S14AHF (PDF version)
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NAG Library Manual

# NAG Library Routine DocumentS14AHF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

S14AHF returns the value of $\mathrm{ln}G\left(x\right)$, the scaled logarithm of the gamma function $\Gamma \left(x\right)$, 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 $\mathrm{ln}G\left(x\right)$, where $G\left(x\right)=\Gamma \left(x+1\right)/{\left(\frac{x}{e}\right)}^{x}$. This is a variant of the $\mathrm{ln}\Gamma \left(x\right)$ function (see also S14ABF), which avoids rounding problems for very large arguments by computing $\mathrm{ln}\Gamma \left(x\right)$ with the Stirling approximation factored out.
For $0, $\mathrm{ln}G\left(x\right)=\mathrm{ln}\Gamma \left(x+1\right)-x\mathrm{ln}x+x$;
and for $15\le x$, $\mathrm{ln}G\left(x\right)=\frac{1}{2}\mathrm{ln}x+\mathrm{ln}\left(\sqrt{2\pi }\right)+\frac{1}{x}R\left(1/{x}^{2}\right)$, where $R$ is a suitable Remez approximation.
For $x\le 0.0$, the value $\mathrm{ln}G\left(x\right)$ is undefined; S14AHF returns zero and exits with ${\mathbf{IFAIL}}={\mathbf{1}}$.

## 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: ${\mathbf{X}}>0.0$.
2:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
On entry, ${\mathbf{X}}\le 0.0$. 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.

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.

### 9.1  Program Text

Program Text (s14ahfe.f90)

### 9.2  Program Data

Program Data (s14ahfe.d)

### 9.3  Program Results

Program Results (s14ahfe.r)

S14AHF (PDF version)
S Chapter Contents
S Chapter Introduction
NAG Library Manual