g01 Chapter Contents
g01 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_gamma_pdf_vector (g01kkc)

## 1  Purpose

nag_gamma_pdf_vector (g01kkc) returns a number of values of the probability density function (PDF), or its logarithm, for the gamma distribution.

## 2  Specification

 #include #include
 void nag_gamma_pdf_vector (Nag_Boolean ilog, Integer lx, const double x[], Integer la, const double a[], Integer lb, const double b[], double pdf[], Integer ivalid[], NagError *fail)

## 3  Description

The gamma distribution with shape parameter ${\alpha }_{i}$ and scale parameter ${\beta }_{i}$ has PDF
 $f xi,αi,βi = 1 βi αi Γαi xi αi-1 e -xi / βi if ​ xi ≥ 0 ; αi , βi > 0 fxi,αi,βi=0 otherwise.$
If $0.01\le {x}_{i},{\alpha }_{i},{\beta }_{i}\le 100$ then an algorithm based directly on the gamma distribution's PDF is used. For values outside this range, the function is calculated via the Poisson distribution's PDF as described in Loader (2000) (see Section 8).
The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See Section 2.6 in the g01 Chapter Introduction for further information.

## 4  References

Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

## 5  Arguments

1:     ilogNag_BooleanInput
On entry: the value of ilog determines whether the logarithmic value is returned in pdf.
${\mathbf{ilog}}=\mathrm{Nag_FALSE}$
$f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)$, the probability density function is returned.
${\mathbf{ilog}}=\mathrm{Nag_TRUE}$
$\mathrm{log}\left(f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)\right)$, the logarithm of the probability density function is returned.
Constraint: ${\mathbf{ilog}}=\mathrm{Nag_FALSE}$ or $\mathrm{Nag_TRUE}$.
2:     lxIntegerInput
On entry: the length of the array x.
Constraint: ${\mathbf{lx}}>0$.
3:     x[lx]const doubleInput
On entry: ${x}_{i}$, the values at which the PDF is to be evaluated with ${x}_{i}={\mathbf{x}}\left[j\right]$, , for $i=1,2,\dots ,\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
4:     laIntegerInput
On entry: the length of the array a.
Constraint: ${\mathbf{la}}>0$.
5:     a[la]const doubleInput
On entry: ${\alpha }_{i}$, the shape parameter with ${\alpha }_{i}={\mathbf{a}}\left[j\right]$, .
Constraint: ${\mathbf{a}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{la}}$.
6:     lbIntegerInput
On entry: the length of the array b.
Constraint: ${\mathbf{lb}}>0$.
7:     b[lb]const doubleInput
On entry: ${\beta }_{i}$, the scale parameter with ${\beta }_{i}={\mathbf{b}}\left[j\right]$, .
Constraint: ${\mathbf{b}}\left[\mathit{j}-1\right]>0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lb}}$.
8:     pdf[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array pdf must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: $f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)$ or $\mathrm{log}\left(f\left({x}_{i},{\alpha }_{i},{\beta }_{i}\right)\right)$.
9:     ivalid[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array ivalid must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{lx}},{\mathbf{la}},{\mathbf{lb}}\right)$.
On exit: ${\mathbf{ivalid}}\left[i-1\right]$ indicates any errors with the input arguments, with
${\mathbf{ivalid}}\left[i-1\right]=0$
No error.
${\mathbf{ivalid}}\left[i-1\right]=1$
${\alpha }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=2$
${\beta }_{i}\le 0.0$.
${\mathbf{ivalid}}\left[i-1\right]=3$
$\frac{{x}_{i}}{{\beta }_{i}}$ overflows, the value returned should be a reasonable approximation.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ARRAY_SIZE
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{la}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lb}}>0$.
On entry, $\text{array size}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lx}}>0$.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
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.
NW_IVALID
On entry, at least one value of x, a or b was invalid.

Not applicable.

## 8  Further Comments

Due to the lack of a stable link to Loader (2000) paper, we give a brief overview of the method, as applied to the Poisson distribution. The Poisson distribution has a continuous mass function given by,
 $px;λ = λx x! e-λ .$ (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
 $log p x;λ = x log⁡λ - log x! - λ .$
For large $x$ and $\lambda$, $x\mathrm{log}\lambda$ and $\mathrm{log}\left(x!\right)$ are very large, of the same order of magnitude and when calculated have rounding errors. The subtraction of these two terms can therefore result in a number, many orders of magnitude smaller and hence we lose accuracy due to subtraction errors. For example for $x=2×{10}^{6}$ and $\lambda =2×{10}^{6}$, $\mathrm{log}\left(x!\right)\approx 2.7×{10}^{7}$ and $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.17326744645834$. But calculated with the method shown later we have $\mathrm{log}\left(p\left(x;\lambda \right)\right)=-8.1732674441334492$. The difference between these two results suggests a loss of about 7 significant figures of precision.
Loader introduces an alternative way of expressing (1) based on the saddle point expansion,
 $log p x;λ = log p x;x - Dx;λ ,$ (2)
where $D\left(x;\lambda \right)$, the deviance for the Poisson distribution is given by,
 $Dx;λ = log p x;x - log p x;λ , = λ D0 x λ ,$ (3)
and
 $D0 ε = ε log⁡ε + 1 - ε .$
For $\epsilon$ close to $1$, ${D}_{0}\left(\epsilon \right)$ can be evaluated through the series expansion
 $λ D0 x λ = x-λ 2 x+λ + 2x ∑ j=1 ∞ v 2j+1 2j+1 , where ​ v = x-λ x+λ ,$
otherwise ${D}_{0}\left(\epsilon \right)$ can be evaluated directly. In addition, Loader suggests evaluating $\mathrm{log}\left(x!\right)$ using the Stirling–De Moivre series,
 $logx! = 12 log⁡ 2πx + x logx -x + δx ,$ (4)
where the error $\delta \left(x\right)$ is given by
 $δx = 112x - 1 360x3 + 1 1260x5 + O x-7 .$
Finally $\mathrm{log}\left(p\left(x;\lambda \right)\right)$ can be evaluated by combining equations (1)(4) to get,
 $p x;λ = 1 2πx e - δx - λ D0 x/λ .$

## 9  Example

This example prints the value of the gamma distribution PDF at six different points ${x}_{i}$ with differing ${\alpha }_{i}$ and ${\beta }_{i}$.

### 9.1  Program Text

Program Text (g01kkce.c)

### 9.2  Program Data

Program Data (g01kkce.d)

### 9.3  Program Results

Program Results (g01kkce.r)