PSI Function

public function PSI(XX)

Explanation of machine-dependent constants

XINF = largest positive machine number XMAX1 = beta (p-1), where beta is the radix for the floating-point system, and p is the number of base-beta digits in the floating-point significand. This is an upper bound on non-integral floating-point numbers, and the negative of the lower bound on acceptable negative arguments for PSI. If rounding is necessary, round this value down. XMIN1 = the smallest in magnitude acceptable argument. We recommend XMIN1 = MAX(1/XINF,xmin) rounded up, where xmin is the smallest positive floating-point number. XSMALL = absolute argument below which PICOTAN(PIX) may be represented by 1/X. We recommend XSMALL < sqrt(3 eps)/pi, where eps is the smallest positive number such that 1+eps > 1. XLARGE = argument beyond which PSI(X) may be represented by LOG(X). The solution to the equation x*ln(x) = beta p is a safe value.

 Approximate values for some important machines are

                    beta  p     eps     xmin       XINF

CDC 7600 (S.P.) 2 48 7.11E-15 3.13E-294 1.26E+322 CRAY-1 (S.P.) 2 48 7.11E-15 4.58E-2467 5.45E+2465 IEEE (IBM/XT, SUN, etc.) (S.P.) 2 24 1.19E-07 1.18E-38 3.40E+38 IEEE (IBM/XT, SUN, etc.) (D.P.) 2 53 1.11D-16 2.23E-308 1.79D+308 IBM 3033 (D.P.) 16 14 1.11D-16 5.40D-79 7.23D+75 SUN 3/160 (D.P.) 2 53 1.11D-16 2.23D-308 1.79D+308 VAX 11/780 (S.P.) 2 24 5.96E-08 2.94E-39 1.70E+38 (D.P.) 2 56 1.39D-17 2.94D-39 1.70D+38 (G Format) (D.P.) 2 53 1.11D-16 5.57D-309 8.98D+307

                     XMIN1      XMAX1     XSMALL    XLARGE

CDC 7600 (S.P.) 3.13E-294 1.40E+14 4.64E-08 9.42E+12 CRAY-1 (S.P.) 1.84E-2466 1.40E+14 4.64E-08 9.42E+12 IEEE (IBM/XT, SUN, etc.) (S.P.) 1.18E-38 8.38E+06 1.90E-04 1.20E+06 IEEE (IBM/XT, SUN, etc.) (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14 IBM 3033 (D.P.) 1.39D-76 4.50D+15 5.80D-09 2.05D+15 SUN 3/160 (D.P.) 2.23D-308 4.50D+15 5.80D-09 2.71D+14 VAX 11/780 (S.P.) 5.89E-39 8.38E+06 1.35E-04 1.20E+06 (D.P.) 5.89D-39 3.60D+16 2.05D-09 2.05D+15 (G Format) (D.P.) 1.12D-308 4.50D+15 5.80D-09 2.71D+14

Error Returns

The program returns XINF for X < -XMAX1, for X zero or a negative integer, or when X lies in (-XMIN1, 0), and returns -XINF when X lies in (0, XMIN1).

Intrinsic functions required are:

 ABS, AINT, DBLE, INT, LOG, REAL, TAN

Author: W. J. Cody Mathematics and Computer Science Division Argonne National Laboratory Argonne, IL 60439

Latest modification: June 8, 1988

Arguments

Return Value real(kind=wp)