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SUBROUTINE DQAGPE (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, LIMIT,
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+ RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, PTS,
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+ IORD, LEVEL, NDIN, LAST)
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C***BEGIN PROLOGUE DQAGPE
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C***PURPOSE Approximate a given definite integral I = Integral of F
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C over (A,B), hopefully satisfying the accuracy claim:
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C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
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C Break points of the integration interval, where local
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C difficulties of the integrand may occur (e.g. singularities
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C or discontinuities) are provided by the user.
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C***LIBRARY SLATEC (QUADPACK)
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C***TYPE DOUBLE PRECISION (QAGPE-S, DQAGPE-D)
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C***KEYWORDS AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
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C GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE,
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C SINGULARITIES AT USER SPECIFIED POINTS
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C***AUTHOR Piessens, Robert
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C Applied Mathematics and Programming Division
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C Applied Mathematics and Programming Division
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C Computation of a definite integral
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C Standard fortran subroutine
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C Double precision version
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C F - Double precision
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C Function subprogram defining the integrand
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C function F(X). The actual name for F needs to be
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C declared E X T E R N A L in the driver program.
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C A - Double precision
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C Lower limit of integration
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C B - Double precision
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C Upper limit of integration
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C Number equal to two more than the number of
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C user-supplied break points within the integration
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C If NPTS2.LT.2, the routine will end with IER = 6.
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C POINTS - Double precision
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C Vector of dimension NPTS2, the first (NPTS2-2)
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C elements of which are the user provided break
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C POINTS. If these POINTS do not constitute an
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C ascending sequence there will be an automatic
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C EPSABS - Double precision
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C Absolute accuracy requested
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C EPSREL - Double precision
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C Relative accuracy requested
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C and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
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C the routine will end with IER = 6.
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C Gives an upper bound on the number of subintervals
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C in the partition of (A,B), LIMIT.GE.NPTS2
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C If LIMIT.LT.NPTS2, the routine will end with
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C RESULT - Double precision
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C Approximation to the integral
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C ABSERR - Double precision
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C Estimate of the modulus of the absolute error,
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C which should equal or exceed ABS(I-RESULT)
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C Number of integrand evaluations
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C IER = 0 Normal and reliable termination of the
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C routine. It is assumed that the requested
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C accuracy has been achieved.
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C IER.GT.0 Abnormal termination of the routine.
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C The estimates for integral and error are
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C less reliable. It is assumed that the
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C requested accuracy has not been achieved.
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C IER = 1 Maximum number of subdivisions allowed
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C has been achieved. One can allow more
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C subdivisions by increasing the value of
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C LIMIT (and taking the according dimension
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C adjustments into account). However, if
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C this yields no improvement it is advised
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C to analyze the integrand in order to
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C determine the integration difficulties. If
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C the position of a local difficulty can be
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C determined (i.e. SINGULARITY,
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C DISCONTINUITY within the interval), it
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C should be supplied to the routine as an
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C element of the vector points. If necessary
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C an appropriate special-purpose integrator
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C must be used, which is designed for
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C handling the type of difficulty involved.
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C = 2 The occurrence of roundoff error is
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C detected, which prevents the requested
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C tolerance from being achieved.
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C The error may be under-estimated.
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C = 3 Extremely bad integrand behaviour occurs
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C At some points of the integration
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C = 4 The algorithm does not converge.
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C Roundoff error is detected in the
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C extrapolation table. It is presumed that
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C the requested tolerance cannot be
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C achieved, and that the returned result is
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C the best which can be obtained.
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C = 5 The integral is probably divergent, or
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C slowly convergent. It must be noted that
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C divergence can occur with any other value
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C = 6 The input is invalid because
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C Break points are specified outside
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C the integration range or
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C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28))
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C RESULT, ABSERR, NEVAL, LAST, RLIST(1),
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C and ELIST(1) are set to zero. ALIST(1) and
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C BLIST(1) are set to A and B respectively.
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C ALIST - Double precision
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C Vector of dimension at least LIMIT, the first
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C LAST elements of which are the left end points
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C of the subintervals in the partition of the given
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C integration range (A,B)
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C BLIST - Double precision
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C Vector of dimension at least LIMIT, the first
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C LAST elements of which are the right end points
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C of the subintervals in the partition of the given
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C integration range (A,B)
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C RLIST - Double precision
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C Vector of dimension at least LIMIT, the first
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C LAST elements of which are the integral
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C approximations on the subintervals
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C ELIST - Double precision
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C Vector of dimension at least LIMIT, the first
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C LAST elements of which are the moduli of the
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C absolute error estimates on the subintervals
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C PTS - Double precision
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C Vector of dimension at least NPTS2, containing the
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C integration limits and the break points of the
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C interval in ascending sequence.
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C Vector of dimension at least LIMIT, containing the
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C subdivision levels of the subinterval, i.e. if
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C (AA,BB) is a subinterval of (P1,P2) where P1 as
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C well as P2 is a user-provided break point or
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C integration limit, then (AA,BB) has level L if
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C ABS(BB-AA) = ABS(P2-P1)*2**(-L).
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C Vector of dimension at least NPTS2, after first
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C integration over the intervals (PTS(I)),PTS(I+1),
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C I = 0,1, ..., NPTS2-2, the error estimates over
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C some of the intervals may have been increased
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C artificially, in order to put their subdivision
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C forward. If this happens for the subinterval
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C numbered K, NDIN(K) is put to 1, otherwise
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C Vector of dimension at least LIMIT, the first K
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C elements of which are pointers to the
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C error estimates over the subintervals,
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C such that ELIST(IORD(1)), ..., ELIST(IORD(K))
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C form a decreasing sequence, with K = LAST
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C If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST
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C Number of subintervals actually produced in the
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C subdivisions process
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C***REFERENCES (NONE)
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C***ROUTINES CALLED D1MACH, DQELG, DQK21, DQPSRT
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C***REVISION HISTORY (YYMMDD)
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C 800101 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C***END PROLOGUE DQAGPE
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DOUBLE PRECISION A,ABSEPS,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1,
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1 A2,B,BLIST,B1,B2,CORREC,DEFABS,DEFAB1,DEFAB2,
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2 DRES,D1MACH,ELIST,EPMACH,EPSABS,EPSREL,ERLARG,ERLAST,ERRBND,
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3 ERRMAX,ERROR1,ERRO12,ERROR2,ERRSUM,ERTEST,F,OFLOW,POINTS,PTS,
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4 RESA,RESABS,RESEPS,RESULT,RES3LA,RLIST,RLIST2,SIGN,TEMP,UFLOW
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INTEGER I,ID,IER,IERRO,IND1,IND2,IORD,IP1,IROFF1,IROFF2,IROFF3,J,
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1 JLOW,JUPBND,K,KSGN,KTMIN,LAST,LEVCUR,LEVEL,LEVMAX,LIMIT,MAXERR,
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2 NDIN,NEVAL,NINT,NINTP1,NPTS,NPTS2,NRES,NRMAX,NUMRL2
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DIMENSION ALIST(*),BLIST(*),ELIST(*),IORD(*),
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1 LEVEL(*),NDIN(*),POINTS(*),PTS(*),RES3LA(3),
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2 RLIST(*),RLIST2(52)
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C THE DIMENSION OF RLIST2 IS DETERMINED BY THE VALUE OF
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C LIMEXP IN SUBROUTINE EPSALG (RLIST2 SHOULD BE OF DIMENSION
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C (LIMEXP+2) AT LEAST).
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C LIST OF MAJOR VARIABLES
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C -----------------------
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C ALIST - LIST OF LEFT END POINTS OF ALL SUBINTERVALS
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C CONSIDERED UP TO NOW
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C BLIST - LIST OF RIGHT END POINTS OF ALL SUBINTERVALS
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C CONSIDERED UP TO NOW
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C RLIST(I) - APPROXIMATION TO THE INTEGRAL OVER
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C (ALIST(I),BLIST(I))
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C RLIST2 - ARRAY OF DIMENSION AT LEAST LIMEXP+2
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C CONTAINING THE PART OF THE EPSILON TABLE WHICH
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C IS STILL NEEDED FOR FURTHER COMPUTATIONS
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C ELIST(I) - ERROR ESTIMATE APPLYING TO RLIST(I)
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C MAXERR - POINTER TO THE INTERVAL WITH LARGEST ERROR
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C ERRMAX - ELIST(MAXERR)
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C ERLAST - ERROR ON THE INTERVAL CURRENTLY SUBDIVIDED
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C (BEFORE THAT SUBDIVISION HAS TAKEN PLACE)
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C AREA - SUM OF THE INTEGRALS OVER THE SUBINTERVALS
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C ERRSUM - SUM OF THE ERRORS OVER THE SUBINTERVALS
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C ERRBND - REQUESTED ACCURACY MAX(EPSABS,EPSREL*
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C *****1 - VARIABLE FOR THE LEFT SUBINTERVAL
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C *****2 - VARIABLE FOR THE RIGHT SUBINTERVAL
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C LAST - INDEX FOR SUBDIVISION
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C NRES - NUMBER OF CALLS TO THE EXTRAPOLATION ROUTINE
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C NUMRL2 - NUMBER OF ELEMENTS IN RLIST2. IF AN APPROPRIATE
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C APPROXIMATION TO THE COMPOUNDED INTEGRAL HAS
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C BEEN OBTAINED, IT IS PUT IN RLIST2(NUMRL2) AFTER
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C NUMRL2 HAS BEEN INCREASED BY ONE.
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C ERLARG - SUM OF THE ERRORS OVER THE INTERVALS LARGER
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C THAN THE SMALLEST INTERVAL CONSIDERED UP TO NOW
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C EXTRAP - LOGICAL VARIABLE DENOTING THAT THE ROUTINE
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C IS ATTEMPTING TO PERFORM EXTRAPOLATION. I.E.
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C BEFORE SUBDIVIDING THE SMALLEST INTERVAL WE
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C TRY TO DECREASE THE VALUE OF ERLARG.
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C NOEXT - LOGICAL VARIABLE DENOTING THAT EXTRAPOLATION IS
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C NO LONGER ALLOWED (TRUE-VALUE)
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C MACHINE DEPENDENT CONSTANTS
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C ---------------------------
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C EPMACH IS THE LARGEST RELATIVE SPACING.
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C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE.
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C OFLOW IS THE LARGEST POSITIVE MAGNITUDE.
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C***FIRST EXECUTABLE STATEMENT DQAGPE
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C TEST ON VALIDITY OF PARAMETERS
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C -----------------------------
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IF(NPTS2.LT.2.OR.LIMIT.LE.NPTS.OR.(EPSABS.LE.0.0D+00.AND.
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1 EPSREL.LT.MAX(0.5D+02*EPMACH,0.5D-28))) IER = 6
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IF(IER.EQ.6) GO TO 999
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C IF ANY BREAK POINTS ARE PROVIDED, SORT THEM INTO AN
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C ASCENDING SEQUENCE.
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IF(A.GT.B) SIGN = -1.0D+00
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IF(NPTS.EQ.0) GO TO 15
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15 PTS(NPTS+2) = MAX(A,B)
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IF(NPTS.EQ.0) GO TO 40
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IF(PTS(I).LE.PTS(J)) GO TO 20
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IF(PTS(1).NE.MIN(A,B).OR.PTS(NINTP1).NE.MAX(A,B)) IER = 6
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IF(IER.EQ.6) GO TO 999
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C COMPUTE FIRST INTEGRAL AND ERROR APPROXIMATIONS.
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C ------------------------------------------------
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CALL DQK21(F,A1,B1,AREA1,ERROR1,DEFABS,RESA)
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ABSERR = ABSERR+ERROR1
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RESULT = RESULT+AREA1
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IF(ERROR1.EQ.RESA.AND.ERROR1.NE.0.0D+00) NDIN(I) = 1
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RESABS = RESABS+DEFABS
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IF(NDIN(I).EQ.1) ELIST(I) = ABSERR
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ERRSUM = ERRSUM+ELIST(I)
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ERRBND = MAX(EPSABS,EPSREL*DRES)
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IF(ABSERR.LE.0.1D+03*EPMACH*RESABS.AND.ABSERR.GT.ERRBND) IER = 2
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IF(NINT.EQ.1) GO TO 80
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IF(ELIST(IND1).GT.ELIST(IND2)) GO TO 60
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IF(IND1.EQ.IORD(I)) GO TO 70
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IF(LIMIT.LT.NPTS2) IER = 1
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80 IF(IER.NE.0.OR.ABSERR.LE.ERRBND) GO TO 999
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ERRMAX = ELIST(MAXERR)
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IF(DRES.GE.(0.1D+01-0.5D+02*EPMACH)*RESABS) KSGN = 1
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DO 160 LAST = NPTS2,LIMIT
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C BISECT THE SUBINTERVAL WITH THE NRMAX-TH LARGEST ERROR
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LEVCUR = LEVEL(MAXERR)+1
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B1 = 0.5D+00*(ALIST(MAXERR)+BLIST(MAXERR))
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CALL DQK21(F,A1,B1,AREA1,ERROR1,RESA,DEFAB1)
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CALL DQK21(F,A2,B2,AREA2,ERROR2,RESA,DEFAB2)
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C IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL
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C AND ERROR AND TEST FOR ACCURACY.
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ERRO12 = ERROR1+ERROR2
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ERRSUM = ERRSUM+ERRO12-ERRMAX
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AREA = AREA+AREA12-RLIST(MAXERR)
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IF(DEFAB1.EQ.ERROR1.OR.DEFAB2.EQ.ERROR2) GO TO 95
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IF(ABS(RLIST(MAXERR)-AREA12).GT.0.1D-04*ABS(AREA12)
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1 .OR.ERRO12.LT.0.99D+00*ERRMAX) GO TO 90
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IF(EXTRAP) IROFF2 = IROFF2+1
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IF(.NOT.EXTRAP) IROFF1 = IROFF1+1
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90 IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF3 = IROFF3+1
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95 LEVEL(MAXERR) = LEVCUR
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RLIST(MAXERR) = AREA1
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ERRBND = MAX(EPSABS,EPSREL*ABS(AREA))
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C TEST FOR ROUNDOFF ERROR AND EVENTUALLY SET ERROR FLAG.
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IF(IROFF1+IROFF2.GE.10.OR.IROFF3.GE.20) IER = 2
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IF(IROFF2.GE.5) IERRO = 3
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C SET ERROR FLAG IN THE CASE THAT THE NUMBER OF
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C SUBINTERVALS EQUALS LIMIT.
438
IF(LAST.EQ.LIMIT) IER = 1
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C SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR
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C AT A POINT OF THE INTEGRATION RANGE
443
IF(MAX(ABS(A1),ABS(B2)).LE.(0.1D+01+0.1D+03*EPMACH)*
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1 (ABS(A2)+0.1D+04*UFLOW)) IER = 4
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C APPEND THE NEWLY-CREATED INTERVALS TO THE LIST.
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IF(ERROR2.GT.ERROR1) GO TO 100
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ELIST(MAXERR) = ERROR1
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100 ALIST(MAXERR) = A2
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RLIST(MAXERR) = AREA2
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ELIST(MAXERR) = ERROR2
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C CALL SUBROUTINE DQPSRT TO MAINTAIN THE DESCENDING ORDERING
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C IN THE LIST OF ERROR ESTIMATES AND SELECT THE SUBINTERVAL
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C WITH NRMAX-TH LARGEST ERROR ESTIMATE (TO BE BISECTED NEXT).
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110 CALL DQPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX)
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C ***JUMP OUT OF DO-LOOP
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IF(ERRSUM.LE.ERRBND) GO TO 190
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C ***JUMP OUT OF DO-LOOP
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IF(IER.NE.0) GO TO 170
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ERLARG = ERLARG-ERLAST
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IF(LEVCUR+1.LE.LEVMAX) ERLARG = ERLARG+ERRO12
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C TEST WHETHER THE INTERVAL TO BE BISECTED NEXT IS THE
480
IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160
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120 IF(IERRO.EQ.3.OR.ERLARG.LE.ERTEST) GO TO 140
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C THE SMALLEST INTERVAL HAS THE LARGEST ERROR.
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C BEFORE BISECTING DECREASE THE SUM OF THE ERRORS OVER
487
C THE LARGER INTERVALS (ERLARG) AND PERFORM EXTRAPOLATION.
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IF(LAST.GT.(2+LIMIT/2)) JUPBND = LIMIT+3-LAST
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ERRMAX = ELIST(MAXERR)
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C ***JUMP OUT OF DO-LOOP
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IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160
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C PERFORM EXTRAPOLATION.
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140 NUMRL2 = NUMRL2+1
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RLIST2(NUMRL2) = AREA
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IF(NUMRL2.LE.2) GO TO 155
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CALL DQELG(NUMRL2,RLIST2,RESEPS,ABSEPS,RES3LA,NRES)
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IF(KTMIN.GT.5.AND.ABSERR.LT.0.1D-02*ERRSUM) IER = 5
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IF(ABSEPS.GE.ABSERR) GO TO 150
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ERTEST = MAX(EPSABS,EPSREL*ABS(RESEPS))
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C ***JUMP OUT OF DO-LOOP
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IF(ABSERR.LT.ERTEST) GO TO 170
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C PREPARE BISECTION OF THE SMALLEST INTERVAL.
519
150 IF(NUMRL2.EQ.1) NOEXT = .TRUE.
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IF(IER.GE.5) GO TO 170
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ERRMAX = ELIST(MAXERR)
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C SET THE FINAL RESULT.
530
C ---------------------
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170 IF(ABSERR.EQ.OFLOW) GO TO 190
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IF((IER+IERRO).EQ.0) GO TO 180
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IF(IERRO.EQ.3) ABSERR = ABSERR+CORREC
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IF(RESULT.NE.0.0D+00.AND.AREA.NE.0.0D+00)GO TO 175
538
IF(ABSERR.GT.ERRSUM)GO TO 190
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IF(AREA.EQ.0.0D+00) GO TO 210
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175 IF(ABSERR/ABS(RESULT).GT.ERRSUM/ABS(AREA))GO TO 190
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C TEST ON DIVERGENCE.
545
180 IF(KSGN.EQ.(-1).AND.MAX(ABS(RESULT),ABS(AREA)).LE.
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1 RESABS*0.1D-01) GO TO 210
547
IF(0.1D-01.GT.(RESULT/AREA).OR.(RESULT/AREA).GT.0.1D+03.OR.
548
1 ERRSUM.GT.ABS(AREA)) IER = 6
551
C COMPUTE GLOBAL INTEGRAL SUM.
555
RESULT = RESULT+RLIST(K)
558
210 IF(IER.GT.2) IER = IER-1
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src/numerical/slatec/fortran/dqagpe.f
