*DECK DQAGPE SUBROUTINE DQAGPE (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, LIMIT, + RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, PTS, + IORD, LEVEL, NDIN, LAST) C***BEGIN PROLOGUE DQAGPE C***PURPOSE Approximate a given definite integral I = Integral of F C over (A,B), hopefully satisfying the accuracy claim: C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)). C Break points of the integration interval, where local C difficulties of the integrand may occur (e.g. singularities C or discontinuities) are provided by the user. C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A2A1 C***TYPE DOUBLE PRECISION (QAGPE-S, DQAGPE-D) C***KEYWORDS AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE, C GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE, C SINGULARITIES AT USER SPECIFIED POINTS 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 NPTS2 - Integer C Number equal to two more than the number of C user-supplied break points within the integration C range, NPTS2.GE.2. C If NPTS2.LT.2, the routine will end with IER = 6. C C POINTS - Double precision C Vector of dimension NPTS2, the first (NPTS2-2) C elements of which are the user provided break C POINTS. If these POINTS do not constitute an C ascending sequence there will be an automatic C sorting. 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.NPTS2 C If LIMIT.LT.NPTS2, the routine will end with C 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 integral 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 of C LIMIT (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 (i.e. SINGULARITY, C DISCONTINUITY within the interval), it C should be supplied to the routine as an C element of the vector points. If necessary C an appropriate special-purpose integrator C must be used, which is designed for C handling the type of difficulty involved. C = 2 The occurrence of roundoff error is C detected, which prevents the requested C tolerance from being achieved. C The error may be under-estimated. C = 3 Extremely bad integrand behaviour occurs C 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. It is presumed that C the requested tolerance cannot be C achieved, and that the returned result is C the best which can be 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.GT.0. C = 6 The input is invalid because C NPTS2.LT.2 or C Break points are specified outside C the integration range or C (EPSABS.LE.0 and C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28)) C or LIMIT.LT.NPTS2. C RESULT, ABSERR, NEVAL, LAST, RLIST(1), C and ELIST(1) are set to zero. ALIST(1) and C BLIST(1) are set to A and B 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 given C 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 PTS - Double precision C Vector of dimension at least NPTS2, containing the C integration limits and the break points of the C interval in ascending sequence. C C LEVEL - Integer C Vector of dimension at least LIMIT, containing the C subdivision levels of the subinterval, i.e. if C (AA,BB) is a subinterval of (P1,P2) where P1 as C well as P2 is a user-provided break point or C integration limit, then (AA,BB) has level L if C ABS(BB-AA) = ABS(P2-P1)*2**(-L). C C NDIN - Integer C Vector of dimension at least NPTS2, after first C integration over the intervals (PTS(I)),PTS(I+1), C I = 0,1, ..., NPTS2-2, the error estimates over C some of the intervals may have been increased C artificially, in order to put their subdivision C forward. If this happens for the subinterval C numbered K, NDIN(K) is put to 1, otherwise C NDIN(K) = 0. 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 subdivisions 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 DQAGPE DOUBLE PRECISION A,ABSEPS,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1, 1 A2,B,BLIST,B1,B2,CORREC,DEFABS,DEFAB1,DEFAB2, 2 DRES,D1MACH,ELIST,EPMACH,EPSABS,EPSREL,ERLARG,ERLAST,ERRBND, 3 ERRMAX,ERROR1,ERRO12,ERROR2,ERRSUM,ERTEST,F,OFLOW,POINTS,PTS, 4 RESA,RESABS,RESEPS,RESULT,RES3LA,RLIST,RLIST2,SIGN,TEMP,UFLOW INTEGER I,ID,IER,IERRO,IND1,IND2,IORD,IP1,IROFF1,IROFF2,IROFF3,J, 1 JLOW,JUPBND,K,KSGN,KTMIN,LAST,LEVCUR,LEVEL,LEVMAX,LIMIT,MAXERR, 2 NDIN,NEVAL,NINT,NINTP1,NPTS,NPTS2,NRES,NRMAX,NUMRL2 LOGICAL EXTRAP,NOEXT C C DIMENSION ALIST(*),BLIST(*),ELIST(*),IORD(*), 1 LEVEL(*),NDIN(*),POINTS(*),PTS(*),RES3LA(3), 2 RLIST(*),RLIST2(52) C EXTERNAL F C C THE DIMENSION OF RLIST2 IS DETERMINED BY THE VALUE OF C LIMEXP IN SUBROUTINE EPSALG (RLIST2 SHOULD BE OF DIMENSION C (LIMEXP+2) AT LEAST). C 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 C CONTAINING THE PART OF THE EPSILON TABLE WHICH C IS STILL 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 SUBINTERVAL C *****2 - VARIABLE FOR THE RIGHT SUBINTERVAL C LAST - INDEX FOR SUBDIVISION C NRES - NUMBER OF CALLS TO THE EXTRAPOLATION ROUTINE C NUMRL2 - NUMBER OF ELEMENTS IN RLIST2. IF AN APPROPRIATE C APPROXIMATION TO THE COMPOUNDED INTEGRAL HAS C BEEN OBTAINED, IT IS PUT IN RLIST2(NUMRL2) AFTER C NUMRL2 HAS BEEN INCREASED BY ONE. 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 C IS ATTEMPTING TO PERFORM EXTRAPOLATION. I.E. C BEFORE SUBDIVIDING THE SMALLEST INTERVAL WE C TRY TO DECREASE THE VALUE OF ERLARG. C NOEXT - LOGICAL VARIABLE DENOTING THAT EXTRAPOLATION IS C 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 DQAGPE EPMACH = D1MACH(4) 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 LEVEL(1) = 0 NPTS = NPTS2-2 IF(NPTS2.LT.2.OR.LIMIT.LE.NPTS.OR.(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 IF ANY BREAK POINTS ARE PROVIDED, SORT THEM INTO AN C ASCENDING SEQUENCE. C SIGN = 1.0D+00 IF(A.GT.B) SIGN = -1.0D+00 PTS(1) = MIN(A,B) IF(NPTS.EQ.0) GO TO 15 DO 10 I = 1,NPTS PTS(I+1) = POINTS(I) 10 CONTINUE 15 PTS(NPTS+2) = MAX(A,B) NINT = NPTS+1 A1 = PTS(1) IF(NPTS.EQ.0) GO TO 40 NINTP1 = NINT+1 DO 20 I = 1,NINT IP1 = I+1 DO 20 J = IP1,NINTP1 IF(PTS(I).LE.PTS(J)) GO TO 20 TEMP = PTS(I) PTS(I) = PTS(J) PTS(J) = TEMP 20 CONTINUE IF(PTS(1).NE.MIN(A,B).OR.PTS(NINTP1).NE.MAX(A,B)) IER = 6 IF(IER.EQ.6) GO TO 999 C C COMPUTE FIRST INTEGRAL AND ERROR APPROXIMATIONS. C ------------------------------------------------ C 40 RESABS = 0.0D+00 DO 50 I = 1,NINT B1 = PTS(I+1) CALL DQK21(F,A1,B1,AREA1,ERROR1,DEFABS,RESA) ABSERR = ABSERR+ERROR1 RESULT = RESULT+AREA1 NDIN(I) = 0 IF(ERROR1.EQ.RESA.AND.ERROR1.NE.0.0D+00) NDIN(I) = 1 RESABS = RESABS+DEFABS LEVEL(I) = 0 ELIST(I) = ERROR1 ALIST(I) = A1 BLIST(I) = B1 RLIST(I) = AREA1 IORD(I) = I A1 = B1 50 CONTINUE ERRSUM = 0.0D+00 DO 55 I = 1,NINT IF(NDIN(I).EQ.1) ELIST(I) = ABSERR ERRSUM = ERRSUM+ELIST(I) 55 CONTINUE C C TEST ON ACCURACY. C LAST = NINT NEVAL = 21*NINT DRES = ABS(RESULT) ERRBND = MAX(EPSABS,EPSREL*DRES) IF(ABSERR.LE.0.1D+03*EPMACH*RESABS.AND.ABSERR.GT.ERRBND) IER = 2 IF(NINT.EQ.1) GO TO 80 DO 70 I = 1,NPTS JLOW = I+1 IND1 = IORD(I) DO 60 J = JLOW,NINT IND2 = IORD(J) IF(ELIST(IND1).GT.ELIST(IND2)) GO TO 60 IND1 = IND2 K = J 60 CONTINUE IF(IND1.EQ.IORD(I)) GO TO 70 IORD(K) = IORD(I) IORD(I) = IND1 70 CONTINUE IF(LIMIT.LT.NPTS2) IER = 1 80 IF(IER.NE.0.OR.ABSERR.LE.ERRBND) GO TO 999 C C INITIALIZATION C -------------- C RLIST2(1) = RESULT MAXERR = IORD(1) ERRMAX = ELIST(MAXERR) AREA = RESULT NRMAX = 1 NRES = 0 NUMRL2 = 1 KTMIN = 0 EXTRAP = .FALSE. NOEXT = .FALSE. ERLARG = ERRSUM ERTEST = ERRBND LEVMAX = 1 IROFF1 = 0 IROFF2 = 0 IROFF3 = 0 IERRO = 0 UFLOW = D1MACH(1) OFLOW = D1MACH(2) ABSERR = OFLOW KSGN = -1 IF(DRES.GE.(0.1D+01-0.5D+02*EPMACH)*RESABS) KSGN = 1 C C MAIN DO-LOOP C ------------ C DO 160 LAST = NPTS2,LIMIT C C BISECT THE SUBINTERVAL WITH THE NRMAX-TH LARGEST ERROR C ESTIMATE. C LEVCUR = LEVEL(MAXERR)+1 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,RESA,DEFAB1) CALL DQK21(F,A2,B2,AREA2,ERROR2,RESA,DEFAB2) C C IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL C AND ERROR AND TEST FOR ACCURACY. C NEVAL = NEVAL+42 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 95 IF(ABS(RLIST(MAXERR)-AREA12).GT.0.1D-04*ABS(AREA12) 1 .OR.ERRO12.LT.0.99D+00*ERRMAX) GO TO 90 IF(EXTRAP) IROFF2 = IROFF2+1 IF(.NOT.EXTRAP) IROFF1 = IROFF1+1 90 IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF3 = IROFF3+1 95 LEVEL(MAXERR) = LEVCUR LEVEL(LAST) = LEVCUR 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 C SUBINTERVALS 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 100 ALIST(LAST) = A2 BLIST(MAXERR) = B1 BLIST(LAST) = B2 ELIST(MAXERR) = ERROR1 ELIST(LAST) = ERROR2 GO TO 110 100 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 110 CALL DQPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX) C ***JUMP OUT OF DO-LOOP IF(ERRSUM.LE.ERRBND) GO TO 190 C ***JUMP OUT OF DO-LOOP IF(IER.NE.0) GO TO 170 IF(NOEXT) GO TO 160 ERLARG = ERLARG-ERLAST IF(LEVCUR+1.LE.LEVMAX) ERLARG = ERLARG+ERRO12 IF(EXTRAP) GO TO 120 C C TEST WHETHER THE INTERVAL TO BE BISECTED NEXT IS THE C SMALLEST INTERVAL. C IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160 EXTRAP = .TRUE. NRMAX = 2 120 IF(IERRO.EQ.3.OR.ERLARG.LE.ERTEST) GO TO 140 C C THE SMALLEST INTERVAL HAS THE LARGEST ERROR. C BEFORE BISECTING DECREASE THE SUM OF THE ERRORS OVER C THE LARGER INTERVALS (ERLARG) AND PERFORM EXTRAPOLATION. C ID = NRMAX JUPBND = LAST IF(LAST.GT.(2+LIMIT/2)) JUPBND = LIMIT+3-LAST DO 130 K = ID,JUPBND MAXERR = IORD(NRMAX) ERRMAX = ELIST(MAXERR) C ***JUMP OUT OF DO-LOOP IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160 NRMAX = NRMAX+1 130 CONTINUE C C PERFORM EXTRAPOLATION. C 140 NUMRL2 = NUMRL2+1 RLIST2(NUMRL2) = AREA IF(NUMRL2.LE.2) GO TO 155 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 150 KTMIN = 0 ABSERR = ABSEPS RESULT = RESEPS CORREC = ERLARG ERTEST = MAX(EPSABS,EPSREL*ABS(RESEPS)) C ***JUMP OUT OF DO-LOOP IF(ABSERR.LT.ERTEST) GO TO 170 C C PREPARE BISECTION OF THE SMALLEST INTERVAL. C 150 IF(NUMRL2.EQ.1) NOEXT = .TRUE. IF(IER.GE.5) GO TO 170 155 MAXERR = IORD(1) ERRMAX = ELIST(MAXERR) NRMAX = 1 EXTRAP = .FALSE. LEVMAX = LEVMAX+1 ERLARG = ERRSUM 160 CONTINUE C C SET THE FINAL RESULT. C --------------------- C C 170 IF(ABSERR.EQ.OFLOW) GO TO 190 IF((IER+IERRO).EQ.0) GO TO 180 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 175 IF(ABSERR.GT.ERRSUM)GO TO 190 IF(AREA.EQ.0.0D+00) GO TO 210 GO TO 180 175 IF(ABSERR/ABS(RESULT).GT.ERRSUM/ABS(AREA))GO TO 190 C C TEST ON DIVERGENCE. C 180 IF(KSGN.EQ.(-1).AND.MAX(ABS(RESULT),ABS(AREA)).LE. 1 DEFABS*0.1D-01) GO TO 210 IF(0.1D-01.GT.(RESULT/AREA).OR.(RESULT/AREA).GT.0.1D+03.OR. 1 ERRSUM.GT.ABS(AREA)) IER = 6 GO TO 210 C C COMPUTE GLOBAL INTEGRAL SUM. C 190 RESULT = 0.0D+00 DO 200 K = 1,LAST RESULT = RESULT+RLIST(K) 200 CONTINUE ABSERR = ERRSUM 210 IF(IER.GT.2) IER = IER-1 RESULT = RESULT*SIGN 999 RETURN END