NAG Library Routine Document

S14AFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

S14AFF returns the value of the

k

th derivative of the psi function

ψ (z)

for complex

z

and

k = 0, 1, \dots, 4

, via the function name.

2 Specification

FUNCTION S14AFF (

Z, K, IFAIL)

COMPLEX (KIND=nag_wp) S14AFF

INTEGER	K, IFAIL
COMPLEX (KIND=nag_wp)	Z

3 Description

S14AFF evaluates an approximation to the

k

th derivative of the psi function

ψ (z)

given by

ψ^{(k)} (z) = \frac{d^{k}}{d z^{k}} ψ (z) = \frac{d^{k}}{d z^{k}} (\frac{d}{d z} \log_{e} Γ (z)),

where

z = x + i y

is complex provided

y \neq 0

and

k = 0, 1, \dots, 4

. If

y = 0

z

is real and thus

ψ^{(k)} (z)

is singular when

z = 0, - 1, - 2, \dots

Note that

ψ^{(k)} (z)

is also known as the polygamma function. Specifically,

ψ^{(0)} (z)

is often referred to as the digamma function and

ψ^{(1)} (z)

as the trigamma function in the literature. Further details can be found in Abramowitz and Stegun (1972).

S14AFF is based on a modification of the method proposed by Kölbig (1972).

To obtain the value of

ψ^{(k)} (z)

when

z

is real, S14AEF can be used.

4 References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Kölbig K S (1972) Programs for computing the logarithm of the gamma function, and the digamma function, for complex arguments Comp. Phys. Comm. 4 221–226

5 Parameters

1: Z – COMPLEX (KIND=nag_wp)Input: On entry: the argument $z$ of the function.
Constraint: $Re (Z)$ must not be ‘too close’ (see Section 6) to a non-positive integer when $Im (Z) = 0.0$ .
2: K – INTEGERInput: On entry: the function $ψ^{(k)} (z)$ to be evaluated.
Constraint: $0 \leq K \leq 4$ .
3: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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,	$K < 0$ ,
or	$K > 4$ ,
or	$Re (Z)$ is ‘too close’ to a non-positive integer when $Im (Z) = 0.0$ . That is, $abs (Re (Z) - nint (Re (Z))) < machine precision \times nint (abs (Re (Z)))$ .

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

7 Accuracy

Empirical tests have shown that the maximum relative error is a loss of approximately two decimal places of precision.

8 Further Comments

None.

9 Example

This example evaluates the psi (trigamma) function

ψ^{(1)} (z)

z = - 1.5 + 2.5 i

, and prints the results.

NAG Library Routine DocumentS14AFF