NAG Library Routine Document

s14adf (polygamma_deriv)

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NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

S (specfun) Chapter Contents

S (specfun) Chapter Introduction

▸▿ 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

s14adf returns a sequence of values of scaled derivatives of the psi function

ψ (x)

(also known as the digamma function).

2

Specification

Fortran Interface

Subroutine s14adf (

x, n, m, ans, ifail)

Integer, Intent (In)	::	n, m
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x
Real (Kind=nag_wp), Intent (Out)	::	ans(m)

C Header Interface

#include <nagmk26.h>

void	s14adf_ (const double x, const Integer n, const Integer m, double ans[], Integer ifail)

3

Description

s14adf 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 routine is derived from the routine 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

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: $x$ – Real (Kind=nag_wp)Input: 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: $ans (m)$ – Real (Kind=nag_wp) arrayOutput: On exit: the first $m$ elements of ans contain the required values $w (k, x)$ , for $k = n, \dots, n + m - 1$ .
5: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .

$ifail = 2$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

$ifail = 3$: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

$ifail = 4$: Computation abandoned due to the likelihood of underflow.

$ifail = 5$: Computation abandoned due to the likelihood of overflow.

$ifail = 6$: There is not enough internal workspace to continue computation. m is probably too large.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

All constants in s14adf 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 s14adf, 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

s14adf is not threaded in any implementation.

9

Further Comments

The time taken for a call of s14adf is approximately proportional to

m

, plus a constant. In general, it is much cheaper to call s14adf 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 s14adf.

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 Routine Document

s14adf (polygamma_deriv)

▸▿ 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