*DECK DQAGE SUBROUTINE DQAGE (F, A, B, EPSABS, EPSREL, KEY, LIMIT, RESULT, + ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST) C***BEGIN PROLOGUE DQAGE C***PURPOSE The routine calculates an approximation result to a given C definite integral I = Integral of F over (A,B), C hopefully satisfying following claim for accuracy C ABS(I-RESLT).LE.MAX(EPSABS,EPSREL*ABS(I)). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A1 C***TYPE DOUBLE PRECISION (QAGE-S, DQAGE-D) C***KEYWORDS AUTOMATIC INTEGRATOR, GAUSS-KRONROD RULES, C GENERAL-PURPOSE, GLOBALLY ADAPTIVE, INTEGRAND EXAMINATOR, C QUADPACK, QUADRATURE 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 Computation of a definite integral 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 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 KEY - Integer C Key for choice of local integration rule C A Gauss-Kronrod pair is used with C 7 - 15 points if KEY.LT.2, C 10 - 21 points if KEY = 2, C 15 - 31 points if KEY = 3, C 20 - 41 points if KEY = 4, C 25 - 51 points if KEY = 5, C 30 - 61 points if KEY.GT.5. C C LIMIT - Integer C Gives an upper bound on the number of subintervals C in the partition of (A,B), LIMIT.GE.1. 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 result and error are C less reliable. It is assumed that the C requested accuracy has not been achieved. C ERROR MESSAGES C IER = 1 Maximum number of subdivisions allowed C has been achieved. One can allow more C subdivisions by increasing the value C of LIMIT. C However, if this yields no improvement it C is rather advised to analyze the integrand C in order to determine the integration C difficulties. If the position of a local C difficulty can be determined(e.g. C SINGULARITY, DISCONTINUITY within the C interval) one will probably gain from C splitting up the interval at this point C and calling the integrator on the C subranges. If possible, an appropriate C special-purpose integrator should be used C which is designed for handling the type of C difficulty involved. 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 (EPSABS.LE.0 and C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28), C RESULT, ABSERR, NEVAL, LAST, RLIST(1) , C ELIST(1) and IORD(1) are set to zero. C ALIST(1) 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 C integral 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 elements of which are pointers to the C error estimates over the subintervals, C such that ELIST(IORD(1)), ..., C ELIST(IORD(K)) form a decreasing sequence, C with K = LAST if LAST.LE.(LIMIT/2+2), and C K = LIMIT+1-LAST otherwise C C LAST - Integer C Number of subintervals actually produced in the C subdivision process C C***REFERENCES (NONE) C***ROUTINES CALLED D1MACH, DQK15, DQK21, DQK31, DQK41, DQK51, DQK61, C 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 DQAGE C DOUBLE PRECISION A,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1,A2,B, 1 BLIST,B1,B2,DEFABS,DEFAB1,DEFAB2,D1MACH,ELIST,EPMACH, 2 EPSABS,EPSREL,ERRBND,ERRMAX,ERROR1,ERROR2,ERRO12,ERRSUM,F, 3 RESABS,RESULT,RLIST,UFLOW INTEGER IER,IORD,IROFF1,IROFF2,K,KEY,KEYF,LAST,LIMIT,MAXERR,NEVAL, 1 NRMAX C DIMENSION ALIST(*),BLIST(*),ELIST(*),IORD(*), 1 RLIST(*) C EXTERNAL F 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 DQAGE EPMACH = D1MACH(4) UFLOW = D1MACH(1) C C TEST ON VALIDITY OF PARAMETERS C ------------------------------ C IER = 0 NEVAL = 0 LAST = 0 RESULT = 0.0D+00 ABSERR = 0.0D+00 ALIST(1) = A BLIST(1) = B RLIST(1) = 0.0D+00 ELIST(1) = 0.0D+00 IORD(1) = 0 IF(EPSABS.LE.0.0D+00.AND. 1 EPSREL.LT.MAX(0.5D+02*EPMACH,0.5D-28)) IER = 6 IF(IER.EQ.6) GO TO 999 C C FIRST APPROXIMATION TO THE INTEGRAL C ----------------------------------- C KEYF = KEY IF(KEY.LE.0) KEYF = 1 IF(KEY.GE.7) KEYF = 6 NEVAL = 0 IF(KEYF.EQ.1) CALL DQK15(F,A,B,RESULT,ABSERR,DEFABS,RESABS) IF(KEYF.EQ.2) CALL DQK21(F,A,B,RESULT,ABSERR,DEFABS,RESABS) IF(KEYF.EQ.3) CALL DQK31(F,A,B,RESULT,ABSERR,DEFABS,RESABS) IF(KEYF.EQ.4) CALL DQK41(F,A,B,RESULT,ABSERR,DEFABS,RESABS) IF(KEYF.EQ.5) CALL DQK51(F,A,B,RESULT,ABSERR,DEFABS,RESABS) IF(KEYF.EQ.6) CALL DQK61(F,A,B,RESULT,ABSERR,DEFABS,RESABS) LAST = 1 RLIST(1) = RESULT ELIST(1) = ABSERR IORD(1) = 1 C C TEST ON ACCURACY. C ERRBND = MAX(EPSABS,EPSREL*ABS(RESULT)) IF(ABSERR.LE.0.5D+02*EPMACH*DEFABS.AND.ABSERR.GT.ERRBND) IER = 2 IF(LIMIT.EQ.1) IER = 1 IF(IER.NE.0.OR.(ABSERR.LE.ERRBND.AND.ABSERR.NE.RESABS) 1 .OR.ABSERR.EQ.0.0D+00) GO TO 60 C C INITIALIZATION C -------------- C C ERRMAX = ABSERR MAXERR = 1 AREA = RESULT ERRSUM = ABSERR NRMAX = 1 IROFF1 = 0 IROFF2 = 0 C C MAIN DO-LOOP C ------------ C DO 30 LAST = 2,LIMIT C C BISECT THE SUBINTERVAL WITH THE LARGEST ERROR ESTIMATE. C A1 = ALIST(MAXERR) B1 = 0.5D+00*(ALIST(MAXERR)+BLIST(MAXERR)) A2 = B1 B2 = BLIST(MAXERR) IF(KEYF.EQ.1) CALL DQK15(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.2) CALL DQK21(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.3) CALL DQK31(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.4) CALL DQK41(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.5) CALL DQK51(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.6) CALL DQK61(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) IF(KEYF.EQ.1) CALL DQK15(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) IF(KEYF.EQ.2) CALL DQK21(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) IF(KEYF.EQ.3) CALL DQK31(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) IF(KEYF.EQ.4) CALL DQK41(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) IF(KEYF.EQ.5) CALL DQK51(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) IF(KEYF.EQ.6) CALL DQK61(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) C C IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL C AND ERROR AND TEST FOR ACCURACY. C NEVAL = NEVAL+1 AREA12 = AREA1+AREA2 ERRO12 = ERROR1+ERROR2 ERRSUM = ERRSUM+ERRO12-ERRMAX AREA = AREA+AREA12-RLIST(MAXERR) IF(DEFAB1.EQ.ERROR1.OR.DEFAB2.EQ.ERROR2) GO TO 5 IF(ABS(RLIST(MAXERR)-AREA12).LE.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 5 RLIST(MAXERR) = AREA1 RLIST(LAST) = AREA2 ERRBND = MAX(EPSABS,EPSREL*ABS(AREA)) IF(ERRSUM.LE.ERRBND) GO TO 8 C C TEST FOR ROUNDOFF ERROR AND EVENTUALLY SET ERROR FLAG. C IF(IROFF1.GE.6.OR.IROFF2.GE.20) IER = 2 C C SET ERROR FLAG IN THE CASE THAT THE NUMBER OF SUBINTERVALS C EQUALS LIMIT. C IF(LAST.EQ.LIMIT) IER = 1 C C SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR C AT A POINT OF THE INTEGRATION RANGE. C IF(MAX(ABS(A1),ABS(B2)).LE.(0.1D+01+0.1D+03* 1 EPMACH)*(ABS(A2)+0.1D+04*UFLOW)) IER = 3 C C APPEND THE NEWLY-CREATED INTERVALS TO THE LIST. C 8 IF(ERROR2.GT.ERROR1) GO TO 10 ALIST(LAST) = A2 BLIST(MAXERR) = B1 BLIST(LAST) = B2 ELIST(MAXERR) = ERROR1 ELIST(LAST) = ERROR2 GO TO 20 10 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 THE LARGEST ERROR ESTIMATE (TO BE BISECTED NEXT). C 20 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 40 30 CONTINUE C C COMPUTE FINAL RESULT. C --------------------- C 40 RESULT = 0.0D+00 DO 50 K=1,LAST RESULT = RESULT+RLIST(K) 50 CONTINUE ABSERR = ERRSUM 60 IF(KEYF.NE.1) NEVAL = (10*KEYF+1)*(2*NEVAL+1) IF(KEYF.EQ.1) NEVAL = 30*NEVAL+15 999 RETURN END