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NAG Library Function Document

nag_polygamma_deriv (s14adc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_polygamma_deriv (s14adc) returns a sequence of values of scaled derivatives of the psi function

ψ (x)

(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

w (k, x) = \frac{{(- 1)}^{k + 1} ψ^{(k)} (x)}{k!},

for

x > 0

k = n

n + 1, \dots, n + m - 1

, where

ψ

is the psi function

ψ (x) = \frac{d}{d x} \ln Γ (x) = \frac{Γ^{'} (x)}{Γ (x)},

and

ψ^{(k)}

denotes the

k

th derivative of

ψ

The function is derived from the function PSIFN in Amos (1983). The basic method of evaluation of

w (k, x)

is the asymptotic series

w (k, x) \sim ε (k, x) + \frac{1}{2 x^{k + 1}} + \frac{1}{x^{k}} \sum_{j = 1}^{\infty} B_{2 j} \frac{(2 j + k - 1)!}{(2 j)! k! x^{2 j}}

for large

x

greater than a machine-dependent value

x_{\min}

, followed by backward recurrence using

w (k, x) = w (k, x + 1) + x^{- k - 1}

for smaller values of

x

, where

ε (k, x) = - \ln x

when

k = 0

ε (k, x) = \frac{1}{k x^{k}}

when

k > 0

, and

B_{2 j}

j = 1, 2, \dots

, are the Bernoulli numbers.

When

k

is large, the above procedure may be inefficient, and the expansion

w (k, x) = \sum_{j = 1}^{\infty} \frac{1}{{(x + j)}^{k + 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: $x$ – doubleInput: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $n$ – IntegerInput: On entry: the index of the first member $n$ of the sequence of functions.

Constraint: $n \geq 0$ .
3: $m$ – IntegerInput: On entry: the number of members $m$ required in the sequence $w (k, x)$ , for $k = n, \dots, n + m - 1$ .

Constraint: $m \geq 1$ .
4: $ans [m]$ – doubleOutput: On exit: the first $m$ elements of ans contain the required values $w (k, x)$ , for $k = n, \dots, n + m - 1$ .
5: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_WORKSPACE: There is not enough internal workspace to continue computation. m is probably too large.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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 = \min (t, 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

1 \leq n \leq 50

, 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

- ψ (x)

, have shown somewhat better accuracy, except at points close to the zero of

ψ (x)

x ≃ 1.461632

, where only absolute accuracy can be obtained.

8 Parallelism and Performance

nag_polygamma_deriv (s14adc) is not threaded in any implementation.

9 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

w (k, x)

, for

k = n, \dots, n + m - 1

, rather than to make

m

separate calls of nag_polygamma_deriv (s14adc).

10 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Function Documentnag_polygamma_deriv (s14adc)