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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

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.
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

Not applicable.

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)