dqagie.f

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      SUBROUTINE dqagie(F,BOUND,INF,EPSABS,EPSREL,LIMIT,RESULT,ABSERR,
     *   neval,ier,alist,blist,rlist,elist,iord,last)
C***BEGIN PROLOGUE  DQAGIE
C***DATE WRITTEN   800101   (YYMMDD)
C***REVISION DATE  830518   (YYMMDD)
C***CATEGORY NO.  H2A3A1,H2A4A1
C***KEYWORDS  AUTOMATIC INTEGRATOR, INFINITE INTERVALS,
C             GENERAL-PURPOSE, TRANSFORMATION, EXTRAPOLATION,
C             GLOBALLY ADAPTIVE
C***AUTHOR  PIESSENS,ROBERT,APPL. MATH & PROGR. DIV - K.U.LEUVEN
C           DE DONCKER,ELISE,APPL. MATH & PROGR. DIV - K.U.LEUVEN
C***PURPOSE  THE ROUTINE CALCULATES AN APPROXIMATION RESULT TO A GIVEN
C            INTEGRAL   I = INTEGRAL OF F OVER (BOUND,+INFINITY)
C            OR I = INTEGRAL OF F OVER (-INFINITY,BOUND)
C            OR I = INTEGRAL OF F OVER (-INFINITY,+INFINITY),
C            HOPEFULLY SATISFYING FOLLOWING CLAIM FOR ACCURACY
C            ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I))
C***DESCRIPTION
C
C INTEGRATION OVER INFINITE INTERVALS
C STANDARD FORTRAN SUBROUTINE
C
C            F      - SUBROUTINE F(X,IERR,RESULT) 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            BOUND  - DOUBLE PRECISION
C                     FINITE BOUND OF INTEGRATION RANGE
C                     (HAS NO MEANING IF INTERVAL IS DOUBLY-INFINITE)
C
C            INF    - DOUBLE PRECISION
C                     INDICATING THE KIND OF INTEGRATION RANGE INVOLVED
C                     INF = 1 CORRESPONDS TO  (BOUND,+INFINITY),
C                     INF = -1            TO  (-INFINITY,BOUND),
C                     INF = 2             TO (-INFINITY,+INFINITY).
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.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. THE
C                             ESTIMATES FOR RESULT AND ERROR ARE LESS
C                             RELIABLE. IT IS ASSUMED THAT THE REQUESTED
C                             ACCURACY HAS NOT BEEN ACHIEVED.
C                     IER.LT.0 EXIT REQUESTED FROM USER-SUPPLIED
C                             FUNCTION.
C
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.
C                             IF THE POSITION OF A LOCAL DIFFICULTY CAN
C                             BE 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
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.
C                             IT IS ASSUMED THAT THE REQUESTED TOLERANCE
C                             CANNOT BE ACHIEVED, AND THAT THE RETURNED
C                             RESULT IS 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.
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 0
C                             AND 1 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 TRANSFORMED INTEGRATION RANGE (0,1).
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 TRANSFORMED INTEGRATION RANGE (0,1).
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 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
C                     IN THE SUBDIVISION PROCESS
C
C***REFERENCES  (NONE)
C***ROUTINES CALLED  D1MACH,DQELG,DQK15I,DQPSRT
C***END PROLOGUE  DQAGIE
      DOUBLE PRECISION abseps,abserr,alist,area,area1,area12,area2,a1,
     *  a2,blist,boun,bound,b1,b2,correc,dabs,defabs,defab1,defab2,
     *  dmax1,dres,d1mach,elist,epmach,epsabs,epsrel,erlarg,erlast,
     *  errbnd,errmax,error1,error2,erro12,errsum,ertest,oflow,resabs,
     *  reseps,result,res3la,rlist,rlist2,small,uflow
      INTEGER id,ier,ierro,inf,iord,iroff1,iroff2,iroff3,jupbnd,k,ksgn,
     *  ktmin,last,limit,maxerr,neval,nres,nrmax,numrl2
      LOGICAL extrap,noext
C
      dimension alist(limit),blist(limit),elist(limit),iord(limit),
     *  res3la(3),rlist(limit),rlist2(52)
C
      EXTERNAL f
C
C            THE DIMENSION OF RLIST2 IS DETERMINED BY THE VALUE OF
C            LIMEXP IN SUBROUTINE DQELG.
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
C                       WICH 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 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
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
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  DQAGIE
       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) = 0.0d+00
      blist(1) = 0.1d+01
      rlist(1) = 0.0d+00
      elist(1) = 0.0d+00
      iord(1) = 0
      IF(epsabs.LE.0.0d+00.AND.epsrel.LT.dmax1(0.5d+02*epmach,0.5d-28))
     *  ier = 6
       IF(ier.EQ.6) go to 999
C
C
C           FIRST APPROXIMATION TO THE INTEGRAL
C           -----------------------------------
C
C           DETERMINE THE INTERVAL TO BE MAPPED ONTO (0,1).
C           IF INF = 2 THE INTEGRAL IS COMPUTED AS I = I1+I2, WHERE
C           I1 = INTEGRAL OF F OVER (-INFINITY,0),
C           I2 = INTEGRAL OF F OVER (0,+INFINITY).
C
      boun = bound
      IF(inf.EQ.2) boun = 0.0d+00
      CALL dqk15i(f,boun,inf,0.0d+00,0.1d+01,result,abserr,
     *  defabs,resabs,ier)
      IF (ier .LT. 0) RETURN
C
C           TEST ON ACCURACY
C
      last = 1
      rlist(1) = result
      elist(1) = abserr
      iord(1) = 1
      dres = dabs(result)
      errbnd = dmax1(epsabs,epsrel*dres)
      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.
     *  abserr.EQ.0.0d+00) go to 130
C
C           INITIALIZATION
C           --------------
C
      uflow = d1mach(1)
      oflow = d1mach(2)
      rlist2(1) = result
      errmax = abserr
      maxerr = 1
      area = result
      errsum = abserr
      abserr = oflow
      nrmax = 1
      nres = 0
      ktmin = 0
      numrl2 = 2
      extrap = .false.
      noext = .false.
      ierro = 0
      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 NRMAX-TH LARGEST ERROR ESTIMATE.
C
        a1 = alist(maxerr)
        b1 = 0.5d+00*(alist(maxerr)+blist(maxerr))
        a2 = b1
        b2 = blist(maxerr)
        erlast = errmax
        CALL dqk15i(f,boun,inf,a1,b1,area1,error1,resabs,defab1,ier)
        IF (ier .LT. 0) RETURN
        CALL dqk15i(f,boun,inf,a2,b2,area2,error2,resabs,defab2,ier)
        IF (ier .LT. 0) RETURN
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(dabs(rlist(maxerr)-area12).GT.0.1d-04*dabs(area12)
     *  .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 = dmax1(epsabs,epsrel*dabs(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 SOME POINTS OF THE INTEGRATION RANGE.
C
        IF(dmax1(dabs(a1),dabs(b2)).LE.(0.1d+01+0.1d+03*epmach)*
     *  (dabs(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)
        IF(errsum.LE.errbnd) go to 115
        IF(ier.NE.0) go to 100
        IF(last.EQ.2) go to 80
        IF(noext) go to 90
        erlarg = erlarg-erlast
        IF(dabs(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(dabs(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)
          IF(dabs(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 = dmax1(epsabs,epsrel*dabs(reseps))
        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 = 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/dabs(result).GT.errsum/dabs(area))go to 115
C
C           TEST ON DIVERGENCE
C
  110 IF(ksgn.EQ.(-1).AND.dmax1(dabs(result),dabs(area)).LE.
     * defabs*0.1d-01) go to 130
      IF(0.1d-01.GT.(result/area).OR.(result/area).GT.0.1d+03.
     *or.errsum.GT.dabs(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 neval = 30*last-15
      IF(inf.EQ.2) neval = 2*neval
      IF(ier.GT.2) ier=ier-1
  999 RETURN
      END