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CS REAL FUNCTION GAMMA(X)
DOUBLE PRECISION FUNCTION DGAMMA(X)
C----------------------------------------------------------------------
C
C This routine calculates the GAMMA function for a real argument X.
C Computation is based on an algorithm outlined in reference 1.
C The program uses rational functions that approximate the GAMMA
C function to at least 20 significant decimal digits. Coefficients
C for the approximation over the interval (1,2) are unpublished.
C Those for the approximation for X .GE. 12 are from reference 2.
C The accuracy achieved depends on the arithmetic system, the
C compiler, the intrinsic functions, and proper selection of the
C machine-dependent constants.
C
C
C*******************************************************************
C*******************************************************************
C
C Explanation of machine-dependent constants
C
C beta - radix for the floating-point representation
C maxexp - the smallest positive power of beta that overflows
C XBIG - the largest argument for which GAMMA(X) is representable
C in the machine, i.e., the solution to the equation
C GAMMA(XBIG) = beta**maxexp
C XINF - the largest machine representable floating-point number;
C approximately beta**maxexp
C EPS - the smallest positive floating-point number such that
C 1.0+EPS .GT. 1.0
C XMININ - the smallest positive floating-point number such that
C 1/XMININ is machine representable
C
C Approximate values for some important machines are:
C
C beta maxexp XBIG
C
C CRAY-1 (S.P.) 2 8191 966.961
C Cyber 180/855
C under NOS (S.P.) 2 1070 177.803
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 2 128 35.040
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2 1024 171.624
C IBM 3033 (D.P.) 16 63 57.574
C VAX D-Format (D.P.) 2 127 34.844
C VAX G-Format (D.P.) 2 1023 171.489
C
C XINF EPS XMININ
C
C CRAY-1 (S.P.) 5.45E+2465 7.11E-15 1.84E-2466
C Cyber 180/855
C under NOS (S.P.) 1.26E+322 3.55E-15 3.14E-294
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 3.40E+38 1.19E-7 1.18E-38
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 1.79D+308 2.22D-16 2.23D-308
C IBM 3033 (D.P.) 7.23D+75 2.22D-16 1.39D-76
C VAX D-Format (D.P.) 1.70D+38 1.39D-17 5.88D-39
C VAX G-Format (D.P.) 8.98D+307 1.11D-16 1.12D-308
C
C*******************************************************************
C*******************************************************************
C
C Error returns
C
C The program returns the value XINF for singularities or
C when overflow would occur. The computation is believed
C to be free of underflow and overflow.
C
C
C Intrinsic functions required are:
C
C INT, DBLE, EXP, LOG, REAL, SIN
C
C
C References: "An Overview of Software Development for Special
C Functions", W. J. Cody, Lecture Notes in Mathematics,
C 506, Numerical Analysis Dundee, 1975, G. A. Watson
C (ed.), Springer Verlag, Berlin, 1976.
C
C Computer Approximations, Hart, Et. Al., Wiley and
C sons, New York, 1968.
C
C Latest modification: October 12, 1989
C
C Authors: W. J. Cody and L. Stoltz
C Applied Mathematics Division
C Argonne National Laboratory
C Argonne, IL 60439
C
C----------------------------------------------------------------------
INTEGER I,N
LOGICAL PARITY
CS REAL
DOUBLE PRECISION
1 C,CONV,EPS,FACT,HALF,ONE,P,PI,Q,RES,SQRTPI,SUM,TWELVE,
2 TWO,X,XBIG,XDEN,XINF,XMININ,XNUM,Y,Y1,YSQ,Z,ZERO
DIMENSION C(7),P(8),Q(8)
C----------------------------------------------------------------------
C Mathematical constants
C----------------------------------------------------------------------
CS DATA ONE,HALF,TWELVE,TWO,ZERO/1.0E0,0.5E0,12.0E0,2.0E0,0.0E0/,
CS 1 SQRTPI/0.9189385332046727417803297E0/,
CS 2 PI/3.1415926535897932384626434E0/
DATA ONE,HALF,TWELVE,TWO,ZERO/1.0D0,0.5D0,12.0D0,2.0D0,0.0D0/,
1 SQRTPI/0.9189385332046727417803297D0/,
2 PI/3.1415926535897932384626434D0/
C----------------------------------------------------------------------
C Machine dependent parameters
C----------------------------------------------------------------------
CS DATA XBIG,XMININ,EPS/35.040E0,1.18E-38,1.19E-7/,
CS 1 XINF/3.4E38/
DATA XBIG,XMININ,EPS/171.624D0,2.23D-308,2.22D-16/,
1 XINF/1.79D308/
C----------------------------------------------------------------------
C Numerator and denominator coefficients for rational minimax
C approximation over (1,2).
C----------------------------------------------------------------------
CS DATA P/-1.71618513886549492533811E+0,2.47656508055759199108314E+1,
CS 1 -3.79804256470945635097577E+2,6.29331155312818442661052E+2,
CS 2 8.66966202790413211295064E+2,-3.14512729688483675254357E+4,
CS 3 -3.61444134186911729807069E+4,6.64561438202405440627855E+4/
CS DATA Q/-3.08402300119738975254353E+1,3.15350626979604161529144E+2,
CS 1 -1.01515636749021914166146E+3,-3.10777167157231109440444E+3,
CS 2 2.25381184209801510330112E+4,4.75584627752788110767815E+3,
CS 3 -1.34659959864969306392456E+5,-1.15132259675553483497211E+5/
DATA P/-1.71618513886549492533811D+0,2.47656508055759199108314D+1,
1 -3.79804256470945635097577D+2,6.29331155312818442661052D+2,
2 8.66966202790413211295064D+2,-3.14512729688483675254357D+4,
3 -3.61444134186911729807069D+4,6.64561438202405440627855D+4/
DATA Q/-3.08402300119738975254353D+1,3.15350626979604161529144D+2,
1 -1.01515636749021914166146D+3,-3.10777167157231109440444D+3,
2 2.25381184209801510330112D+4,4.75584627752788110767815D+3,
3 -1.34659959864969306392456D+5,-1.15132259675553483497211D+5/
C----------------------------------------------------------------------
C Coefficients for minimax approximation over (12, INF).
C----------------------------------------------------------------------
CS DATA C/-1.910444077728E-03,8.4171387781295E-04,
CS 1 -5.952379913043012E-04,7.93650793500350248E-04,
CS 2 -2.777777777777681622553E-03,8.333333333333333331554247E-02,
CS 3 5.7083835261E-03/
DATA C/-1.910444077728D-03,8.4171387781295D-04,
1 -5.952379913043012D-04,7.93650793500350248D-04,
2 -2.777777777777681622553D-03,8.333333333333333331554247D-02,
3 5.7083835261D-03/
C----------------------------------------------------------------------
C Statement functions for conversion between integer and float
C----------------------------------------------------------------------
CS CONV(I) = REAL(I)
CONV(I) = DBLE(I)
PARITY = .FALSE.
FACT = ONE
N = 0
Y = X
IF (Y .LE. ZERO) THEN
C----------------------------------------------------------------------
C Argument is negative
C----------------------------------------------------------------------
Y = -X
Y1 = AINT(Y)
RES = Y - Y1
IF (RES .NE. ZERO) THEN
IF (Y1 .NE. AINT(Y1*HALF)*TWO) PARITY = .TRUE.
FACT = -PI / SIN(PI*RES)
Y = Y + ONE
ELSE
RES = XINF
GO TO 900
END IF
END IF
C----------------------------------------------------------------------
C Argument is positive
C----------------------------------------------------------------------
IF (Y .LT. EPS) THEN
C----------------------------------------------------------------------
C Argument .LT. EPS
C----------------------------------------------------------------------
IF (Y .GE. XMININ) THEN
RES = ONE / Y
ELSE
RES = XINF
GO TO 900
END IF
ELSE IF (Y .LT. TWELVE) THEN
Y1 = Y
IF (Y .LT. ONE) THEN
C----------------------------------------------------------------------
C 0.0 .LT. argument .LT. 1.0
C----------------------------------------------------------------------
Z = Y
Y = Y + ONE
ELSE
C----------------------------------------------------------------------
C 1.0 .LT. argument .LT. 12.0, reduce argument if necessary
C----------------------------------------------------------------------
N = INT(Y) - 1
Y = Y - CONV(N)
Z = Y - ONE
END IF
C----------------------------------------------------------------------
C Evaluate approximation for 1.0 .LT. argument .LT. 2.0
C----------------------------------------------------------------------
XNUM = ZERO
XDEN = ONE
DO 260 I = 1, 8
XNUM = (XNUM + P(I)) * Z
XDEN = XDEN * Z + Q(I)
260 CONTINUE
RES = XNUM / XDEN + ONE
IF (Y1 .LT. Y) THEN
C----------------------------------------------------------------------
C Adjust result for case 0.0 .LT. argument .LT. 1.0
C----------------------------------------------------------------------
RES = RES / Y1
ELSE IF (Y1 .GT. Y) THEN
C----------------------------------------------------------------------
C Adjust result for case 2.0 .LT. argument .LT. 12.0
C----------------------------------------------------------------------
DO 290 I = 1, N
RES = RES * Y
Y = Y + ONE
290 CONTINUE
END IF
ELSE
C----------------------------------------------------------------------
C Evaluate for argument .GE. 12.0,
C----------------------------------------------------------------------
IF (Y .LE. XBIG) THEN
YSQ = Y * Y
SUM = C(7)
DO 350 I = 1, 6
SUM = SUM / YSQ + C(I)
350 CONTINUE
SUM = SUM/Y - Y + SQRTPI
SUM = SUM + (Y-HALF)*LOG(Y)
RES = EXP(SUM)
ELSE
RES = XINF
GO TO 900
END IF
END IF
C----------------------------------------------------------------------
C Final adjustments and return
C----------------------------------------------------------------------
IF (PARITY) RES = -RES
IF (FACT .NE. ONE) RES = FACT / RES
CS900 GAMMA = RES
900 DGAMMA = RES
RETURN
C ---------- Last line of GAMMA ----------
END
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