1: $x$ – double scalar: The argument $x$ of the function.

Constraint: $x$ must not be ‘too close’ (see Error Indicators and Warnings) to a non-positive integer.
2: $k$ – int64int32nag_int scalar: The function $ψ^{(k)} (x)$ to be evaluated.

Constraint: $0 \leq k \leq 6$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

On entry,	$k < 0$ ,
or	$k > 6$ ,
or	x is ‘too close’ to a non-positive integer. That is, $abs (x - nint (x)) < machine precision \times nint (abs (x))$ .

$ifail = 2$: The evaluation has been abandoned due to the likelihood of underflow. The result is returned as zero.

$ifail = 3$: The evaluation has been abandoned due to the likelihood of overflow. The result is returned as zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

All constants in nag_specfun_polygamma_deriv (s14ad) 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 = \min (t, 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)

x = 1.46 \dots

, where only absolute accuracy can be obtained.

Further Comments

None.

Example

This example evaluates

ψ^{(2)} (x)

x = 2.5

, and prints the results.

Open in the MATLAB editor: s14ae_example

function s14ae_example


fprintf('s14ae example results\n\n');

x = 2.5;
k = int64(2);
[result, ifail] = s14ae(x, k);

disp('    x     k   (d^K/dx^K)psi(x)');
fprintf('%6.1f%5d    %12.4e\n',x,k,result);

s14ae example results

    x     k   (d^K/dx^K)psi(x)
   2.5    2     -2.3620e-01

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox