C $ f2py -m erfcore -h erfcore.pyf erfcore.f
C f2py erfcore.pyf erfcore.f -c --fcompiler=gnu95 --compiler=mingw32 -lmsvcr71
C $ f2py --fcompiler=gnu95 --compiler=mingw32 -lmsvcr71 -m erfcore -c erfcore.f
C
C--------------------------------------------------------------------
C
C DERF subprogram computes approximate values for erf(x).
C (see comments heading CALERF).
C
C Author/date: W. J. Cody, January 8, 1985
C
C--------------------------------------------------------------------
FUNCTION DERF( X ) RESULT (VALUE)
IMPLICIT NONE
DOUBLE PRECISION, INTENT(IN) :: X
DOUBLE PRECISION :: VALUE
INTEGER, PARAMETER :: JINT = 0
CALL CALERF(X,VALUE,JINT)
RETURN
END FUNCTION DERF
C--------------------------------------------------------------------
C
C DERFC subprogram computes approximate values for erfc(x).
C (see comments heading CALERF).
C
C Author/date: W. J. Cody, January 8, 1985
C
C--------------------------------------------------------------------
FUNCTION DERFC( X ) RESULT (VALUE)
IMPLICIT NONE
DOUBLE PRECISION, INTENT(IN) :: X
DOUBLE PRECISION :: VALUE
INTEGER, PARAMETER :: JINT = 1
CALL CALERF(X,VALUE,JINT)
RETURN
END FUNCTION DERFC
C------------------------------------------------------------------
C
C DERFCX subprogram computes approximate values for exp(x*x) * erfc(x).
C (see comments heading CALERF).
C
C Author/date: W. J. Cody, March 30, 1987
C
C------------------------------------------------------------------
FUNCTION DERFCX( X ) RESULT (VALUE)
IMPLICIT NONE
DOUBLE PRECISION, INTENT(IN) :: X
DOUBLE PRECISION :: VALUE
INTEGER, PARAMETER :: JINT = 2
CALL CALERF(X,VALUE,JINT)
RETURN
END FUNCTION DERFCX
SUBROUTINE CALERF(ARG,RESULT,JINT)
IMPLICIT NONE
C------------------------------------------------------------------
C
C CALERF packet evaluates erf(x), erfc(x), and exp(x*x)*erfc(x)
C for a real argument x. It contains three FUNCTION type
C subprograms: ERF, ERFC, and ERFCX (or DERF, DERFC, and DERFCX),
C and one SUBROUTINE type subprogram, CALERF. The calling
C statements for the primary entries are:
C
C Y=ERF(X) (or Y=DERF(X)),
C
C Y=ERFC(X) (or Y=DERFC(X)),
C and
C Y=ERFCX(X) (or Y=DERFCX(X)).
C
C The routine CALERF is intended for internal packet use only,
C all computations within the packet being concentrated in this
C routine. The function subprograms invoke CALERF with the
C statement
C
C CALL CALERF(ARG,RESULT,JINT)
C
C where the parameter usage is as follows
C
C Function Parameters for CALERF
C call ARG Result JINT
C
C ERF(ARG) ANY REAL ARGUMENT ERF(ARG) 0
C ERFC(ARG) ABS(ARG) .LT. XBIG ERFC(ARG) 1
C ERFCX(ARG) XNEG .LT. ARG .LT. XMAX ERFCX(ARG) 2
C
C The main computation evaluates near-minimax approximations
C from "Rational Chebyshev approximations for the error function"
C by W. J. Cody, Math. Comp., 1969, PP. 631-638. This
C transportable program uses rational functions that theoretically
C approximate erf(x) and erfc(x) to at least 18 significant
C decimal digits. The accuracy achieved depends on the arithmetic
C system, the compiler, the intrinsic functions, and proper
C selection of the machine-dependent constants.
C
C*******************************************************************
C*******************************************************************
C
C Explanation of machine-dependent constants
C
C XMIN = the smallest positive floating-point number.
C XINF = the largest positive finite floating-point number.
C XNEG = the largest negative argument acceptable to ERFCX;
C the negative of the solution to the equation
C 2*exp(x*x) = XINF.
C XSMALL = argument below which erf(x) may be represented by
C 2*x/sqrt(pi) and above which x*x will not underflow.
C A conservative value is the largest machine number X
C such that 1.0 + X = 1.0 to machine precision.
C XBIG = largest argument acceptable to ERFC; solution to
C the equation: W(x) * (1-0.5/x**2) = XMIN, where
C W(x) = exp(-x*x)/[x*sqrt(pi)].
C XHUGE = argument above which 1.0 - 1/(2*x*x) = 1.0 to
C machine precision. A conservative value is
C 1/[2*sqrt(XSMALL)]
C XMAX = largest acceptable argument to ERFCX; the minimum
C of XINF and 1/[sqrt(pi)*XMIN].
C
C Approximate values for some important machines are:
C
C XMIN XINF XNEG XSMALL
C
C C 7600 (S.P.) 3.13E-294 1.26E+322 -27.220 7.11E-15
C CRAY-1 (S.P.) 4.58E-2467 5.45E+2465 -75.345 7.11E-15
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 1.18E-38 3.40E+38 -9.382 5.96E-8
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2.23D-308 1.79D+308 -26.628 1.11D-16
C IBM 195 (D.P.) 5.40D-79 7.23E+75 -13.190 1.39D-17
C UNIVAC 1108 (D.P.) 2.78D-309 8.98D+307 -26.615 1.73D-18
C VAX D-Format (D.P.) 2.94D-39 1.70D+38 -9.345 1.39D-17
C VAX G-Format (D.P.) 5.56D-309 8.98D+307 -26.615 1.11D-16
C
C
C XBIG XHUGE XMAX
C
C C 7600 (S.P.) 25.922 8.39E+6 1.80X+293
C CRAY-1 (S.P.) 75.326 8.39E+6 5.45E+2465
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 9.194 2.90E+3 4.79E+37
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 26.543 6.71D+7 2.53D+307
C IBM 195 (D.P.) 13.306 1.90D+8 7.23E+75
C UNIVAC 1108 (D.P.) 26.582 5.37D+8 8.98D+307
C VAX D-Format (D.P.) 9.269 1.90D+8 1.70D+38
C VAX G-Format (D.P.) 26.569 6.71D+7 8.98D+307
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C The program returns ERFC = 0 for ARG .GE. XBIG;
C
C ERFCX = XINF for ARG .LT. XNEG;
C and
C ERFCX = 0 for ARG .GE. XMAX.
C
C
C Intrinsic functions required are:
C
C ABS, AINT, EXP
C
C
C Author: W. J. Cody
C Mathematics and Computer Science Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C Latest modification: March 19, 1990
C Updated to F90 by pab 23.03.2003
C
C------------------------------------------------------------------
DOUBLE PRECISION, INTENT(IN) :: ARG
INTEGER, INTENT(IN) :: JINT
DOUBLE PRECISION, INTENT(INOUT):: RESULT
! Local variables
INTEGER :: I
DOUBLE PRECISION :: DEL,X,XDEN,XNUM,Y,YSQ
C------------------------------------------------------------------
C Mathematical constants
C------------------------------------------------------------------
DOUBLE PRECISION, PARAMETER :: ZERO = 0.0D0
DOUBLE PRECISION, PARAMETER :: HALF = 0.05D0
DOUBLE PRECISION, PARAMETER :: ONE = 1.0D0
DOUBLE PRECISION, PARAMETER :: TWO = 2.0D0
DOUBLE PRECISION, PARAMETER :: FOUR = 4.0D0
DOUBLE PRECISION, PARAMETER :: SIXTEN = 16.0D0
DOUBLE PRECISION, PARAMETER :: SQRPI = 5.6418958354775628695D-1
DOUBLE PRECISION, PARAMETER :: THRESH = 0.46875D0
C------------------------------------------------------------------
C Machine-dependent constants
C------------------------------------------------------------------
DOUBLE PRECISION, PARAMETER :: XNEG = -26.628D0
DOUBLE PRECISION, PARAMETER :: XSMALL = 1.11D-16
DOUBLE PRECISION, PARAMETER :: XBIG = 26.543D0
DOUBLE PRECISION, PARAMETER :: XHUGE = 6.71D7
DOUBLE PRECISION, PARAMETER :: XMAX = 2.53D307
DOUBLE PRECISION, PARAMETER :: XINF = 1.79D308
!---------------------------------------------------------------
! Coefficents to the rational polynomials
!--------------------------------------------------------------
DOUBLE PRECISION, DIMENSION(5) :: A, Q
DOUBLE PRECISION, DIMENSION(4) :: B
DOUBLE PRECISION, DIMENSION(9) :: C
DOUBLE PRECISION, DIMENSION(8) :: D
DOUBLE PRECISION, DIMENSION(6) :: P
C------------------------------------------------------------------
C Coefficients for approximation to erf in first interval
C------------------------------------------------------------------
PARAMETER (A = (/ 3.16112374387056560D00,
& 1.13864154151050156D02,3.77485237685302021D02,
& 3.20937758913846947D03, 1.85777706184603153D-1/))
PARAMETER ( B = (/2.36012909523441209D01,2.44024637934444173D02,
& 1.28261652607737228D03,2.84423683343917062D03/))
C------------------------------------------------------------------
C Coefficients for approximation to erfc in second interval
C------------------------------------------------------------------
PARAMETER ( C=(/5.64188496988670089D-1,8.88314979438837594D0,
1 6.61191906371416295D01,2.98635138197400131D02,
2 8.81952221241769090D02,1.71204761263407058D03,
3 2.05107837782607147D03,1.23033935479799725D03,
4 2.15311535474403846D-8/))
PARAMETER ( D =(/1.57449261107098347D01,1.17693950891312499D02,
1 5.37181101862009858D02,1.62138957456669019D03,
2 3.29079923573345963D03,4.36261909014324716D03,
3 3.43936767414372164D03,1.23033935480374942D03/))
C------------------------------------------------------------------
C Coefficients for approximation to erfc in third interval
C------------------------------------------------------------------
PARAMETER ( P =(/3.05326634961232344D-1,3.60344899949804439D-1,
1 1.25781726111229246D-1,1.60837851487422766D-2,
2 6.58749161529837803D-4,1.63153871373020978D-2/))
PARAMETER (Q =(/2.56852019228982242D00,1.87295284992346047D00,
1 5.27905102951428412D-1,6.05183413124413191D-2,
2 2.33520497626869185D-3/))
C------------------------------------------------------------------
X = ARG
Y = ABS(X)
IF (Y .LE. THRESH) THEN
C------------------------------------------------------------------
C Evaluate erf for |X| <= 0.46875
C------------------------------------------------------------------
!YSQ = ZERO
IF (Y .GT. XSMALL) THEN
YSQ = Y * Y
XNUM = A(5)*YSQ
XDEN = YSQ
DO I = 1, 3
XNUM = (XNUM + A(I)) * YSQ
XDEN = (XDEN + B(I)) * YSQ
END DO
RESULT = X * (XNUM + A(4)) / (XDEN + B(4))
ELSE
RESULT = X * A(4) / B(4)
ENDIF
IF (JINT .NE. 0) RESULT = ONE - RESULT
IF (JINT .EQ. 2) RESULT = EXP(YSQ) * RESULT
GO TO 800
C------------------------------------------------------------------
C Evaluate erfc for 0.46875 <= |X| <= 4.0
C------------------------------------------------------------------
ELSE IF (Y .LE. FOUR) THEN
XNUM = C(9)*Y
XDEN = Y
DO I = 1, 7
XNUM = (XNUM + C(I)) * Y
XDEN = (XDEN + D(I)) * Y
END DO
RESULT = (XNUM + C(8)) / (XDEN + D(8))
IF (JINT .NE. 2) THEN
YSQ = AINT(Y*SIXTEN)/SIXTEN
DEL = (Y-YSQ)*(Y+YSQ)
RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT
END IF
C------------------------------------------------------------------
C Evaluate erfc for |X| > 4.0
C------------------------------------------------------------------
ELSE
RESULT = ZERO
IF (Y .GE. XBIG) THEN
IF ((JINT .NE. 2) .OR. (Y .GE. XMAX)) GO TO 300
IF (Y .GE. XHUGE) THEN
RESULT = SQRPI / Y
GO TO 300
END IF
END IF
YSQ = ONE / (Y * Y)
XNUM = P(6)*YSQ
XDEN = YSQ
DO I = 1, 4
XNUM = (XNUM + P(I)) * YSQ
XDEN = (XDEN + Q(I)) * YSQ
ENDDO
RESULT = YSQ *(XNUM + P(5)) / (XDEN + Q(5))
RESULT = (SQRPI - RESULT) / Y
IF (JINT .NE. 2) THEN
YSQ = AINT(Y*SIXTEN)/SIXTEN
DEL = (Y-YSQ)*(Y+YSQ)
RESULT = EXP(-YSQ*YSQ) * EXP(-DEL) * RESULT
END IF
END IF
C------------------------------------------------------------------
C Fix up for negative argument, erf, etc.
C------------------------------------------------------------------
300 IF (JINT .EQ. 0) THEN
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
FUNCTION FIINV(P) RESULT (VAL)
IMPLICIT NONE
*
* ALGORITHM AS241 APPL. STATIST. (1988) VOL. 37, NO. 3
*
* Produces the normal deviate Z corresponding to a given lower
* tail area of P.
* Absolute error less than 1e-13
* Relative error less than 1e-15 for abs(VAL)>0.1
*
* The hash sums below are the sums of the mantissas of the
* coefficients. They are included for use in checking
* transcription.
*
DOUBLE PRECISION, INTENT(in) :: P
DOUBLE PRECISION :: VAL
!local variables
DOUBLE PRECISION SPLIT1, SPLIT2, CONST1, CONST2, ONE, ZERO, HALF,
& A0, A1, A2, A3, A4, A5, A6, A7, B1, B2, B3, B4, B5, B6, B7,
& C0, C1, C2, C3, C4, C5, C6, C7, D1, D2, D3, D4, D5, D6, D7,
& E0, E1, E2, E3, E4, E5, E6, E7, F1, F2, F3, F4, F5, F6, F7,
& Q, R
PARAMETER ( SPLIT1 = 0.425D0, SPLIT2 = 5.D0,
& CONST1 = 0.180625D0, CONST2 = 1.6D0,
& ONE = 1.D0, ZERO = 0.D0, HALF = 0.5D0 )
*
* Coefficients for P close to 0.5
*
PARAMETER (
* A0 = 3.38713 28727 96366 6080D0,
* A1 = 1.33141 66789 17843 7745D+2,
* A2 = 1.97159 09503 06551 4427D+3,
* A3 = 1.37316 93765 50946 1125D+4,
* A4 = 4.59219 53931 54987 1457D+4,
* A5 = 6.72657 70927 00870 0853D+4,
* A6 = 3.34305 75583 58812 8105D+4,
* A7 = 2.50908 09287 30122 6727D+3,
* B1 = 4.23133 30701 60091 1252D+1,
* B2 = 6.87187 00749 20579 0830D+2,
* B3 = 5.39419 60214 24751 1077D+3,
* B4 = 2.12137 94301 58659 5867D+4,
* B5 = 3.93078 95800 09271 0610D+4,
* B6 = 2.87290 85735 72194 2674D+4,
* B7 = 5.22649 52788 52854 5610D+3 )
* HASH SUM AB 55.88319 28806 14901 4439
*
* Coefficients for P not close to 0, 0.5 or 1.
*
PARAMETER (
* C0 = 1.42343 71107 49683 57734D0,
* C1 = 4.63033 78461 56545 29590D0,
* C2 = 5.76949 72214 60691 40550D0,
* C3 = 3.64784 83247 63204 60504D0,
* C4 = 1.27045 82524 52368 38258D0,
* C5 = 2.41780 72517 74506 11770D-1,
* C6 = 2.27238 44989 26918 45833D-2,
* C7 = 7.74545 01427 83414 07640D-4,
* D1 = 2.05319 16266 37758 82187D0,
* D2 = 1.67638 48301 83803 84940D0,
* D3 = 6.89767 33498 51000 04550D-1,
* D4 = 1.48103 97642 74800 74590D-1,
* D5 = 1.51986 66563 61645 71966D-2,
* D6 = 5.47593 80849 95344 94600D-4,
* D7 = 1.05075 00716 44416 84324D-9 )
* HASH SUM CD 49.33206 50330 16102 89036
*
* Coefficients for P near 0 or 1.
*
PARAMETER (
* E0 = 6.65790 46435 01103 77720D0,
* E1 = 5.46378 49111 64114 36990D0,
* E2 = 1.78482 65399 17291 33580D0,
* E3 = 2.96560 57182 85048 91230D-1,
* E4 = 2.65321 89526 57612 30930D-2,
* E5 = 1.24266 09473 88078 43860D-3,
* E6 = 2.71155 55687 43487 57815D-5,
* E7 = 2.01033 43992 92288 13265D-7,
* F1 = 5.99832 20655 58879 37690D-1,
* F2 = 1.36929 88092 27358 05310D-1,
* F3 = 1.48753 61290 85061 48525D-2,
* F4 = 7.86869 13114 56132 59100D-4,
* F5 = 1.84631 83175 10054 68180D-5,
* F6 = 1.42151 17583 16445 88870D-7,
* F7 = 2.04426 31033 89939 78564D-15 )
* HASH SUM EF 47.52583 31754 92896 71629
*
Q = ( P - HALF)
IF ( ABS(Q) .LE. SPLIT1 ) THEN ! Central range.
R = CONST1 - Q*Q
VAL = Q*( ( ( ((((A7*R + A6)*R + A5)*R + A4)*R + A3)
* *R + A2 )*R + A1 )*R + A0 )
* /( ( ( ((((B7*R + B6)*R + B5)*R + B4)*R + B3)
* *R + B2 )*R + B1 )*R + ONE)
ELSE ! near the endpoints
R = MIN( P, ONE - P )
IF (R .GT.ZERO) THEN ! ( 2.d0*R .GT. CFxCutOff) THEN ! R .GT.0.d0
R = SQRT( -LOG(R) )
IF ( R .LE. SPLIT2 ) THEN
R = R - CONST2
VAL = ( ( ( ((((C7*R + C6)*R + C5)*R + C4)*R + C3)
* *R + C2 )*R + C1 )*R + C0 )
* /( ( ( ((((D7*R + D6)*R + D5)*R + D4)*R + D3)
* *R + D2 )*R + D1 )*R + ONE )
ELSE
R = R - SPLIT2
VAL = ( ( ( ((((E7*R + E6)*R + E5)*R + E4)*R + E3)
* *R + E2 )*R + E1 )*R + E0 )
* /( ( ( ((((F7*R + F6)*R + F5)*R + F4)*R + F3)
* *R + F2 )*R + F1 )*R + ONE )
END IF
ELSE
VAL = 37.D0 !XMAX 9.d0
END IF
IF ( Q < ZERO ) VAL = - VAL
END IF
RETURN
END FUNCTION FIINV
FUNCTION FI2( Z ) RESULT (VALUE)
! USE GLOBALDATA, ONLY : XMAX
IMPLICIT NONE
DOUBLE PRECISION, INTENT(in) :: Z
DOUBLE PRECISION :: VALUE
*
* Normal distribution probabilities accurate to 1.e-15.
* relative error less than 1e-8;
* Z = no. of standard deviations from the mean.
*
* Based upon algorithm 5666 for the error function, from:
* Hart, J.F. et al, 'Computer Approximations', Wiley 1968
*
* Programmer: Alan Miller
*
* Latest revision - 30 March 1986
*
DOUBLE PRECISION :: P0, P1, P2, P3, P4, P5, P6,
* Q0, Q1, Q2, Q3, Q4, Q5, Q6, Q7,XMAX,
* P, EXPNTL, CUTOFF, ROOTPI, ZABS, Z2
PARAMETER(
* P0 = 220.20 68679 12376 1D0,
* P1 = 221.21 35961 69931 1D0,
* P2 = 112.07 92914 97870 9D0,
* P3 = 33.912 86607 83830 0D0,
* P4 = 6.3739 62203 53165 0D0,
* P5 = 0.70038 30644 43688 1D0,
* P6 = 0.035262 49659 98910 9D0 )
PARAMETER(
* Q0 = 440.41 37358 24752 2D0,
* Q1 = 793.82 65125 19948 4D0,
* Q2 = 637.33 36333 78831 1D0,
* Q3 = 296.56 42487 79673 7D0,
* Q4 = 86.780 73220 29460 8D0,
* Q5 = 16.064 17757 92069 5D0,
* Q6 = 1.7556 67163 18264 2D0,
* Q7 = 0.088388 34764 83184 4D0 )
PARAMETER( ROOTPI = 2.5066 28274 63100 1D0 )
PARAMETER( CUTOFF = 7.0710 67811 86547 5D0 )
PARAMETER( XMAX = 8.25D0 )
*
ZABS = ABS(Z)
*
* |Z| > 37 (or XMAX)
*
IF ( ZABS .GT. XMAX ) THEN
P = 0.d0
ELSE
*
* |Z| <= 37
*
Z2 = ZABS * ZABS
EXPNTL = EXP( -Z2 * 0.5D0 )
*
* |Z| < CUTOFF = 10/SQRT(2)
*
IF ( ZABS < CUTOFF ) THEN
P = EXPNTL*( (((((P6*ZABS + P5)*ZABS + P4)*ZABS + P3)*ZABS
* + P2)*ZABS + P1)*ZABS + P0)/(((((((Q7*ZABS + Q6)*ZABS
* + Q5)*ZABS + Q4)*ZABS + Q3)*ZABS + Q2)*ZABS + Q1)*ZABS
* + Q0 )
*
* |Z| >= CUTOFF.
*
ELSE
P = EXPNTL/( ZABS + 1.d0/( ZABS + 2.d0/( ZABS + 3.d0/( ZABS
* + 4.d0/( ZABS + 0.65D0 ) ) ) ) )/ROOTPI
END IF
END IF
IF ( Z .GT. 0.d0 ) P = 1.d0 - P
VALUE = P
RETURN
END FUNCTION FI2
FUNCTION FI( Z ) RESULT (VALUE)
IMPLICIT NONE
DOUBLE PRECISION, INTENT(in) :: Z
DOUBLE PRECISION :: VALUE
! Local variables
DOUBLE PRECISION, PARAMETER:: SQ2M1 = 0.70710678118655D0 ! 1/SQRT(2)
DOUBLE PRECISION, PARAMETER:: HALF = 0.5D0
VALUE = DERFC(-Z*SQ2M1)*HALF
RETURN
END FUNCTION FI