NAG Library Routine Document

S14AGF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

S14AGF returns the value of the logarithm of the gamma function

\ln Γ (z)

for complex

z

, via the function name.

2 Specification

FUNCTION S14AGF (

Z, IFAIL)

COMPLEX (KIND=nag_wp) S14AGF

INTEGER	IFAIL
COMPLEX (KIND=nag_wp)	Z

3 Description

S14AGF evaluates an approximation to the logarithm of the gamma function

\ln Γ (z)

defined for

Re (z) > 0

\ln Γ (z) = \ln \int_{0}^{\infty} e^{- t} t^{z - 1} d t

where

z = x + i y

is complex. It is extended to the rest of the complex plane by analytic continuation unless

y = 0

, in which case

z

is real and each of the points

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

is a singularity and a branch point.

S14AGF is based on the method proposed by Kölbig (1972) in which the value of

\ln Γ (z)

is computed in the different regions of the

z

plane by means of the formulae

\begin{array}{l} \ln Γ (z) & = & (z - \frac{1}{2}) \ln z - z + \frac{1}{2} \ln 2 π + z \sum_{k = 1}^{K} \frac{B_{2 k}}{2 k (2 k - 1)} z^{- 2 k} + R_{K} (z) & if ​ x \geq x_{0} \geq 0, \\ = & \ln Γ (z + n) - \ln \prod_{ν = 0}^{n - 1} (z + ν) & if ​ x_{0} > x \geq 0, \\ = & \ln π - \ln Γ (1 - z) - \ln (\sin π z) & if ​ x < 0, \end{array}

where

n = [x_{0}] - [x]

\{B_{2 k}\}

are Bernoulli numbers (see Abramowitz and Stegun (1972)) and

[x]

is the largest integer

\leq x

. Note that care is taken to ensure that the imaginary part is computed correctly, and not merely modulo

2 π

The routine uses the values

K = 10

and

x_{0} = 7

. The remainder term

R_{K} (z)

is discussed in Section 7.

To obtain the value of

\ln Γ (z)

when

z

is real and positive, S14ABF 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: $IFAIL$ – 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, $Re (Z)$ 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 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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

Not applicable.

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 DocumentS14AGF