1 ERROR1,RESAS1,INTEGR,NEV) NEVAL = NEVAL+NEV CALL DQC25S(F,A,B,A2,B2,ALFA,BETA,RI,RJ,RG,RH,AREA2, 1 ERROR2,RESAS2,INTEGR,NEV) NEVAL = NEVAL+NEV C C IMPROVE PREVIOUS APPROXIMATIONS INTEGRAL AND ERROR C AND TEST FOR ACCURACY. C AREA12 = AREA1+AREA2 ERRO12 = ERROR1+ERROR2 ERRSUM = ERRSUM+ERRO12-ERRMAX AREA = AREA+AREA12-RLIST(MAXERR) IF(A.EQ.A1.OR.B.EQ.B2) GO TO 30 IF(RESAS1.EQ.ERROR1.OR.RESAS2.EQ.ERROR2) GO TO 30 C C TEST FOR ROUNDOFF ERROR. C IF(ABS(RLIST(MAXERR)-AREA12).LT.0.1D-04*ABS(AREA12) 1 .AND.ERRO12.GE.0.99D+00*ERRMAX) IROFF1 = IROFF1+1 IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF2 = IROFF2+1 30 RLIST(MAXERR) = AREA1 RLIST(LAST) = AREA2 C C TEST ON ACCURACY. C ERRBND = MAX(EPSABS,EPSREL*ABS(AREA)) IF(ERRSUM.LE.ERRBND) GO TO 35 C C SET ERROR FLAG IN THE CASE THAT THE NUMBER OF INTERVAL C BISECTIONS EXCEEDS LIMIT. C IF(LAST.EQ.LIMIT) IER = 1 C C C SET ERROR FLAG IN THE CASE OF ROUNDOFF ERROR. C IF(IROFF1.GE.6.OR.IROFF2.GE.20) IER = 2 C C SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR C AT INTERIOR POINTS OF INTEGRATION RANGE. C IF(MAX(ABS(A1),ABS(B2)).LE.(0.1D+01+0.1D+03*EPMACH)* 1 (ABS(A2)+0.1D+04*UFLOW)) IER = 3 C C APPEND THE NEWLY-CREATED INTERVALS TO THE LIST. C 35 IF(ERROR2.GT.ERROR1) GO TO 40 ALIST(LAST) = A2 BLIST(MAXERR) = B1 BLIST(LAST) = B2 ELIST(MAXERR) = ERROR1 ELIST(LAST) = ERROR2 GO TO 50 40 ALIST(MAXERR) = A2 ALIST(LAST) = A1 BLIST(LAST) = B1 RLIST(MAXERR) = AREA2 RLIST(LAST) = AREA1 ELIST(MAXERR) = ERROR2 ELIST(LAST) = ERROR1 C C CALL SUBROUTINE DQPSRT TO MAINTAIN THE DESCENDING ORDERING C IN THE LIST OF ERROR ESTIMATES AND SELECT THE SUBINTERVAL C WITH LARGEST ERROR ESTIMATE (TO BE BISECTED NEXT). C 50 CALL DQPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX) C ***JUMP OUT OF DO-LOOP IF (IER.NE.0.OR.ERRSUM.LE.ERRBND) GO TO 70 60 CONTINUE C C COMPUTE FINAL RESULT. C --------------------- C 70 RESULT = 0.0D+00 DO 80 K=1,LAST RESULT = RESULT+RLIST(K) 80 CONTINUE ABSERR = ERRSUM 999 RETURN END