nag_real_polygamma (s14aec) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_real_polygamma (s14aec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_real_polygamma (s14aec) returns the value of the kth derivative of the psi function ψx for real x and k=0,1,,6.

2  Specification

#include <nag.h>
#include <nags.h>
double  nag_real_polygamma (double x, Integer k, NagError *fail)

3  Description

nag_real_polygamma (s14aec) evaluates an approximation to the kth derivative of the psi function ψx given by
ψ k x=dkdxk ψx=dkdxk ddx logeΓx ,
where x is real with x0,-1,-2, and k=0,1,,6. For negative noninteger values of x, the recurrence relationship
ψ k x+1=ψ k x+dkdxk 1x
is used. The value of -1k+1ψ k x k!  is obtained by a call to nag_polygamma_deriv (s14adc), which is based on the function PSIFN in Amos (1983).
Note that ψ k x is also known as the polygamma function. Specifically, ψ 0 x is often referred to as the digamma function and ψ 1 x as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

4  References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications
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:     xdoubleInput
On entry: the argument x of the function.
Constraint: x must not be ‘too close’ (see Section 6) to a non-positive integer.
2:     kIntegerInput
On entry: the function ψkx to be evaluated.
Constraint: 0k6.
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, k=value.
Constraint: k6.
On entry, k=value.
Constraint: k0.
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.
Evaluation abandoned due to likelihood of overflow.
On entry, x is ‘too close’ to a non-positive integer: x=value and nintx=value.
Evaluation abandoned due to likelihood of underflow.

7  Accuracy

All constants in nag_polygamma_deriv (s14adc) are given to approximately 18 digits of precision. If t denotes the number of digits of precision in the floating-point arithmetic being used, then clearly the maximum number in the results obtained is limited by p=mint,18. Empirical tests by Amos (1983) have shown that the maximum relative error is a loss of approximately two decimal places of precision. Further tests with the function -ψ 0 x have shown somewhat improved accuracy, except at points near the positive zero of ψ 0 x at x=1.46, where only absolute accuracy can be obtained.

8  Parallelism and Performance

Not applicable.

9  Further Comments


10  Example

This example evaluates ψ 2 x at x=2.5, and prints the results.

10.1  Program Text

Program Text (s14aece.c)

10.2  Program Data

Program Data (s14aece.d)

10.3  Program Results

Program Results (s14aece.r)

nag_real_polygamma (s14aec) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014