# NAG CL Interfaces14anc (gamma_​vector)

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## 1Purpose

s14anc returns an array of values of the gamma function $\Gamma \left(x\right)$.

## 2Specification

 #include
 void s14anc (Integer n, const double x[], double f[], Integer ivalid[], NagError *fail)
The function may be called by the names: s14anc, nag_specfun_gamma_vector or nag_gamma_vector.

## 3Description

s14anc evaluates an approximation to the gamma function $\Gamma \left(x\right)$ for an array of arguments ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. The function is based on the Chebyshev expansion:
 $Γ(1+u) = ∑r=0′ ar Tr (t)$
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 function:
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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left[{\mathbf{n}}\right]$const double Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left[\mathit{i}-1\right]\notin {ℤ}_{0}^{-}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left[{\mathbf{n}}\right]$double Output
On exit: $\Gamma \left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left[{\mathbf{n}}\right]$Integer Output
On exit: ${\mathbf{ivalid}}\left[\mathit{i}-1\right]$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${x}_{i}$ is too large and positive. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains the approximate value of $\Gamma \left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_GT in s14aac , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=2$
${x}_{i}$ is too large and negative. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains zero. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_LT in s14aac , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=3$
${x}_{i}$ is too close to zero. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains the approximate value of $\Gamma \left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_REAL_ARG_LT in s14aac , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left[i-1\right]=4$
${x}_{i}$ is a negative integer, at which values $\Gamma \left({x}_{i}\right)$ are infinite. ${\mathbf{f}}\left[\mathit{i}-1\right]$ contains a large positive value.
5: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NW_IVALID
On entry, at least one value of x was invalid.

## 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:
 $ε≃|xΨ(x)|δ$
(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 $|x\Psi \left(x\right)|$.
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 $\text{}\simeq 60$.)

## 8Parallelism and Performance

s14anc is not threaded in any implementation.

None.

## 10Example

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

### 10.1Program Text

Program Text (s14ance.c)

### 10.2Program Data

Program Data (s14ance.d)

### 10.3Program Results

Program Results (s14ance.r)