*DECK QAGIE SUBROUTINE QAGIE (F, BOUND, INF, EPSABS, EPSREL, LIMIT, RESULT, + ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST) C***BEGIN PROLOGUE QAGIE 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***LIBRARY SLATEC (QUADPACK) C***CATEGORY H2A3A1, H2A4A1 C***TYPE SINGLE PRECISION (QAGIE-S, DQAGIE-D) C***KEYWORDS AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE, C GLOBALLY ADAPTIVE, INFINITE INTERVALS, QUADPACK, C QUADRATURE, TRANSFORMATION 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 Integration over infinite intervals C Standard fortran subroutine C 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 BOUND - Real C Finite bound of integration range C (has no meaning if interval is doubly-infinite) C C INF - Real 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 - 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.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. The C estimates for result and error are less C reliable. It is assumed that the requested C 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. 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 - 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 transformed integration range (0,1). 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 transformed integration range (0,1). 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 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 QELG, QK15I, 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 QAGIE