# NAG CL Interfaceg01kfc (pdf_​gamma)

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## 1Purpose

g01kfc returns the value of the probability density function (PDF) for the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$ at a point $x$.

## 2Specification

 #include
 double g01kfc (double x, double a, double b, NagError *fail)
The function may be called by the names: g01kfc, nag_stat_pdf_gamma or nag_gamma_pdf.

## 3Description

The gamma distribution has PDF
 $f(x)= 1βαΓ(α) xα-1e-x/β if ​x≥0; α,β>0 f(x)=0 otherwise.$
If $0.01\le x,\alpha ,\beta \le 100$ 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 9).
Loader C (2000) Fast and accurate computation of binomial probabilities (not yet published)

## 5Arguments

1: $\mathbf{x}$double Input
On entry: $x$, the value at which the PDF is to be evaluated.
2: $\mathbf{a}$double Input
On entry: $\alpha$, the shape parameter of the gamma distribution.
Constraint: ${\mathbf{a}}>0.0$.
3: $\mathbf{b}$double Input
On entry: $\beta$, the scale parameter of the gamma distribution.
Constraints:
• ${\mathbf{b}}>0.0$;
• $\frac{{\mathbf{x}}}{{\mathbf{b}}}<\frac{1}{{\mathbf{nag_real_safe_small_number}}\left(\right)}$.
4: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

If ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, then g01kfc returns $0.0$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 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.
NE_OVERFLOW
Computation abandoned owing to overflow due to extreme parameter values.
NE_REAL
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{b}}>0.0$.

Not applicable.

## 8Parallelism and Performance

g01kfc is not threaded in any implementation.

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,
 $p(x;λ) = λx x! e-λ .$ (1)
The usual way of computing this quantity would be to take the logarithm and calculate,
 $log(x;λ) = x log⁡λ - log(x!) - λ .$
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,
 $log(p(x;λ)) = log(p(x;x)) - D(x;λ) ,$ (2)
where $D\left(x;\lambda \right)$, the deviance for the Poisson distribution is given by,
 $D(x;λ) = log(p(x;x)) - log(p(x;λ)) , = λ D0 ( x λ ) ,$ (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,
 $log(x!) = 12 log⁡ (2πx) + x log(x) -x + δ(x) ,$ (4)
where the error $\delta \left(x\right)$ is given by
 $δ(x) = 112x - 1 360x3 + 1 1260x5 + O (x−7) .$
Finally $\mathrm{log}\left(p\left(x;\lambda \right)\right)$ can be evaluated by combining equations (1)(4) to get,
 $p (x;λ) = 1 2πx e - δ(x) - λ D0 (x/λ) .$

## 10Example

This example prints the value of the gamma distribution PDF at six different points x with differing a and b.

### 10.1Program Text

Program Text (g01kfce.c)

### 10.2Program Data

Program Data (g01kfce.d)

### 10.3Program Results

Program Results (g01kfce.r)