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Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_specfun_gamma_log_scaled_real (s14ah)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_specfun_gamma_log_scaled_real (s14ah) returns the value of

\ln G (x)

, the scaled logarithm of the gamma function

Γ (x)

, via the function name.

Syntax

[result, ifail] = s14ah(x)

[result, ifail] = nag_specfun_gamma_log_scaled_real(x)

Description

nag_specfun_gamma_log_scaled_real (s14ah) 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_specfun_gamma_log_real (s14ab)), 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_specfun_gamma_log_scaled_real (s14ah) returns zero and exits with

ifail = 1

References

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

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: The argument $x$ of the function.

Constraint: $x > 0.0$ .

Optional Input Parameters

None.

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, $x \leq 0.0$ . On soft failure, the function value returned is zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_specfun_gamma_log_scaled_real (s14ah) has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

Further Comments

None.

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

Open in the MATLAB editor: s14ah_example

function s14ah_example


fprintf('s14ah example results\n\n');

x = [1, 1.25, 1.5, 1.75, 2, 5, 10, 20, 1000];
n = size(x,2);
result = x;

for j=1:n
  [result(j), ifail] = s14ah(x(j));
end

disp('      x        ln G(x)');
fprintf('%12.3e%12.3e\n',[x; result]);

s14ah example results

      x        ln G(x)
   1.000e+00   1.000e+00
   1.250e+00   1.096e+00
   1.500e+00   1.176e+00
   1.750e+00   1.246e+00
   2.000e+00   1.307e+00
   5.000e+00   1.740e+00
   1.000e+01   2.079e+00
   2.000e+01   2.421e+00
   1.000e+03   4.373e+00

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox