# qk51.f

          SUBROUTINE QK51 (F, A, B, RESULT, ABSERR, RESABS, RESASC)
    C***BEGIN PROLOGUE  QK51
    C***PURPOSE  To compute I = Integral of F over (A,B) with error
    C                           estimate
    C                       J = Integral of ABS(F) over (A,B)
    C***LIBRARY   SLATEC (QUADPACK)
    C***CATEGORY  H2A1A2
    C***TYPE      SINGLE PRECISION (QK51-S, DQK51-D)
    C***KEYWORDS  51-POINT GAUSS-KRONROD RULES, 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           Integration rules
    C           Standard fortran subroutine
    C           Real version
    C
    C           PARAMETERS
    C            ON ENTRY
    C              F      - Real
    C                       Function subroutine 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 calling program.
    C
    C              A      - Real
    C                       Lower limit of integration
    C
    C              B      - Real
    C                       Upper limit of integration
    C
    C            ON RETURN
    C              RESULT - Real
    C                       Approximation to the integral I
    C                       RESULT is computed by applying the 51-point
    C                       Kronrod rule (RESK) obtained by optimal addition
    C                       of abscissae to the 25-point Gauss rule (RESG).
    C
    C              ABSERR - Real
    C                       Estimate of the modulus of the absolute error,
    C                       which should not exceed ABS(I-RESULT)
    C
    C              RESABS - Real
    C                       Approximation to the integral J
    C
    C              RESASC - Real
    C                       Approximation to the integral of ABS(F-I/(B-A))
    C                       over (A,B)
    C
    C***REFERENCES  (NONE)
    C***ROUTINES CALLED  R1MACH
    C***REVISION HISTORY  (YYMMDD)
    C   800101  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   890531  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C***END PROLOGUE  QK51