*DECK QAGE SUBROUTINE QAGE (F, A, B, EPSABS, EPSREL, KEY, LIMIT, RESULT, + ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST) C***BEGIN PROLOGUE QAGE C***PURPOSE The routine calculates an approximation result to a given C definite integral I = Integral of F over (A,B), C hopefully satisfying following claim for accuracy C ABS(I-RESLT).LE.MAX(EPSABS,EPSREL*ABS(I)). C***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A1A1 C***TYPE SINGLE PRECISION (QAGE-S, DQAGE-D) C***KEYWORDS AUTOMATIC INTEGRATOR, GAUSS-KRONROD RULES, C GENERAL-PURPOSE, GLOBALLY ADAPTIVE, INTEGRAND EXAMINATOR, C QUADPACK, QUADRATURE 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 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 KEY - Integer C Key for choice of local integration rule C A Gauss-Kronrod pair is used with C 7 - 15 points if KEY.LT.2, C 10 - 21 points if KEY = 2, C 15 - 31 points if KEY = 3, C 20 - 41 points if KEY = 4, C 25 - 51 points if KEY = 5, C 30 - 61 points if KEY.GT.5. 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 - 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 result 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 C of LIMIT. C However, if this yields no improvement it C is rather advised to analyze the integrand C in order to determine the integration C difficulties. If the position of a local C difficulty can be determined(e.g. C SINGULARITY, DISCONTINUITY within the C interval) one will probably gain from C splitting up the interval at this point C and calling the integrator on the C subranges. If possible, an appropriate C special-purpose integrator should be used C which is designed for handling the type of C difficulty involved. C = 2 The occurrence of roundoff error is C detected, which prevents the requested C tolerance from being achieved. C = 3 Extremely bad integrand behaviour occurs C at some points of the integration C interval. 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 A and B C respectively. C C ALIST - Real 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 given 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 C end points of the subintervals in the partition C of the given integration range (A,B) C C RLIST - Real C Vector of dimension at least LIMIT, the first C LAST elements of which are the C integral 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 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)), ..., C ELIST(IORD(K)) form a decreasing sequence, C with K = LAST if LAST.LE.(LIMIT/2+2), and C K = LIMIT+1-LAST otherwise C C LAST - Integer C Number of subintervals actually produced in the C subdivision process C C***REFERENCES (NONE) C***ROUTINES CALLED QK15, QK21, QK31, QK41, QK51, QK61, 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 QAGE