nag_polygamma_deriv (s14adc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_polygamma_deriv (s14adc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function ψx (also known as the digamma function).

2  Specification

#include <nag.h>
#include <nags.h>
void  nag_polygamma_deriv (double x, Integer n, Integer m, double ans[], NagError *fail)

3  Description

nag_polygamma_deriv (s14adc) computes m values of the function
wk,x=-1k+1ψ k x k! ,
for x>0, k=n, n+1,,n+m-1, where ψ is the psi function
ψx=ddx lnΓx=Γx Γx ,
and ψ k  denotes the kth derivative of ψ.
The function is derived from the function PSIFN in Amos (1983). The basic method of evaluation of wk,x is the asymptotic series
wk,xεk,x+12xk+1 +1xkj=1B2j2j+k-1! 2j!k!x2j
for large x greater than a machine-dependent value xmin, followed by backward recurrence using
wk,x=wk,x+1+x-k-1
for smaller values of x, where εk,x=-lnx when k=0, εk,x= 1kxk  when k>0, and B2j, j=1,2,, are the Bernoulli numbers.
When k is large, the above procedure may be inefficient, and the expansion
wk,x=j=11x+jk+1,
which converges rapidly for large k, is used instead.

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>0.0.
2:     nIntegerInput
On entry: the index of the first member n of the sequence of functions.
Constraint: n0.
3:     mIntegerInput
On entry: the number of members m required in the sequence wk,x, for k=n,,n+m-1.
Constraint: m1.
4:     ans[m]doubleOutput
On exit: the first m elements of ans contain the required values wk,x, for k=n,,n+m-1.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
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.
NE_INTERNAL_WORKSPACE
There is not enough internal workspace to continue computation. m is probably too large.
NE_OVERFLOW_LIKELY
Computation abandoned due to the likelihood of overflow.
NE_REAL
On entry, x=value.
Constraint: x>0.0.
NE_UNDERFLOW_LIKELY
Computation abandoned due to the likelihood of underflow.

7  Accuracy

All constants in nag_polygamma_deriv (s14adc) 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=mint,18. Empirical tests of nag_polygamma_deriv (s14adc), taking values of x in the range 0.0<x<50.0, and n in the range 1n50, have shown that the maximum relative error is a loss of approximately two decimal places of precision. Tests with n=0, i.e., testing the function -ψx, have shown somewhat better accuracy, except at points close to the zero of ψx, x1.461632, where only absolute accuracy can be obtained.

8  Further Comments

The time taken for a call of nag_polygamma_deriv (s14adc) is approximately proportional to m, plus a constant. In general, it is much cheaper to call nag_polygamma_deriv (s14adc) with m greater than 1 to evaluate the function wk,x, for k=n,,n+m-1, rather than to make m separate calls of nag_polygamma_deriv (s14adc).

9  Example

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

9.1  Program Text

Program Text (s14adce.c)

9.2  Program Data

Program Data (s14adce.d)

9.3  Program Results

Program Results (s14adce.r)


nag_polygamma_deriv (s14adc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012