NAG Library Routine Document

S14AHF

+− 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

S14AHF returns the value of

\ln G (x)

, the scaled logarithm of the gamma function

Γ (x)

, via the function name.

2 Specification

FUNCTION S14AHF (

X, IFAIL)

REAL (KIND=nag_wp) S14AHF

INTEGER	IFAIL
REAL (KIND=nag_wp)	X

3 Description

S14AHF calculates an approximate value for

\ln G (x)

, where

G (x) = Γ (x + 1) / {(\frac{x}{e})}^{x}

. This is a variant of the

\ln Γ (x)

function (see also S14ABF), which avoids rounding problems for very large arguments by computing

\ln Γ (x)

with the Stirling approximation factored out.

For

0 < x < 15

\ln G (x) = \ln Γ (x + 1) - x \ln x + x

;

and for

15 \leq x

\ln G (x) = \frac{1}{2} \ln x + \ln (\sqrt{2 π}) + \frac{1}{x} R (1 / x^{2})

, where

R

is a suitable Remez approximation.

For

x \leq 0.0

, the value

\ln G (x)

is undefined; S14AHF returns zero and exits with

IFAIL = 1

4 References

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

5 Parameters

1: X – REAL (KIND=nag_wp)Input: On entry: the argument $x$ of the function.
Constraint: $X > 0.0$ .
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, $X \leq 0.0$ . On soft failure, the function value returned is zero.

7 Accuracy

S14AHF has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

8 Further Comments

None.

9 Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine DocumentS14AHF