nag_scaled_log_gamma (s14ahc) (PDF version)
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NAG C Library Manual

# NAG Library Function Documentnag_scaled_log_gamma (s14ahc)

## 1  Purpose

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

## 2  Specification

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

## 3  Description

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 NE_REAL_ARG_LE.

## 4  References

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

## 5  Arguments

1:     xdoubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}>0.0$.
2:     failNagError *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, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}>0.0$.

## 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.

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 (s14ahce.c)

### 9.2  Program Data

Program Data (s14ahce.d)

### 9.3  Program Results

Program Results (s14ahce.r)

nag_scaled_log_gamma (s14ahc) (PDF version)
s Chapter Contents
s Chapter Introduction
NAG C Library Manual