S Chapter Contents
S Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentS14AAF

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

S14AAF returns the value of the gamma function $\Gamma \left(x\right)$, via the function name.

## 2  Specification

 FUNCTION S14AAF ( X, IFAIL)
 REAL (KIND=nag_wp) S14AAF
 INTEGER IFAIL REAL (KIND=nag_wp) X

## 3  Description

S14AAF evaluates an approximation to the gamma function $\Gamma \left(x\right)$. The routine is based on the Chebyshev expansion:
 $Γ1+u=∑r=0′arTrt, where ​ 0≤u<1,t=2u-1,$
and uses the property $\Gamma \left(1+x\right)=x\Gamma \left(x\right)$. If $x=N+1+u$ where $N$ is integral and $0\le u<1$ then it follows that:
• for $N>0$, $\text{ }\Gamma \left(x\right)=\left(x-1\right)\left(x-2\right)\cdots \left(x-N\right)\Gamma \left(1+u\right)$,
• for $N=0$, $\text{ }\Gamma \left(x\right)=\Gamma \left(1+u\right)$,
• for $N<0$, $\text{ }\Gamma \left(x\right)=\frac{\Gamma \left(1+u\right)}{x\left(x+1\right)\left(x+2\right)\cdots \left(x-N-1\right)}$.
There are four possible failures for this routine:
 (i) if $x$ is too large, there is a danger of overflow since $\Gamma \left(x\right)$ could become too large to be represented in the machine; (ii) if $x$ is too large and negative, there is a danger of underflow; (iii) if $x$ is equal to a negative integer, $\Gamma \left(x\right)$ would overflow since it has poles at such points; (iv) if $x$ is too near zero, there is again the danger of overflow on some machines. For small $x$, $\Gamma \left(x\right)\simeq 1/x$, and on some machines there exists a range of nonzero but small values of $x$ for which $1/x$ is larger than the greatest representable value.

## 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: ${\mathbf{X}}$ must not be zero or a negative integer.
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:
${\mathbf{IFAIL}}=1$
The argument is too large. On soft failure the routine returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.
${\mathbf{IFAIL}}=2$
The argument is too large and negative. On soft failure the routine returns zero.
${\mathbf{IFAIL}}=3$
The argument is too close to zero. On soft failure the routine returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.
${\mathbf{IFAIL}}=4$
The argument is a negative integer, at which value $\Gamma \left(x\right)$ is infinite. On soft failure the routine returns a large positive value.

## 7  Accuracy

Let $\delta$ and $\epsilon$ be the relative errors in the argument and the result respectively. If $\delta$ is somewhat larger than the machine precision (i.e., is due to data errors etc.), then $\epsilon$ and $\delta$ are approximately related by:
 $ε≃xΨxδ$
(provided $\epsilon$ is also greater than the representation error). Here $\Psi \left(x\right)$ is the digamma function $\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}$. Figure 1 shows the behaviour of the error amplification factor $\left|x\Psi \left(x\right)\right|$.
If $\delta$ is of the same order as machine precision, then rounding errors could make $\epsilon$ slightly larger than the above relation predicts.
There is clearly a severe, but unavoidable, loss of accuracy for arguments close to the poles of $\Gamma \left(x\right)$ at negative integers. However relative accuracy is preserved near the pole at $x=0$ right up to the point of failure arising from the danger of overflow.
Also accuracy will necessarily be lost as $x$ becomes large since in this region
 $ε≃δxln⁡x.$
However since $\Gamma \left(x\right)$ increases rapidly with $x$, the routine must fail due to the danger of overflow before this loss of accuracy is too great. (For example, for $x=20$, the amplification factor $\text{}\simeq 60$.)
Figure 1

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 (s14aafe.f90)

### 9.2  Program Data

Program Data (s14aafe.d)

### 9.3  Program Results

Program Results (s14aafe.r)