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NAG Library Function Document

nag_scaled_log_gamma (s14ahc)

▸▿ 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

nag_scaled_log_gamma (s14ahc) returns the value of

\ln G (x)

, the scaled logarithm of the gamma function

Γ (x)

2 Specification

#include <nag.h>

#include <nags.h>

double

nag_scaled_log_gamma (double x, NagError *fail)

3 Description

nag_scaled_log_gamma (s14ahc) 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 nag_log_gamma (s14abc)), 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; nag_scaled_log_gamma (s14ahc) returns zero and exits with

fail . code =

NE_REAL_ARG_LE.

4 References

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

5 Arguments

1: $x$ – doubleInput: On entry: the argument $x$ of the function.

Constraint: $x > 0.0$ .
2: $fail$ – NagError *Input/Output: The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_LE: On entry, $x = 〈value〉$ .
Constraint: $x > 0.0$ .

7 Accuracy

nag_scaled_log_gamma (s14ahc) has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

8 Parallelism and Performance

nag_scaled_log_gamma (s14ahc) is not threaded in any implementation.

9 Further Comments

None.

10 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 Function Documentnag_scaled_log_gamma (s14ahc)