*DECK DQAGSE SUBROUTINE DQAGSE (F, A, B, EPSABS, EPSREL, LIMIT, RESULT, ABSERR, + NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST) C***BEGIN PROLOGUE DQAGSE 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-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A1 C***TYPE DOUBLE PRECISION (QAGSE-S, DQAGSE-D) C***KEYWORDS AUTOMATIC INTEGRATOR, END POINT SINGULARITIES, C EXTRAPOLATION, GENERAL-PURPOSE, GLOBALLY ADAPTIVE, 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 LIMIT - Integer C Gives an upper bound on the number of subintervals C in the partition of (A,B) 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 integral and error are C 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 sub- C divisions by increasing the value of LIMIT C (and taking the according dimension C adjustments into account). However, if C this yields no improvement it is advised C to analyze the integrand in order to C determine the integration difficulties. If C the position of a local difficulty can be C determined (e.g. singularity, C discontinuity within the interval) one C will probably gain from splitting up the C interval at this point and calling the C integrator on the subranges. If possible, C an appropriate special-purpose integrator C should be used, which is designed for C handling the type of difficulty involved. C = 2 The occurrence of roundoff error is detec- C ted, which prevents the requested C tolerance from being achieved. C The error may be under-estimated. C = 3 Extremely bad integrand behaviour C occurs at some points of the integration C interval. C = 4 The algorithm does not converge. C Roundoff error is detected in the C extrapolation table. C It is presumed that the requested C tolerance cannot be achieved, and that the C returned result is the best which can be C obtained. C = 5 The integral is probably divergent, or C slowly convergent. It must be noted that C divergence can occur with any other value C of IER. 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 IORD(1) and ELIST(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 end points C of the subintervals in the partition of the C 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 end points C of the subintervals in the partition of the given C 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 elements of which are pointers to the C error estimates over the subintervals, C such that ELIST(IORD(1)), ..., ELIST(IORD(K)) C form a decreasing sequence, with K = LAST C If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST C otherwise C C LAST - Integer C Number of subintervals actually produced in the C subdivision process C C***REFERENCES (NONE) C***ROUTINES CALLED D1MACH, DQELG, DQK21, 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 DQAGSE C DOUBLE PRECISION A,ABSEPS,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1, 1 A2,B,BLIST,B1,B2,CORREC,DEFABS,DEFAB1,DEFAB2,D1MACH, 2 DRES,ELIST,EPMACH,EPSABS,EPSREL,ERLARG,ERLAST,ERRBND,ERRMAX, 3 ERROR1,ERROR2,ERRO12,ERRSUM,ERTEST,F,OFLOW,RESABS,RESEPS,RESULT, 4 RES3LA,RLIST,RLIST2,SMALL,UFLOW INTEGER ID,IER,IERRO,IORD,IROFF1,IROFF2,IROFF3,JUPBND,K,KSGN, 1 KTMIN,LAST,LIMIT,MAXERR,NEVAL,NRES,NRMAX,NUMRL2 LOGICAL EXTRAP,NOEXT C DIMENSION ALIST(*),BLIST(*),ELIST(*),IORD(*), 1 RES3LA(3),RLIST(*),RLIST2(52) C EXTERNAL F C C THE DIMENSION OF RLIST2 IS DETERMINED BY THE VALUE OF C LIMEXP IN SUBROUTINE DQELG (RLIST2 SHOULD BE OF DIMENSION C (LIMEXP+2) AT LEAST). 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 RLIST2 - ARRAY OF DIMENSION AT LEAST LIMEXP+2 CONTAINING C THE PART OF THE EPSILON TABLE WHICH IS STILL C NEEDED FOR FURTHER COMPUTATIONS C ELIST(I) - ERROR ESTIMATE APPLYING TO RLIST(I) C MAXERR - POINTER TO THE INTERVAL WITH LARGEST ERROR C ESTIMATE C ERRMAX - ELIST(MAXERR) C ERLAST - ERROR ON THE INTERVAL CURRENTLY SUBDIVIDED C (BEFORE THAT SUBDIVISION HAS TAKEN PLACE) 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 INTERVAL C *****2 - VARIABLE FOR THE RIGHT INTERVAL C LAST - INDEX FOR SUBDIVISION C NRES - NUMBER OF CALLS TO THE EXTRAPOLATION ROUTINE C NUMRL2 - NUMBER OF ELEMENTS CURRENTLY IN RLIST2. IF AN C APPROPRIATE APPROXIMATION TO THE COMPOUNDED C INTEGRAL HAS BEEN OBTAINED IT IS PUT IN C RLIST2(NUMRL2) AFTER NUMRL2 HAS BEEN INCREASED C BY ONE. C SMALL - LENGTH OF THE SMALLEST INTERVAL CONSIDERED UP C TO NOW, MULTIPLIED BY 1.5 C ERLARG - SUM OF THE ERRORS OVER THE INTERVALS LARGER C THAN THE SMALLEST INTERVAL CONSIDERED UP TO NOW C EXTRAP - LOGICAL VARIABLE DENOTING THAT THE ROUTINE IS C ATTEMPTING TO PERFORM EXTRAPOLATION I.E. BEFORE C SUBDIVIDING THE SMALLEST INTERVAL WE TRY TO C DECREASE THE VALUE OF ERLARG. C NOEXT - LOGICAL VARIABLE DENOTING THAT EXTRAPOLATION C IS NO LONGER ALLOWED (TRUE VALUE) C C MACHINE DEPENDENT CONSTANTS C --------------------------- C C EPMACH IS THE LARGEST RELATIVE SPACING. C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE. C OFLOW IS THE LARGEST POSITIVE MAGNITUDE. C C***FIRST EXECUTABLE STATEMENT DQAGSE EPMACH = D1MACH(4) C C TEST ON VALIDITY OF PARAMETERS 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 IF(EPSABS.LE.0.0D+00.AND.EPSREL.LT.MAX(0.5D+02*EPMACH,0.5D-28)) 1 IER = 6 IF(IER.EQ.6) GO TO 999 C C FIRST APPROXIMATION TO THE INTEGRAL C ----------------------------------- C UFLOW = D1MACH(1) OFLOW = D1MACH(2) IERRO = 0 CALL DQK21(F,A,B,RESULT,ABSERR,DEFABS,RESABS) C C TEST ON ACCURACY. C DRES = ABS(RESULT) ERRBND = MAX(EPSABS,EPSREL*DRES) LAST = 1 RLIST(1) = RESULT ELIST(1) = ABSERR IORD(1) = 1 IF(ABSERR.LE.1.0D+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).OR. 1 ABSERR.EQ.0.0D+00) GO TO 140 C C INITIALIZATION C -------------- C RLIST2(1) = RESULT ERRMAX = ABSERR MAXERR = 1 AREA = RESULT ERRSUM = ABSERR ABSERR = OFLOW NRMAX = 1 NRES = 0 NUMRL2 = 2 KTMIN = 0 EXTRAP = .FALSE. NOEXT = .FALSE. IROFF1 = 0 IROFF2 = 0 IROFF3 = 0 KSGN = -1 IF(DRES.GE.(0.1D+01-0.5D+02*EPMACH)*DEFABS) KSGN = 1 C C MAIN DO-LOOP C ------------ C DO 90 LAST = 2,LIMIT C C BISECT THE SUBINTERVAL WITH THE NRMAX-TH LARGEST ERROR C ESTIMATE. C A1 = ALIST(MAXERR) B1 = 0.5D+00*(ALIST(MAXERR)+BLIST(MAXERR)) A2 = B1 B2 = BLIST(MAXERR) ERLAST = ERRMAX CALL DQK21(F,A1,B1,AREA1,ERROR1,RESABS,DEFAB1) CALL DQK21(F,A2,B2,AREA2,ERROR2,RESABS,DEFAB2) C C IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL C AND ERROR AND TEST FOR ACCURACY. C 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 15 IF(ABS(RLIST(MAXERR)-AREA12).GT.0.1D-04*ABS(AREA12) 1 .OR.ERRO12.LT.0.99D+00*ERRMAX) GO TO 10 IF(EXTRAP) IROFF2 = IROFF2+1 IF(.NOT.EXTRAP) IROFF1 = IROFF1+1 10 IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF3 = IROFF3+1 15 RLIST(MAXERR) = AREA1 RLIST(LAST) = AREA2 ERRBND = MAX(EPSABS,EPSREL*ABS(AREA)) C C TEST FOR ROUNDOFF ERROR AND EVENTUALLY SET ERROR FLAG. C IF(IROFF1+IROFF2.GE.10.OR.IROFF3.GE.20) IER = 2 IF(IROFF2.GE.5) IERRO = 3 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*EPMACH)* 1 (ABS(A2)+0.1D+04*UFLOW)) IER = 4 C C APPEND THE NEWLY-CREATED INTERVALS TO THE LIST. C IF(ERROR2.GT.ERROR1) GO TO 20 ALIST(LAST) = A2 BLIST(MAXERR) = B1 BLIST(LAST) = B2 ELIST(MAXERR) = ERROR1 ELIST(LAST) = ERROR2 GO TO 30 20 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 NRMAX-TH LARGEST ERROR ESTIMATE (TO BE BISECTED NEXT). C 30 CALL DQPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX) C ***JUMP OUT OF DO-LOOP IF(ERRSUM.LE.ERRBND) GO TO 115 C ***JUMP OUT OF DO-LOOP IF(IER.NE.0) GO TO 100 IF(LAST.EQ.2) GO TO 80 IF(NOEXT) GO TO 90 ERLARG = ERLARG-ERLAST IF(ABS(B1-A1).GT.SMALL) ERLARG = ERLARG+ERRO12 IF(EXTRAP) GO TO 40 C C TEST WHETHER THE INTERVAL TO BE BISECTED NEXT IS THE C SMALLEST INTERVAL. C IF(ABS(BLIST(MAXERR)-ALIST(MAXERR)).GT.SMALL) GO TO 90 EXTRAP = .TRUE. NRMAX = 2 40 IF(IERRO.EQ.3.OR.ERLARG.LE.ERTEST) GO TO 60 C C THE SMALLEST INTERVAL HAS THE LARGEST ERROR. C BEFORE BISECTING DECREASE THE SUM OF THE ERRORS OVER THE C LARGER INTERVALS (ERLARG) AND PERFORM EXTRAPOLATION. C ID = NRMAX JUPBND = LAST IF(LAST.GT.(2+LIMIT/2)) JUPBND = LIMIT+3-LAST DO 50 K = ID,JUPBND MAXERR = IORD(NRMAX) ERRMAX = ELIST(MAXERR) C ***JUMP OUT OF DO-LOOP IF(ABS(BLIST(MAXERR)-ALIST(MAXERR)).GT.SMALL) GO TO 90 NRMAX = NRMAX+1 50 CONTINUE C C PERFORM EXTRAPOLATION. C 60 NUMRL2 = NUMRL2+1 RLIST2(NUMRL2) = AREA CALL DQELG(NUMRL2,RLIST2,RESEPS,ABSEPS,RES3LA,NRES) KTMIN = KTMIN+1 IF(KTMIN.GT.5.AND.ABSERR.LT.0.1D-02*ERRSUM) IER = 5 IF(ABSEPS.GE.ABSERR) GO TO 70 KTMIN = 0 ABSERR = ABSEPS RESULT = RESEPS CORREC = ERLARG ERTEST = MAX(EPSABS,EPSREL*ABS(RESEPS)) C ***JUMP OUT OF DO-LOOP IF(ABSERR.LE.ERTEST) GO TO 100 C C PREPARE BISECTION OF THE SMALLEST INTERVAL. C 70 IF(NUMRL2.EQ.1) NOEXT = .TRUE. IF(IER.EQ.5) GO TO 100 MAXERR = IORD(1) ERRMAX = ELIST(MAXERR) NRMAX = 1 EXTRAP = .FALSE. SMALL = SMALL*0.5D+00 ERLARG = ERRSUM GO TO 90 80 SMALL = ABS(B-A)*0.375D+00 ERLARG = ERRSUM ERTEST = ERRBND RLIST2(2) = AREA 90 CONTINUE C C SET FINAL RESULT AND ERROR ESTIMATE. C ------------------------------------ C 100 IF(ABSERR.EQ.OFLOW) GO TO 115 IF(IER+IERRO.EQ.0) GO TO 110 IF(IERRO.EQ.3) ABSERR = ABSERR+CORREC IF(IER.EQ.0) IER = 3 IF(RESULT.NE.0.0D+00.AND.AREA.NE.0.0D+00) GO TO 105 IF(ABSERR.GT.ERRSUM) GO TO 115 IF(AREA.EQ.0.0D+00) GO TO 130 GO TO 110 105 IF(ABSERR/ABS(RESULT).GT.ERRSUM/ABS(AREA)) GO TO 115 C C TEST ON DIVERGENCE. C 110 IF(KSGN.EQ.(-1).AND.MAX(ABS(RESULT),ABS(AREA)).LE. 1 DEFABS*0.1D-01) GO TO 130 IF(0.1D-01.GT.(RESULT/AREA).OR.(RESULT/AREA).GT.0.1D+03 1 .OR.ERRSUM.GT.ABS(AREA)) IER = 6 GO TO 130 C C COMPUTE GLOBAL INTEGRAL SUM. C 115 RESULT = 0.0D+00 DO 120 K = 1,LAST RESULT = RESULT+RLIST(K) 120 CONTINUE ABSERR = ERRSUM 130 IF(IER.GT.2) IER = IER-1 140 NEVAL = 42*LAST-21 999 RETURN END