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: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $−1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $−1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $−1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.

s14ag: FL CL CPP AD

NAG FL Interfaces14agf (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

NAG FL Interface
s14agf (gamma_log_complex)