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339 lines
13 KiB
FortranFixed
339 lines
13 KiB
FortranFixed
15 years ago
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MODULE ERFCOREMOD
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IMPLICIT NONE
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INTERFACE CALERF
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MODULE PROCEDURE CALERF
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END INTERFACE
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INTERFACE DERF
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MODULE PROCEDURE DERF
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END INTERFACE
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INTERFACE DERFC
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MODULE PROCEDURE DERFC
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END INTERFACE
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INTERFACE DERFCX
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MODULE PROCEDURE DERFCX
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END INTERFACE
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CONTAINS
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!C--------------------------------------------------------------------
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!C
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!C DERF subprogram computes approximate values for erf(x).
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!C (see comments heading CALERF).
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!C
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!C Author/date: W. J. Cody, January 8, 1985
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!C
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!C--------------------------------------------------------------------
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FUNCTION DERF( X ) RESULT (VALUE)
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IMPLICIT NONE
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DOUBLE PRECISION, INTENT(IN) :: X
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DOUBLE PRECISION :: VALUE
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INTEGER, PARAMETER :: JINT = 0
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CALL CALERF(X,VALUE,JINT)
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RETURN
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END FUNCTION DERF
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!C--------------------------------------------------------------------
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!C
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!C DERFC subprogram computes approximate values for erfc(x).
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!C (see comments heading CALERF).
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!C
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!C Author/date: W. J. Cody, January 8, 1985
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!C
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!C--------------------------------------------------------------------
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FUNCTION DERFC( X ) RESULT (VALUE)
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IMPLICIT NONE
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DOUBLE PRECISION, INTENT(IN) :: X
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DOUBLE PRECISION :: VALUE
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INTEGER, PARAMETER :: JINT = 1
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CALL CALERF(X,VALUE,JINT)
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RETURN
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END FUNCTION DERFC
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!C------------------------------------------------------------------
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!C
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!C DERFCX subprogram computes approximate values for exp(x*x) * erfc(x).
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!C (see comments heading CALERF).
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!C
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!C Author/date: W. J. Cody, March 30, 1987
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!C
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!C------------------------------------------------------------------
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FUNCTION DERFCX( X ) RESULT (VALUE)
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IMPLICIT NONE
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DOUBLE PRECISION, INTENT(IN) :: X
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DOUBLE PRECISION :: VALUE
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INTEGER, PARAMETER :: JINT = 2
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CALL CALERF(X,VALUE,JINT)
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RETURN
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END FUNCTION DERFCX
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SUBROUTINE CALERF(ARG,RESULT,JINT)
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IMPLICIT NONE
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!C------------------------------------------------------------------
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!C
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!C CALERF packet evaluates erf(x), erfc(x), and exp(x*x)*erfc(x)
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!C for a real argument x. It contains three FUNCTION type
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!C subprograms: ERF, ERFC, and ERFCX (or DERF, DERFC, and DERFCX),
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!C and one SUBROUTINE type subprogram, CALERF. The calling
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!C statements for the primary entries are:
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!C
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!C Y=ERF(X) (or Y=DERF(X)),
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!C
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!C Y=ERFC(X) (or Y=DERFC(X)),
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!C and
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!C Y=ERFCX(X) (or Y=DERFCX(X)).
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!C
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!C The routine CALERF is intended for internal packet use only,
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!C all computations within the packet being concentrated in this
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!C routine. The function subprograms invoke CALERF with the
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!C statement
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!C
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!C CALL CALERF(ARG,RESULT,JINT)
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!C
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!C where the parameter usage is as follows
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!C
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!C Function Parameters for CALERF
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!C call ARG Result JINT
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!C
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!C ERF(ARG) ANY REAL ARGUMENT ERF(ARG) 0
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!C ERFC(ARG) ABS(ARG) .LT. XBIG ERFC(ARG) 1
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!C ERFCX(ARG) XNEG .LT. ARG .LT. XMAX ERFCX(ARG) 2
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!C
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!C The main computation evaluates near-minimax approximations
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!C from "Rational Chebyshev approximations for the error function"
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!C by W. J. Cody, Math. Comp., 1969, PP. 631-638. This
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!C transportable program uses rational functions that theoretically
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!C approximate erf(x) and erfc(x) to at least 18 significant
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!C decimal digits. The accuracy achieved depends on the arithmetic
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!C system, the compiler, the intrinsic functions, and proper
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!C selection of the machine-dependent constants.
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!C
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!C*******************************************************************
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!C*******************************************************************
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!C
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!C Explanation of machine-dependent constants
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!C
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!C XMIN = the smallest positive floating-point number.
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!C XINF = the largest positive finite floating-point number.
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!C XNEG = the largest negative argument acceptable to ERFCX;
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!C the negative of the solution to the equation
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!C 2*exp(x*x) = XINF.
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!C XSMALL = argument below which erf(x) may be represented by
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!C 2*x/sqrt(pi) and above which x*x will not underflow.
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!C A conservative value is the largest machine number X
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!C such that 1.0 + X = 1.0 to machine precision.
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!C XBIG = largest argument acceptable to ERFC; solution to
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!C the equation: W(x) * (1-0.5/x**2) = XMIN, where
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!C W(x) = exp(-x*x)/[x*sqrt(pi)].
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!C XHUGE = argument above which 1.0 - 1/(2*x*x) = 1.0 to
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!C machine precision. A conservative value is
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!C 1/[2*sqrt(XSMALL)]
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!C XMAX = largest acceptable argument to ERFCX; the minimum
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!C of XINF and 1/[sqrt(pi)*XMIN].
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!C
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!C Approximate values for some important machines are:
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!C
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!C XMIN XINF XNEG XSMALL
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!C
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!C C 7600 (S.P.) 3.13E-294 1.26E+322 -27.220 7.11E-15
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!C CRAY-1 (S.P.) 4.58E-2467 5.45E+2465 -75.345 7.11E-15
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!C IEEE (IBM/XT,
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!C SUN, etc.) (S.P.) 1.18E-38 3.40E+38 -9.382 5.96E-8
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!C IEEE (IBM/XT,
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!C SUN, etc.) (D.P.) 2.23D-308 1.79D+308 -26.628 1.11D-16
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!C IBM 195 (D.P.) 5.40D-79 7.23E+75 -13.190 1.39D-17
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!C UNIVAC 1108 (D.P.) 2.78D-309 8.98D+307 -26.615 1.73D-18
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!C VAX D-Format (D.P.) 2.94D-39 1.70D+38 -9.345 1.39D-17
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!C VAX G-Format (D.P.) 5.56D-309 8.98D+307 -26.615 1.11D-16
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!C
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!C
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!C XBIG XHUGE XMAX
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!C
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!C C 7600 (S.P.) 25.922 8.39E+6 1.80X+293
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!C CRAY-1 (S.P.) 75.326 8.39E+6 5.45E+2465
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!C IEEE (IBM/XT,
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!C SUN, etc.) (S.P.) 9.194 2.90E+3 4.79E+37
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!C IEEE (IBM/XT,
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!C SUN, etc.) (D.P.) 26.543 6.71D+7 2.53D+307
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!C IBM 195 (D.P.) 13.306 1.90D+8 7.23E+75
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!C UNIVAC 1108 (D.P.) 26.582 5.37D+8 8.98D+307
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!C VAX D-Format (D.P.) 9.269 1.90D+8 1.70D+38
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!C VAX G-Format (D.P.) 26.569 6.71D+7 8.98D+307
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!C
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!C*******************************************************************
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!C*******************************************************************
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!C
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!C Error returns
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!C
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!C The program returns ERFC = 0 for ARG .GE. XBIG;
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!C
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!C ERFCX = XINF for ARG .LT. XNEG;
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!C and
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!C ERFCX = 0 for ARG .GE. XMAX.
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!C
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!C
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!C Intrinsic functions required are:
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!C
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!C ABS, AINT, EXP
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!C
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!C
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!C Author: W. J. Cody
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!C Mathematics and Computer Science Division
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!C Argonne National Laboratory
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!C Argonne, IL 60439
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!C
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!C Latest modification: March 19, 1990
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!C Updated to F90 by pab 23.03.2003
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!C
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!C------------------------------------------------------------------
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DOUBLE PRECISION, INTENT(IN) :: ARG
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INTEGER, INTENT(IN) :: JINT
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DOUBLE PRECISION, INTENT(INOUT):: RESULT
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! Local variables
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INTEGER :: I
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DOUBLE PRECISION :: DEL,X,XDEN,XNUM,Y,YSQ
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!C------------------------------------------------------------------
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!C Mathematical constants
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!C------------------------------------------------------------------
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DOUBLE PRECISION, PARAMETER :: ZERO = 0.0D0
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DOUBLE PRECISION, PARAMETER :: HALF = 0.05D0
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DOUBLE PRECISION, PARAMETER :: ONE = 1.0D0
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DOUBLE PRECISION, PARAMETER :: TWO = 2.0D0
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DOUBLE PRECISION, PARAMETER :: FOUR = 4.0D0
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DOUBLE PRECISION, PARAMETER :: SIXTEN = 16.0D0
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DOUBLE PRECISION, PARAMETER :: SQRPI = 5.6418958354775628695D-1
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DOUBLE PRECISION, PARAMETER :: THRESH = 0.46875D0
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!C------------------------------------------------------------------
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!C Machine-dependent constants
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!C------------------------------------------------------------------
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DOUBLE PRECISION, PARAMETER :: XNEG = -26.628D0
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DOUBLE PRECISION, PARAMETER :: XSMALL = 1.11D-16
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DOUBLE PRECISION, PARAMETER :: XBIG = 26.543D0
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DOUBLE PRECISION, PARAMETER :: XHUGE = 6.71D7
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DOUBLE PRECISION, PARAMETER :: XMAX = 2.53D307
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DOUBLE PRECISION, PARAMETER :: XINF = 1.79D308
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!---------------------------------------------------------------
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! Coefficents to the rational polynomials
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!--------------------------------------------------------------
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DOUBLE PRECISION, DIMENSION(5) :: A, Q
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DOUBLE PRECISION, DIMENSION(4) :: B
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DOUBLE PRECISION, DIMENSION(9) :: C
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DOUBLE PRECISION, DIMENSION(8) :: D
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DOUBLE PRECISION, DIMENSION(6) :: P
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!C------------------------------------------------------------------
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!C Coefficients for approximation to erf in first interval
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!C------------------------------------------------------------------
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PARAMETER (A = (/ 3.16112374387056560D00,
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& 1.13864154151050156D02,3.77485237685302021D02,
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& 3.20937758913846947D03, 1.85777706184603153D-1/))
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PARAMETER ( B = (/2.36012909523441209D01,2.44024637934444173D02,
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& 1.28261652607737228D03,2.84423683343917062D03/))
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!C------------------------------------------------------------------
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!C Coefficients for approximation to erfc in second interval
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!C------------------------------------------------------------------
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PARAMETER ( C=(/5.64188496988670089D-1,8.88314979438837594D0,
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1 6.61191906371416295D01,2.98635138197400131D02,
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2 8.81952221241769090D02,1.71204761263407058D03,
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3 2.05107837782607147D03,1.23033935479799725D03,
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4 2.15311535474403846D-8/))
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PARAMETER ( D =(/1.57449261107098347D01,1.17693950891312499D02,
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1 5.37181101862009858D02,1.62138957456669019D03,
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2 3.29079923573345963D03,4.36261909014324716D03,
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3 3.43936767414372164D03,1.23033935480374942D03/))
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!C------------------------------------------------------------------
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!C Coefficients for approximation to erfc in third interval
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!C------------------------------------------------------------------
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PARAMETER ( P =(/3.05326634961232344D-1,3.60344899949804439D-1,
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1 1.25781726111229246D-1,1.60837851487422766D-2,
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2 6.58749161529837803D-4,1.63153871373020978D-2/))
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PARAMETER (Q =(/2.56852019228982242D00,1.87295284992346047D00,
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1 5.27905102951428412D-1,6.05183413124413191D-2,
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2 2.33520497626869185D-3/))
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!C------------------------------------------------------------------
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X = ARG
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Y = ABS(X)
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IF (Y .LE. THRESH) THEN
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!C------------------------------------------------------------------
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!C Evaluate erf for |X| <= 0.46875
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!C------------------------------------------------------------------
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YSQ = ZERO
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IF (Y .GT. XSMALL) THEN
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YSQ = Y * Y
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XNUM = A(5)*YSQ
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XDEN = YSQ
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DO I = 1, 3
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XNUM = (XNUM + A(I)) * YSQ
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XDEN = (XDEN + B(I)) * YSQ
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END DO
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RESULT = X * (XNUM + A(4)) / (XDEN + B(4))
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ELSE
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RESULT = X * A(4) / B(4)
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ENDIF
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IF (JINT .NE. 0) RESULT = ONE - RESULT
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IF (JINT .EQ. 2) RESULT = EXP(YSQ) * RESULT
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GO TO 800
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!C------------------------------------------------------------------
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!C Evaluate erfc for 0.46875 <= |X| <= 4.0
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!C------------------------------------------------------------------
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ELSE IF (Y .LE. FOUR) THEN
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XNUM = C(9)*Y
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XDEN = Y
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DO I = 1, 7
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XNUM = (XNUM + C(I)) * Y
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XDEN = (XDEN + D(I)) * Y
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END DO
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RESULT = (XNUM + C(8)) / (XDEN + D(8))
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IF (JINT .NE. 2) THEN
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YSQ = AINT(Y*SIXTEN)/SIXTEN
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DEL = (Y-YSQ)*(Y+YSQ)
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RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT
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END IF
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!C------------------------------------------------------------------
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!C Evaluate erfc for |X| > 4.0
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!C------------------------------------------------------------------
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ELSE
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RESULT = ZERO
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IF (Y .GE. XBIG) THEN
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IF ((JINT .NE. 2) .OR. (Y .GE. XMAX)) GO TO 300
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IF (Y .GE. XHUGE) THEN
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RESULT = SQRPI / Y
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GO TO 300
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END IF
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END IF
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YSQ = ONE / (Y * Y)
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XNUM = P(6)*YSQ
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XDEN = YSQ
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DO I = 1, 4
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XNUM = (XNUM + P(I)) * YSQ
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XDEN = (XDEN + Q(I)) * YSQ
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ENDDO
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RESULT = YSQ *(XNUM + P(5)) / (XDEN + Q(5))
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RESULT = (SQRPI - RESULT) / Y
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IF (JINT .NE. 2) THEN
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YSQ = AINT(Y*SIXTEN)/SIXTEN
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DEL = (Y-YSQ)*(Y+YSQ)
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RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT
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END IF
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END IF
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!C------------------------------------------------------------------
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!C Fix up for negative argument, erf, etc.
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!C------------------------------------------------------------------
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300 IF (JINT .EQ. 0) THEN
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RESULT = (HALF - RESULT) + HALF
|
||
|
|
IF (X .LT. ZERO) RESULT = -RESULT
|
||
|
|
ELSE IF (JINT .EQ. 1) THEN
|
||
|
|
IF (X .LT. ZERO) RESULT = TWO - RESULT
|
||
|
|
ELSE
|
||
|
|
IF (X .LT. ZERO) THEN
|
||
|
|
IF (X .LT. XNEG) THEN
|
||
|
|
RESULT = XINF
|
||
|
|
ELSE
|
||
|
|
YSQ = AINT(X*SIXTEN)/SIXTEN
|
||
|
|
DEL = (X-YSQ)*(X+YSQ)
|
||
|
|
Y = EXP(YSQ*YSQ) * EXP(DEL)
|
||
|
|
RESULT = (Y+Y) - RESULT
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
END IF
|
||
|
|
800 RETURN
|
||
|
|
END SUBROUTINE CALERF
|
||
|
|
END MODULE ERFCOREMOD
|