# NAG Library Function Document

## 1Purpose

nag_scaled_log_gamma (s14ahc) returns the value of $\mathrm{ln}G\left(x\right)$, the scaled logarithm of the gamma function $\Gamma \left(x\right)$.

## 2Specification

 #include #include
 double nag_scaled_log_gamma (double x, NagError *fail)

## 3Description

nag_scaled_log_gamma (s14ahc) 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 nag_log_gamma (s14abc)), 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; nag_scaled_log_gamma (s14ahc) returns zero and exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_LE.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
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, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.

## 7Accuracy

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.

## 8Parallelism and Performance

nag_scaled_log_gamma (s14ahc) 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 (s14ahce.c)

### 10.2Program Data

Program Data (s14ahce.d)

### 10.3Program Results

Program Results (s14ahce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017