NAG CL Interface
s14acc (polygamma)

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1 Purpose

s14acc returns a value of the function ψ(x)-lnx, where ψ is the psi function ψ(x)= ddx lnΓ(x)= Γ(x) Γ(x) .

2 Specification

#include <nag.h>
double  s14acc (double x, NagError *fail)
The function may be called by the names: s14acc, nag_specfun_polygamma or nag_polygamma_fun.

3 Description

s14acc returns a value of the function ψ(x)-lnx. The psi function is computed without the logarithmic term so that when x is large, sums or differences of psi functions may be computed without unnecessary loss of precision, by analytically combining the logarithmic terms. For example, the difference d=ψ (x+12)-ψ(x) has an asymptotic behaviour for large x given by dln(x+12)-lnx+O( 1x2)ln(1+ 12x ) 12x .
Computing d directly would amount to subtracting two large numbers which are close to ln(x+12) and lnx to produce a small number close to 12x , resulting in a loss of significant digits. However, using this function to compute f(x)=ψ(x)-lnx, we can compute d=f (x+12)-f(x)+ln(1+12x ) , and the dominant logarithmic term may be computed accurately from its power series when x is large. Thus we avoid the unnecessary loss of precision.
The function is derived from the function PSIFN in Amos (1983).

4 References

NIST Digital Library of Mathematical Functions
Amos D E (1983) Algorithm 610: A portable FORTRAN subroutine for derivatives of the psi function ACM Trans. Math. Software 9 494–502

5 Arguments

1: x double Input
On entry: the argument x of the function.
Constraint: x>0.0.
2: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Computation halted due to likelihood of overflow. x may be too small. x=value.
On entry, x=value.
Constraint: x>0.0.
Computation halted due to likelihood of underflow. x may be too large. x=value.

7 Accuracy

All constants in s14acc are given to approximately 18 digits of precision. Calling the number of digits of precision in the floating-point arithmetic being used t, then clearly the maximum number of correct digits in the results obtained is limited by p=min(t,18).
With the above proviso, results returned by this function should be accurate almost to full precision, except at points close to the zero of ψ(x), x1.461632, where only absolute rather than relative accuracy can be obtained.

8 Parallelism and Performance

s14acc is not threaded in any implementation.

9 Further Comments


10 Example

The example program reads values of the argument x from a file, evaluates the function at each value of x and prints the results.

10.1 Program Text

Program Text (s14acce.c)

10.2 Program Data

Program Data (s14acce.d)

10.3 Program Results

Program Results (s14acce.r)
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 −7 −6 −5 −4 −3 −2 −1 0 0 1 2 3 4 5 6 7 8 ψ(x)ln x x Example Program Returns a Value of the Function ψ(x)ln x gnuplot_plot_1