*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