375 lines
19 KiB
Fortran
375 lines
19 KiB
Fortran
|
!
|
||
|
|
!=======================================================================
|
||
|
|
!
|
||
|
|
MODULE CONFLUENT_HYPGEOM_REAL
|
||
|
|
!
|
||
|
|
! This module provides several subroutines to compute
|
||
|
|
! the Gauss hypergeometric function
|
||
|
|
!
|
||
|
|
! 2F1(a,b,c;x)
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! --> <--
|
||
|
|
! --> x real <--
|
||
|
|
! --> <--
|
||
|
|
!
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! 1) SUBROUTINE HYGFX(A,B,C,X,HF)
|
||
|
|
!
|
||
|
|
! 2) SUBROUTINE HYP(Z,A,B,C,RE,IM)
|
||
|
|
!
|
||
|
|
!
|
||
|
|
!
|
||
|
|
USE ACCURACY_REAL
|
||
|
|
!
|
||
|
|
CONTAINS
|
||
|
|
!
|
||
|
|
!=======================================================================
|
||
|
|
!
|
||
|
|
SUBROUTINE HYGFX(A,B,C,X,HF)
|
||
|
|
!
|
||
|
|
! HYGFX computes the hypergeometric function F(a,b,c,x).
|
||
|
|
!
|
||
|
|
! Licensing:
|
||
|
|
!
|
||
|
|
! This routine is copyrighted by Shanjie Zhang and Jianming Jin. However,
|
||
|
|
! they give permission to incorporate this routine into a user program
|
||
|
|
! provided that the copyright is acknowledged.
|
||
|
|
!
|
||
|
|
! Modified:
|
||
|
|
!
|
||
|
|
! 04 April 2012
|
||
|
|
!
|
||
|
|
! Author:
|
||
|
|
!
|
||
|
|
! Shanjie Zhang, Jianming Jin
|
||
|
|
!
|
||
|
|
! Reference:
|
||
|
|
!
|
||
|
|
! Shanjie Zhang, Jianming Jin,
|
||
|
|
! Computation of Special Functions,
|
||
|
|
! Wiley, 1996,
|
||
|
|
! ISBN: 0-471-11963-6,
|
||
|
|
! LC: QA351.C45.
|
||
|
|
!
|
||
|
|
! ==========================================================
|
||
|
|
!
|
||
|
|
! Purpose: Compute hypergeometric function F(a,b,c,x)
|
||
|
|
!
|
||
|
|
! Input : A --- Parameter
|
||
|
|
! B --- Parameter
|
||
|
|
! C --- Parameter, C <> 0,-1,-2,...
|
||
|
|
! X --- Argument (X < 1)
|
||
|
|
!
|
||
|
|
! Output: HF --- F(A,B,C,X)
|
||
|
|
!
|
||
|
|
! Routines called:
|
||
|
|
! (1) GAMMA for computing gamma function
|
||
|
|
! (2) PSI for computing psi function
|
||
|
|
!
|
||
|
|
! ==========================================================
|
||
|
|
!
|
||
|
|
!
|
||
|
|
! Last modified (DS) : 1 Sep 2020
|
||
|
|
!
|
||
|
|
!
|
||
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,HALF
|
||
|
|
USE PI_ETC, ONLY : PI
|
||
|
|
USE GAMMA_FUNCTION, ONLY : GAMMA
|
||
|
|
USE DIGAMMA_FUNCTION, ONLY : PSI
|
||
|
|
!
|
||
|
|
IMPLICIT NONE
|
||
|
|
!
|
||
|
|
INTEGER :: J,K,M,NM
|
||
|
|
!
|
||
|
|
REAL (WP), INTENT(IN) :: C
|
||
|
|
REAL (WP) :: A,B,X
|
||
|
|
REAL (WP), INTENT(OUT) :: HF
|
||
|
|
!
|
||
|
|
REAL (WP) :: C0
|
||
|
|
REAL (WP) :: EPS
|
||
|
|
REAL (WP) :: G0,G1,G2,G3
|
||
|
|
REAL (WP) :: GC,R
|
||
|
|
REAL (WP) :: A0,AA,BB,C1
|
||
|
|
REAL (WP) :: F0,F1
|
||
|
|
REAL (WP) :: GA,GABC,GAM,GB,GBM,GCA,GCAB,GCB,GM
|
||
|
|
REAL (WP) :: HW
|
||
|
|
REAL (WP) :: PA,PB,R1,RM,RP
|
||
|
|
REAL (WP) :: SM,SP,SP0
|
||
|
|
REAL (WP) :: R0,X1
|
||
|
|
!
|
||
|
|
REAL, PARAMETER :: EL = 0.5772156649015329E+00_WP
|
||
|
|
!
|
||
|
|
LOGICAL :: L0,L1,L2,L3,L4,L5
|
||
|
|
!
|
||
|
|
L0 = C == INT(C) .AND. C < ZERO !
|
||
|
|
L1 = ONE - X < 1.0E-15_WP .AND. C - A - B <= ZERO !
|
||
|
|
L2 = A == INT(A) .AND. A < ZERO !
|
||
|
|
L3 = B == INT(B) .AND. B < ZERO !
|
||
|
|
L4 = C - A == INT(C - A) .AND. C - A <= ZERO !
|
||
|
|
L5 = C - B == INT(C - B) .AND. C - B <= ZERO !
|
||
|
|
!
|
||
|
|
IF(L0 .OR. L1) THEN !
|
||
|
|
WRITE(6,110) !
|
||
|
|
WRITE(6,111) !
|
||
|
|
WRITE(6,112) !
|
||
|
|
STOP !
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
IF(0.95E+00_WP < X) THEN !
|
||
|
|
EPS = 1.0E-08_WP !
|
||
|
|
ELSE !
|
||
|
|
EPS = 1.0E-15_WP !
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
IF(X == ZERO .OR. A == ZERO .OR. B == ZERO) THEN !
|
||
|
|
HF = ONE !
|
||
|
|
RETURN !
|
||
|
|
ELSE IF(ONE - X == EPS .AND. ZERO < C - A - B) THEN !
|
||
|
|
CALL GAMMA(C, GC) !
|
||
|
|
CALL GAMMA(C - A - B, GCAB) !
|
||
|
|
CALL GAMMA(C - A, GCA) !
|
||
|
|
CALL GAMMA(C - B, GCB) !
|
||
|
|
HF = GC * GCAB / (GCA * GCB) !
|
||
|
|
RETURN !
|
||
|
|
ELSE IF(ONE + X <= EPS .AND. & !
|
||
|
|
DABS(C - A + B -ONE) <= EPS) THEN !
|
||
|
|
G0 = DSQRT(PI) * TWO ** (- A) !
|
||
|
|
CALL GAMMA(C, G1) !
|
||
|
|
CALL GAMMA(ONE + A / TWO - B, G2) !
|
||
|
|
CALL GAMMA(HALF + HALF * A, G3) !
|
||
|
|
HF = G0 * G1 / (G2 * G3) !
|
||
|
|
RETURN !
|
||
|
|
ELSE IF(L2 .OR. L3) THEN !
|
||
|
|
IF(L2) THEN !
|
||
|
|
NM = INT(DABS(A)) !
|
||
|
|
END IF !
|
||
|
|
IF(L3) THEN !
|
||
|
|
NM = INT(DABS(B)) !
|
||
|
|
END IF !
|
||
|
|
HF = ONE !
|
||
|
|
R = ONE !
|
||
|
|
DO K = 1, NM !
|
||
|
|
R = R * (A + K - ONE) * (B + K - ONE) / & !
|
||
|
|
(K * (C + K - ONE)) * X !
|
||
|
|
HF = HF + R !
|
||
|
|
END DO !
|
||
|
|
RETURN !
|
||
|
|
ELSE IF(L4 .OR. L5) THEN !
|
||
|
|
IF(L4) THEN !
|
||
|
|
NM = INT(DABS(C - A)) !
|
||
|
|
END IF !
|
||
|
|
IF(L5) THEN !
|
||
|
|
NM = INT(DABS(C - B)) !
|
||
|
|
END IF !
|
||
|
|
HF = ONE !
|
||
|
|
R = ONE !
|
||
|
|
DO K = 1, NM !
|
||
|
|
R = R * (C - A + K - ONE) * (C - B + K - ONE) / & !
|
||
|
|
(K * (C + K - ONE)) * X !
|
||
|
|
HF = HF + R !
|
||
|
|
END DO !
|
||
|
|
HF = (ONE - X) ** (C - A - B) * HF !
|
||
|
|
RETURN !
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
AA = A !
|
||
|
|
BB = B !
|
||
|
|
X1 = X !
|
||
|
|
!
|
||
|
|
IF(X < ZERO) THEN !
|
||
|
|
X = X / (X - ONE) !
|
||
|
|
IF(A < C .AND. B < A .AND. ZERO < B) THEN !
|
||
|
|
A = BB !
|
||
|
|
B = AA !
|
||
|
|
END IF !
|
||
|
|
B = C - B !
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
IF(0.75E+00_WP <= X) THEN !
|
||
|
|
GM = ZERO !
|
||
|
|
IF(DABS(C - A - B - INT(C - A - B)) < 1.0E-15_WP) THEN !
|
||
|
|
M = INT(C - A - B) !
|
||
|
|
CALL GAMMA(A, GA) !
|
||
|
|
CALL GAMMA(B, GB) !
|
||
|
|
CALL GAMMA(C, GC) !
|
||
|
|
CALL GAMMA(A + M, GAM) !
|
||
|
|
CALL GAMMA(B + M, GBM) !
|
||
|
|
CALL PSI(A, PA) !
|
||
|
|
CALL PSI(B, PB) !
|
||
|
|
IF(M .NE. 0) THEN !
|
||
|
|
GM = ONE !
|
||
|
|
END IF !
|
||
|
|
DO J = 1, ABS (M) - 1 !
|
||
|
|
GM = GM * J !
|
||
|
|
END DO !
|
||
|
|
RM = ONE !
|
||
|
|
DO J = 1, ABS (M) !
|
||
|
|
RM = RM * J !
|
||
|
|
END DO !
|
||
|
|
F0 = ONE !
|
||
|
|
R0 = ONE !
|
||
|
|
R1 = ONE !
|
||
|
|
SP0 = ZERO !
|
||
|
|
SP = ZERO !
|
||
|
|
IF(0 <= M) THEN !
|
||
|
|
C0 = GM * GC / (GAM * GBM) !
|
||
|
|
C1 = - GC * (X - ONE) ** M / (GA * GB * RM) !
|
||
|
|
DO K = 1, M - 1 !
|
||
|
|
R0 = R0 * (A + K - ONE) * (B + K - ONE) / & !
|
||
|
|
(K * (K - M)) * (ONE - X) !
|
||
|
|
F0 = F0 + R0 !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
DO K = 1, M !
|
||
|
|
SP0 = SP0 + ONE / (A + K - ONE) + & !
|
||
|
|
ONE / (B + K - ONE) - ONE / K !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
F1 = PA + PB + SP0 + TWO * EL + DLOG(ONE - X) !
|
||
|
|
!
|
||
|
|
DO K = 1, 250 !
|
||
|
|
SP = SP + (ONE - A) / (K * (A + K - ONE)) + & !
|
||
|
|
(ONE - B) / (K * (B + K - ONE)) !
|
||
|
|
SM = ZERO !
|
||
|
|
DO J = 1,M !
|
||
|
|
SM = SM + (ONE - A) / ((J + K) * & !
|
||
|
|
(A + J + K - ONE)) + ONE / & !
|
||
|
|
(B + J + K - ONE) !
|
||
|
|
END DO !
|
||
|
|
RP = PA + PB + TWO * EL + SP + SM + DLOG(ONE - X) !
|
||
|
|
R1 = R1 * (A + M + K - ONE) * (B + M + K - ONE) / & !
|
||
|
|
(K * (M + K)) * (ONE - X) !
|
||
|
|
F1 = F1 + R1 * RP !
|
||
|
|
IF(DABS(F1 - HW) < DABS(F1) * EPS) THEN !
|
||
|
|
GO TO 10 !
|
||
|
|
END IF !
|
||
|
|
HW = F1 !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
10 CONTINUE
|
||
|
|
!
|
||
|
|
HF = F0 *C0 + F1 * C1 !
|
||
|
|
!
|
||
|
|
ELSE IF(M < 0) THEN !
|
||
|
|
!
|
||
|
|
M = - M !
|
||
|
|
C0 = GM * GC / (GA * GB * (ONE - X)**M) !
|
||
|
|
C1 = - (-1)**M * GC / (GAM * GBM * RM) !
|
||
|
|
!
|
||
|
|
DO K = 1,M-1 !
|
||
|
|
R0 = R0 * (A - M + K - ONE) * & !
|
||
|
|
(B - M + K - ONE) / & !
|
||
|
|
(K * (K - M)) * (ONE - X) !
|
||
|
|
F0 = F0 + R0 !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
DO K = 1,M !
|
||
|
|
SP0 = SP0 + ONE / K !
|
||
|
|
END DO !
|
||
|
|
F1 = PA + PB - SP0 + TWO * EL + DLOG(ONE - X) !
|
||
|
|
DO K = 1,250 !
|
||
|
|
SP = SP + (ONE - A) / (K * (A + K - ONE)) + & !
|
||
|
|
(ONE - B) / (K * (B + K - ONE)) !
|
||
|
|
SM = ZERO !
|
||
|
|
DO J = 1,M !
|
||
|
|
SM = SM+ ONE /(J + K) !
|
||
|
|
END DO !
|
||
|
|
RP = PA + PB + TWO * EL + SP - SM + & !
|
||
|
|
DLOG(ONE - X) !
|
||
|
|
R1 = R1 * (A + K - ONE) * (B + K - ONE) / & !
|
||
|
|
(K * (M + K)) * (ONE - X) !
|
||
|
|
F1 = F1 + R1 * RP !
|
||
|
|
IF(DABS(F1 - HW) < DABS(F1) * EPS) THEN !
|
||
|
|
GO TO 20 !
|
||
|
|
END IF !
|
||
|
|
HW = F1 !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
20 CONTINUE !
|
||
|
|
!
|
||
|
|
HF = F0 * C0 + F1 * C1 !
|
||
|
|
END IF !
|
||
|
|
ELSE !
|
||
|
|
CALL GAMMA(A,GA) !
|
||
|
|
CALL GAMMA(B,GB) !
|
||
|
|
CALL GAMMA(C,GC) !
|
||
|
|
CALL GAMMA(C-A,GCA) !
|
||
|
|
CALL GAMMA(C-B,GCB) !
|
||
|
|
CALL GAMMA(C-A-B,GCAB) !
|
||
|
|
CALL GAMMA(A+B-C,GABC) !
|
||
|
|
C0 = GC * GCAB / (GCA * GCB) !
|
||
|
|
C1 = GC * GABC / (GA * GB) * (ONE - X)**(C - A - B) !
|
||
|
|
HF = ZERO !
|
||
|
|
R0 = C0 !
|
||
|
|
R1 = C1 !
|
||
|
|
DO K = 1,250 !
|
||
|
|
R0 = R0 * (A + K - ONE) * (B + K - ONE) / & !
|
||
|
|
(K * (A + B - C + K)) * (ONE - X) !
|
||
|
|
R1 = R1 * (C - A + K - ONE) * (C - B + K - ONE) /& !
|
||
|
|
(K * (C - A - B + K)) * (ONE - X) !
|
||
|
|
HF = HF + R0 + R1 !
|
||
|
|
IF(DABS(HF - HW) < DABS(HF) * EPS) THEN !
|
||
|
|
GO TO 30 !
|
||
|
|
END IF !
|
||
|
|
HW = HF !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
30 CONTINUE !
|
||
|
|
!
|
||
|
|
HF = HF + C0 + C1 !
|
||
|
|
!
|
||
|
|
END IF !
|
||
|
|
ELSE !
|
||
|
|
A0 = ONE !
|
||
|
|
IF(C > A .AND. C < TWO * A .AND. & !
|
||
|
|
C > B .AND. C < TWO*B) THEN !
|
||
|
|
A0 = (ONE - X) ** (C - A - B) !
|
||
|
|
A = C - A !
|
||
|
|
B = C - B !
|
||
|
|
END IF !
|
||
|
|
HF = ONE !
|
||
|
|
R = ONE !
|
||
|
|
DO K = 1,250 !
|
||
|
|
R = R *(A + K - ONE) * (B + K - ONE) / & !
|
||
|
|
(K * (C + K - ONE)) * X !
|
||
|
|
HF = HF + R !
|
||
|
|
IF(DABS(HF - HW) <= DABS(HF) * EPS) THEN !
|
||
|
|
GO TO 40 !
|
||
|
|
END IF !
|
||
|
|
HW = HF !
|
||
|
|
END DO !
|
||
|
|
!
|
||
|
|
40 CONTINUE !
|
||
|
|
!
|
||
|
|
HF = A0 * HF !
|
||
|
|
!
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
IF(X1 < ZERO) THEN !
|
||
|
|
X = X1 !
|
||
|
|
C0 = ONE / (ONE - X) ** AA !
|
||
|
|
HF = C0 * HF !
|
||
|
|
END IF
|
||
|
|
!
|
||
|
|
A = AA !
|
||
|
|
B = BB !
|
||
|
|
!
|
||
|
|
IF(120 < K) THEN !
|
||
|
|
WRITE(*,115) !
|
||
|
|
END IF !
|
||
|
|
!
|
||
|
|
! Formats:
|
||
|
|
!
|
||
|
|
110 FORMAT(' ')
|
||
|
|
111 FORMAT('HYGFX - Fatal error!')
|
||
|
|
112 FORMAT('The hypergeometric series is divergent.')
|
||
|
|
115 FORMAT('Warning! You should check the accuracy')
|
||
|
|
!
|
||
|
|
RETURN !
|
||
|
|
!
|
||
|
|
END SUBROUTINE HYGFX
|
||
|
|
!
|
||
|
|
END MODULE CONFLUENT_HYPGEOM_REAL
|