# NAG Library Function Document

## 1Purpose

nag_gamma (s14aac) returns the value of the gamma function $\Gamma \left(x\right)$.

## 2Specification

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

## 3Description

nag_gamma (s14aac) evaluates an approximation to the gamma function $\Gamma \left(x\right)$. The function 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 function:
 (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.

## 4References

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

## 5Arguments

1:    $\mathbf{x}$doubleInput
On entry: the argument $x$ of the function.
Constraint: ${\mathbf{x}}$ must not be zero or a negative integer.
2:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_REAL_ARG_GT
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.
NE_REAL_ARG_LT
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.
NE_REAL_ARG_NEG_INT
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.
NE_REAL_ARG_TOO_SMALL
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.

## 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 function 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

## 8Parallelism and Performance

nag_gamma (s14aac) 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 (s14aace.c)

### 10.2Program Data

Program Data (s14aace.d)

### 10.3Program Results

Program Results (s14aace.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017