# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Function s14ahf ( x,
 Real (Kind=nag_wp) :: s14ahf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nagmk26.h>
 double s14ahf_ (const double *x, Integer *ifail)

## 3Description

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 ${\mathbf{ifail}}={\mathbf{1}}$.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

s14ahf has been designed to produce full relative accuracy for all input arguments. Empirical results obtained by comparing with multiprecision software confirm this.

## 8Parallelism and Performance

s14ahf is not threaded in any implementation.

None.

## 10Example

This example reads values of the argument $x$ from a file, evaluates the function at each value of $x$ and prints the results.

### 10.1Program Text

Program Text (s14ahfe.f90)

### 10.2Program Data

Program Data (s14ahfe.d)

### 10.3Program Results

Program Results (s14ahfe.r)