NAG Library Routine Document

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

Arguments

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: $ifail$ – 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, $Re (z)$ is ‘too close’ to a non-positive integer when $Im (z) = 0.0$ . That is, $abs (Re (z) - nint (Re (z))) < machine precision \times nint (abs (Re (z)))$ .

$ifail = - 99$: 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

The remainder term

R_{K} (z)

satisfies the following error bound:

\begin{array}{lcl} |R_{K} (z)| & \leq & \frac{|B_{2 K}|}{|(2 K - 1)|} z^{1 - 2 K} \\ \leq & \frac{|B_{2 K}|}{|(2 K - 1)|} x^{1 - 2 K} if ​ x \geq 0 . \end{array}

Thus

|R_{10} (7)| < 2.5 \times 10^{- 15}

and hence in theory the routine is capable of achieving an accuracy of approximately

15

significant digits.

8

Parallelism and Performance

s14agf is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example evaluates the logarithm of the gamma function

\ln Γ (z)

z = - 1.5 + 2.5 i

, and prints the results.

NAG Library Routine Document

s14agf (gamma_log_complex)

▸▿ 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

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