SLATEC Routines --- QAGPE ---
*DECK QAGPE
SUBROUTINE QAGPE (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, LIMIT,
+ RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, PTS,
+ IORD, LEVEL, NDIN, LAST)
C***BEGIN PROLOGUE QAGPE
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 SINGLE 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 Real version
C
C PARAMETERS
C ON ENTRY
C F - Real
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 - Real
C Lower limit of integration
C
C B - Real
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 - Real
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 - Real
C Absolute accuracy requested
C EPSREL - Real
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 - Real
C Approximation to the integral
C
C ABSERR - Real
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 - Real
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 - Real
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 - Real
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 - Real
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 - Real
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 QELG, QK21, QPSRT, R1MACH
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 QAGPE