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FUNCTION PSI(XX)
C----------------------------------------------------------------------
C
C This function program evaluates the logarithmic derivative of the
C gamma function,
C
C psi(x) = d/dx (gamma(x)) / gamma(x) = d/dx (ln gamma(x))
C
C for real x, where either
C
C -xmax1 < x < -xmin (x not a negative integer), or
C xmin < x.
C
C The calling sequence for this function is
C
C Y = PSI(X)
C
C The main computation uses rational Chebyshev approximations
C published in Math. Comp. 27, 123-127 (1973) by Cody, Strecok and
C Thacher. This transportable program is patterned after the
C machine-dependent FUNPACK program PSI(X), but cannot match that
C version for efficiency or accuracy. This version uses rational
C approximations that are theoretically accurate to 20 significant
C decimal digits. The accuracy achieved depends on the arithmetic
C system, the compiler, the intrinsic functions, and proper selection
C of the machine-dependent constants.
C
C*******************************************************************
C*******************************************************************
C
C Explanation of machine-dependent constants
C
C XINF = largest positive machine number
C XMAX1 = beta ** (p-1), where beta is the radix for the
C floating-point system, and p is the number of base-beta
C digits in the floating-point significand. This is an
C upper bound on non-integral floating-point numbers, and
C the negative of the lower bound on acceptable negative
C arguments for PSI. If rounding is necessary, round this
C value down.
C XMIN1 = the smallest in magnitude acceptable argument. We
C recommend XMIN1 = MAX(1/XINF,xmin) rounded up, where
C xmin is the smallest positive floating-point number.
C XSMALL = absolute argument below which PI*COTAN(PI*X) may be
C represented by 1/X. We recommend XSMALL < sqrt(3 eps)/pi,
C where eps is the smallest positive number such that
C 1+eps > 1.
C XLARGE = argument beyond which PSI(X) may be represented by
C LOG(X). The solution to the equation
C x*ln(x) = beta ** p
C is a safe value.
C
C Approximate values for some important machines are
C
C beta p eps xmin XINF
C
C CDC 7600 (S.P.) 2 48 7.11E-15 3.13E-294 1.26E+322
C CRAY-1 (S.P.) 2 48 7.11E-15 4.58E-2467 5.45E+2465
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 2 24 1.19E-07 1.18E-38 3.40E+38
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2 53 1.11D-16 2.23E-308 1.79D+308
C IBM 3033 (D.P.) 16 14 1.11D-16 5.40D-79 7.23D+75
C SUN 3/160 (D.P.) 2 53 1.11D-16 2.23D-308 1.79D+308
C VAX 11/780 (S.P.) 2 24 5.96E-08 2.94E-39 1.70E+38
C (D.P.) 2 56 1.39D-17 2.94D-39 1.70D+38
C (G Format) (D.P.) 2 53 1.11D-16 5.57D-309 8.98D+307
C
C XMIN1 XMAX1 XSMALL XLARGE
C
C CDC 7600 (S.P.) 3.13E-294 1.40E+14 4.64E-08 9.42E+12
C CRAY-1 (S.P.) 1.84E-2466 1.40E+14 4.64E-08 9.42E+12
C IEEE (IBM/XT,
C SUN, etc.) (S.P.) 1.18E-38 8.38E+06 1.90E-04 1.20E+06
C IEEE (IBM/XT,
C SUN, etc.) (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14
C IBM 3033 (D.P.) 1.39D-76 4.50D+15 5.80D-09 2.05D+15
C SUN 3/160 (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14
C VAX 11/780 (S.P.) 5.89E-39 8.38E+06 1.35E-04 1.20E+06
C (D.P.) 5.89D-39 3.60D+16 2.05D-09 2.05D+15
C (G Format) (D.P.) 1.12D-308 4.50D+15 5.80D-09 2.71D+14
C
C*******************************************************************
C*******************************************************************
C
C Error Returns
C
C The program returns XINF for X < -XMAX1, for X zero or a negative
C integer, or when X lies in (-XMIN1, 0), and returns -XINF
C when X lies in (0, XMIN1).
C
C Intrinsic functions required are:
C
C ABS, AINT, DBLE, INT, LOG, REAL, TAN
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: June 8, 1988
C
C----------------------------------------------------------------------
INTEGER I,N,NQ
CS REAL
DOUBLE PRECISION
1 AUG,CONV,DEN,PSI,FOUR,FOURTH,HALF,ONE,P1,P2,PIOV4,Q1,Q2,
2 SGN,THREE,XLARGE,UPPER,W,X,XINF,XMAX1,XMIN1,XSMALL,X01,
3 X01D,X02,XX,Z,ZERO
DIMENSION P1(9),P2(7),Q1(8),Q2(6)
C----------------------------------------------------------------------
C Mathematical constants. PIOV4 = pi / 4
C----------------------------------------------------------------------
CS DATA ZERO,FOURTH,HALF,ONE/0.0E0,0.25E0,0.5E0,1.0E0/
CS DATA THREE,FOUR/3.0E0,4.0E0/,PIOV4/7.8539816339744830962E-01/
DATA ZERO,FOURTH,HALF,ONE/0.0D0,0.25D0,0.5D0,1.0D0/
DATA THREE,FOUR/3.0D0,4.0D0/,PIOV4/7.8539816339744830962D-01/
C----------------------------------------------------------------------
C Machine-dependent constants
C----------------------------------------------------------------------
CS DATA XINF/1.70E+38/, XMIN1/5.89E-39/, XMAX1/8.38E+06/,
CS 1 XSMALL/1.35E-04/, XLARGE/1.20E+06/
DATA XINF/1.79D+308/, XMIN1/2.23D-308/, XMAX1/4.50D+15/,
1 XSMALL/5.80D-09/, XLARGE/2.71D+14/
C----------------------------------------------------------------------
C Zero of psi(x)
C----------------------------------------------------------------------
CS DATA X01/187.0E0/,X01D/128.0E0/,X02/6.9464496836234126266E-04/
DATA X01/187.0D0/,X01D/128.0D0/,X02/6.9464496836234126266D-04/
C----------------------------------------------------------------------
C Coefficients for approximation to psi(x)/(x-x0) over [0.5, 3.0]
C----------------------------------------------------------------------
CS DATA P1/4.5104681245762934160E-03,5.4932855833000385356E+00,
CS 1 3.7646693175929276856E+02,7.9525490849151998065E+03,
CS 2 7.1451595818951933210E+04,3.0655976301987365674E+05,
CS 3 6.3606997788964458797E+05,5.8041312783537569993E+05,
CS 4 1.6585695029761022321E+05/
CS DATA Q1/9.6141654774222358525E+01,2.6287715790581193330E+03,
CS 1 2.9862497022250277920E+04,1.6206566091533671639E+05,
CS 2 4.3487880712768329037E+05,5.4256384537269993733E+05,
CS 3 2.4242185002017985252E+05,6.4155223783576225996E-08/
DATA P1/4.5104681245762934160D-03,5.4932855833000385356D+00,
1 3.7646693175929276856D+02,7.9525490849151998065D+03,
2 7.1451595818951933210D+04,3.0655976301987365674D+05,
3 6.3606997788964458797D+05,5.8041312783537569993D+05,
4 1.6585695029761022321D+05/
DATA Q1/9.6141654774222358525D+01,2.6287715790581193330D+03,
1 2.9862497022250277920D+04,1.6206566091533671639D+05,
2 4.3487880712768329037D+05,5.4256384537269993733D+05,
3 2.4242185002017985252D+05,6.4155223783576225996D-08/
C----------------------------------------------------------------------
C Coefficients for approximation to psi(x) - ln(x) + 1/(2x)
C for x > 3.0
C----------------------------------------------------------------------
CS DATA P2/-2.7103228277757834192E+00,-1.5166271776896121383E+01,
CS 1 -1.9784554148719218667E+01,-8.8100958828312219821E+00,
CS 2 -1.4479614616899842986E+00,-7.3689600332394549911E-02,
CS 3 -6.5135387732718171306E-21/
CS DATA Q2/ 4.4992760373789365846E+01, 2.0240955312679931159E+02,
CS 1 2.4736979003315290057E+02, 1.0742543875702278326E+02,
CS 2 1.7463965060678569906E+01, 8.8427520398873480342E-01/
DATA P2/-2.7103228277757834192D+00,-1.5166271776896121383D+01,
1 -1.9784554148719218667D+01,-8.8100958828312219821D+00,
2 -1.4479614616899842986D+00,-7.3689600332394549911D-02,
3 -6.5135387732718171306D-21/
DATA Q2/ 4.4992760373789365846D+01, 2.0240955312679931159D+02,
1 2.4736979003315290057D+02, 1.0742543875702278326D+02,
2 1.7463965060678569906D+01, 8.8427520398873480342D-01/
C----------------------------------------------------------------------
CS CONV(I) = REAL(I)
CONV(I) = DBLE(I)
X = XX
W = ABS(X)
AUG = ZERO
C----------------------------------------------------------------------
C Check for valid arguments, then branch to appropriate algorithm
C----------------------------------------------------------------------
IF ((-X .GE. XMAX1) .OR. (W .LT. XMIN1)) THEN
GO TO 410
ELSE IF (X .GE. HALF) THEN
GO TO 200
C----------------------------------------------------------------------
C X < 0.5, use reflection formula: psi(1-x) = psi(x) + pi * cot(pi*x)
C Use 1/X for PI*COTAN(PI*X) when XMIN1 < |X| <= XSMALL.
C----------------------------------------------------------------------
ELSE IF (W .LE. XSMALL) THEN
AUG = -ONE / X
GO TO 150
END IF
C----------------------------------------------------------------------
C Argument reduction for cot
C----------------------------------------------------------------------
100 IF (X .LT. ZERO) THEN
SGN = PIOV4
ELSE
SGN = -PIOV4
END IF
W = W - AINT(W)
NQ = INT(W * FOUR)
W = FOUR * (W - CONV(NQ) * FOURTH)
C----------------------------------------------------------------------
C W is now related to the fractional part of 4.0 * X.
C Adjust argument to correspond to values in the first
C quadrant and determine the sign.
C----------------------------------------------------------------------
N = NQ / 2
IF ((N+N) .NE. NQ) W = ONE - W
Z = PIOV4 * W
IF (MOD(N,2) .NE. 0) SGN = - SGN
C----------------------------------------------------------------------
C determine the final value for -pi * cotan(pi*x)
C----------------------------------------------------------------------
N = (NQ + 1) / 2
IF (MOD(N,2) .EQ. 0) THEN
C----------------------------------------------------------------------
C Check for singularity
C----------------------------------------------------------------------
IF (Z .EQ. ZERO) GO TO 410
AUG = SGN * (FOUR / TAN(Z))
ELSE
AUG = SGN * (FOUR * TAN(Z))
END IF
150 X = ONE - X
200 IF (X .GT. THREE) GO TO 300
C----------------------------------------------------------------------
C 0.5 <= X <= 3.0
C----------------------------------------------------------------------
DEN = X
UPPER = P1(1) * X
DO 210 I = 1, 7
DEN = (DEN + Q1(I)) * X
UPPER = (UPPER + P1(I+1)) * X
210 CONTINUE
DEN = (UPPER + P1(9)) / (DEN + Q1(8))
X = (X-X01/X01D) - X02
PSI = DEN * X + AUG
GO TO 500
C----------------------------------------------------------------------
C 3.0 < X
C----------------------------------------------------------------------
300 IF (X .LT. XLARGE) THEN
W = ONE / (X * X)
DEN = W
UPPER = P2(1) * W
DO 310 I = 1, 5
DEN = (DEN + Q2(I)) * W
UPPER = (UPPER + P2(I+1)) * W
310 CONTINUE
AUG = (UPPER + P2(7)) / (DEN + Q2(6)) - HALF / X + AUG
END IF
PSI = AUG + LOG(X)
GO TO 500
C----------------------------------------------------------------------
C Error return
C----------------------------------------------------------------------
410 PSI = XINF
IF (X .GT. ZERO) PSI = -XINF
500 RETURN
C---------- Last card of PSI ----------
END
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