CS REAL FUNCTION GAMMA(X) CD 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 Coefficients C for the approximation over the interval (1,2) are unpublished.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 CD 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/ CD DATA ONE,HALF,TWELVE,TWO,ZERO/1.0D0,0.5D0,12.0D0,2.0D0,0.0D0/, CD 1 SQRTPI/0.9189385332046727417803297D0/, CD 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/ CD DATA XBIG,XMININ,EPS/171.624D0,2.23D-308,2.22D-16/, CD 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/ CD DATA P/-1.71618513886549492533811D+0,2.47656508055759199108314D+1, CD 1 -3.79804256470945635097577D+2,6.29331155312818442661052D+2, CD 2 8.66966202790413211295064D+2,-3.14512729688483675254357D+4, CD 3 -3.61444134186911729807069D+4,6.64561438202405440627855D+4/ CD DATA Q/-3.08402300119738975254353D+1,3.15350626979604161529144D+2, CD 1 -1.01515636749021914166146D+3,-3.10777167157231109440444D+3, CD 2 2.25381184209801510330112D+4,4.75584627752788110767815D+3, CD 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/ CD DATA C/-1.910444077728D-03,8.4171387781295D-04, CD 1 -5.952379913043012D-04,7.93650793500350248D-04, CD 2 -2.777777777777681622553D-03,8.333333333333333331554247D-02, CD 3 5.7083835261D-03/ C---------------------------------------------------------------------- C Statement functions for conversion between integer and float C---------------------------------------------------------------------- CS CONV(I) = REAL(I) CD 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 CD900 DGAMMA = RES RETURN C ---------- Last line of GAMMA ---------- END