# NAG FL Interfaces14aaf (gamma)

## 1Purpose

s14aaf returns the value of the gamma function $\Gamma \left(x\right)$, via the function name.

## 2Specification

Fortran Interface
 Function s14aaf ( x,
 Real (Kind=nag_wp) :: s14aaf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x
#include <nag.h>
 double s14aaf_ (const double *x, Integer *ifail)
The routine may be called by the names s14aaf or nagf_specfun_gamma.

## 3Description

s14aaf evaluates an approximation to the gamma function $\Gamma \left(x\right)$. The routine is based on the Chebyshev expansion:
 $Γ1+u = ∑r=0′ ar Tr t$
where $0\le u<1,t=2u-1\text{,}$ 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 $N>0$, $\Gamma \left(x\right)=\left(x-1\right)\left(x-2\right)\cdots \left(x-N\right)\Gamma \left(1+u\right)$, for $N=0$, $\Gamma \left(x\right)=\Gamma \left(1+u\right)$, for $N<0$, $\Gamma \left(x\right)=\frac{\Gamma \left(1+u\right)}{x\left(x+1\right)\left(x+2\right)\cdots \left(x-N-1\right)}$.
There are four possible failures for this routine:
1. (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;
2. (ii)if $x$ is too large and negative, there is a danger of underflow;
3. (iii)if $x$ is equal to a negative integer, $\Gamma \left(x\right)$ would overflow since it has poles at such points;
4. (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.

## 4References

NIST Digital Library of Mathematical Functions

## 5Arguments

1: $\mathbf{x}$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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}\le 〈\mathit{\text{value}}〉$.
The argument is too large, the function returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$. The function returns zero.
Constraint: ${\mathbf{x}}\ge 〈\mathit{\text{value}}〉$.
The argument is too large and negative, the function returns zero.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: $\left|{\mathbf{x}}\right|\ge 〈\mathit{\text{value}}〉$.
The argument is too close to zero, the function returns the approximate value of $\Gamma \left(x\right)$ at the nearest valid argument.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{x}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{x}}$ must not be a negative integer.
The argument is a negative integer, at which values $\Gamma \left(x\right)$ is infinite. The function returns a large positive value.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

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 $\text{}\simeq 60$.)

## 8Parallelism and Performance

s14aaf 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 (s14aafe.f90)

### 10.2Program Data

Program Data (s14aafe.d)

### 10.3Program Results

Program Results (s14aafe.r)