# NAG FL Interfaces14anf (gamma_​vector)

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## 1Purpose

s14anf returns an array of values of the gamma function $\Gamma \left(x\right)$.

## 2Specification

Fortran Interface
 Subroutine s14anf ( n, x, f,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ivalid(n) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Out) :: f(n)
#include <nag.h>
 void s14anf_ (const Integer *n, const double x[], double f[], Integer ivalid[], Integer *ifail)
The routine may be called by the names s14anf or nagf_specfun_gamma_vector.

## 3Description

s14anf 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 routine 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 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of points.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the argument ${x}_{\mathit{i}}$ of the function, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraint: ${\mathbf{x}}\left(\mathit{i}\right)\notin {ℤ}_{0}^{-}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{f}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: $\Gamma \left({x}_{i}\right)$, the function values.
4: $\mathbf{ivalid}\left({\mathbf{n}}\right)$Integer array Output
On exit: ${\mathbf{ivalid}}\left(\mathit{i}\right)$ contains the error code for ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ivalid}}\left(i\right)=0$
No error.
${\mathbf{ivalid}}\left(i\right)=1$
${x}_{i}$ is too large and positive. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of $\Gamma \left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{1}}$ in s14aaf , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=2$
${x}_{i}$ is too large and negative. ${\mathbf{f}}\left(\mathit{i}\right)$ contains zero. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{2}}$ in s14aaf , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=3$
${x}_{i}$ is too close to zero. ${\mathbf{f}}\left(\mathit{i}\right)$ contains the approximate value of $\Gamma \left({x}_{i}\right)$ at the nearest valid argument. The threshold value is the same as for ${\mathbf{ifail}}={\mathbf{2}}$ in s14aaf , as defined in the Users' Note for your implementation.
${\mathbf{ivalid}}\left(i\right)=4$
${x}_{i}$ is a negative integer, at which values $\Gamma \left({x}_{i}\right)$ are infinite. ${\mathbf{f}}\left(\mathit{i}\right)$ contains a large positive value.
5: $\mathbf{ifail}$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, at least one value of x was invalid.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-99$
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:
 $ε≃|xΨ(x)|δ$
(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 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

s14anf 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 (s14anfe.f90)

### 10.2Program Data

Program Data (s14anfe.d)

### 10.3Program Results

Program Results (s14anfe.r)