﻿ s14aa Method
s14aa returns the value of the gamma function $\Gamma \left(x\right)$.

# Syntax

C#
```public static double s14aa(
double x,
out int ifail
)```
Visual Basic
```Public Shared Function s14aa ( _
x As Double, _
<OutAttribute> ByRef ifail As Integer _
) As Double```
Visual C++
```public:
static double s14aa(
double x,
[OutAttribute] int% ifail
)```
F#
```static member s14aa :
x : float *
ifail : int byref -> float
```

#### Parameters

x
Type: System..::..Double
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}$ must not be zero or a negative integer.
ifail
Type: System..::..Int32%
On exit: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

s14aa returns the value of the gamma function $\Gamma \left(x\right)$.

# Description

s14aa evaluates an approximation to the gamma function $\Gamma \left(x\right)$. The method 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 method:
 (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.

# References

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

# Error Indicators and Warnings

Errors or warnings detected by the method:
${\mathbf{ifail}}=1$
The argument is too large. On failure the method 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 failure the method returns zero.
${\mathbf{ifail}}=3$
The argument is too close to zero. On failure the method 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 failure the method returns a large positive value.
${\mathbf{ifail}}=-9000$
An error occured, see message report.

# 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 method 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.

None.

# Example

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

Example program (C#): s14aae.cs

Example program data: s14aae.d

Example program results: s14aae.r