# NAG CL Interfaces14ahc (gamma_​log_​scaled_​real)

Settings help

CL Name Style:

## 1Purpose

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

## 2Specification

 #include
 double s14ahc (double x, NagError *fail)
The function may be called by the names: s14ahc, nag_specfun_gamma_log_scaled_real or nag_scaled_log_gamma.

## 3Description

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 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; s14ahc returns zero and exits with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_LE.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$double Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARG_LE
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}>0.0$.

## 7Accuracy

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

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)