# NAG CL Interfaceg01ffc (inv_​cdf_​gamma)

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## 1Purpose

g01ffc returns the deviate associated with the given lower tail probability of the gamma distribution.

## 2Specification

 #include
 double g01ffc (double p, double a, double b, double tol, NagError *fail)
The function may be called by the names: g01ffc, nag_stat_inv_cdf_gamma or nag_deviates_gamma_dist.

## 3Description

The deviate, ${g}_{p}$, associated with the lower tail probability, $p$, of the gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$, is defined as the solution to
 $P(G≤gp:α,β)=p=1βαΓ(α) ∫0gpe-G/βGα-1dG, 0≤gp<∞;α,β>0.$
The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the ${\chi }^{2}$-distribution.
Let $y=2\frac{{g}_{p}}{\beta }$. The required $y$ is found from the Taylor series expansion
 $y=y0+∑rCr(y0) r! (Eϕ(y0) ) r,$
where ${y}_{0}$ is a starting approximation
• ${C}_{1}\left(u\right)=1$,
• ${C}_{r+1}\left(u\right)=\left(r\Psi +\frac{d}{du}\right){C}_{r}\left(u\right)$,
• $\Psi =\frac{1}{2}-\frac{\alpha -1}{u}$,
• $E=p-\underset{0}{\overset{{y}_{0}}{\int }}\varphi \left(u\right)du$,
• $\varphi \left(u\right)=\frac{1}{{2}^{\alpha }\Gamma \left(\alpha \right)}{e}^{-u/2}{u}^{\alpha -1}$.
For most values of $p$ and $\alpha$ the starting value
 $y01=2α (z⁢19α +1-19α ) 3$
is used, where $z$ is the deviate associated with a lower tail probability of $p$ for the standard Normal distribution.
For $p$ close to zero,
 $y02= (pα2αΓ(α)) 1/α$
is used.
For large $p$ values, when ${y}_{01}>4.4\alpha +6.0$,
 $y03=−2⁢[ln(1-p)-(α-1)ln(12y01)+ln(Γ(α))]$
is found to be a better starting value than ${y}_{01}$.
For small $\alpha$ $\left(\alpha \le 0.16\right)$, $p$ is expressed in terms of an approximation to the exponential integral and ${y}_{04}$ is found by Newton–Raphson iterations.
Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.
Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388

## 5Arguments

1: $\mathbf{p}$double Input
On entry: $p$, the lower tail probability from the required gamma distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2: $\mathbf{a}$double Input
On entry: $\alpha$, the shape parameter of the gamma distribution.
Constraint: $0.0<{\mathbf{a}}\le {10}^{6}$.
3: $\mathbf{b}$double Input
On entry: $\beta$, the scale parameter of the gamma distribution.
Constraint: ${\mathbf{b}}>0.0$.
4: $\mathbf{tol}$double Input
On entry: the relative accuracy required by you in the results. The smallest recommended value is $50×\delta$, where . If g01ffc is entered with tol less than $50×\delta$ or greater or equal to $1.0$, then $50×\delta$ is used instead.
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

If on exit ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_GAM_NOT_CONV, NE_PROBAB_CLOSE_TO_TAIL, NE_REAL_ARG_GE, NE_REAL_ARG_GT, NE_REAL_ARG_LE or NE_REAL_ARG_LT, then g01ffc returns $0.0$.
On any of the error conditions listed below, except ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_ALG_NOT_CONV, g01ffc returns $0.0$.
NE_ALG_NOT_CONV
The algorithm has failed to converge in $100$ iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
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_GAM_NOT_CONV
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
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_PROBAB_CLOSE_TO_TAIL
The probability is too close to $0.0$ for the given a to enable the result to be calculated.
NE_REAL_ARG_GE
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}<1.0$.
NE_REAL_ARG_GT
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}\le {10}^{6}$.
NE_REAL_ARG_LE
On entry, ${\mathbf{a}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{a}}>0.0$.
On entry, ${\mathbf{b}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{b}}>0.0$.
NE_REAL_ARG_LT
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}\ge 0.0$.

## 7Accuracy

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of $\alpha$ or very small values of $p$ there may be some loss of accuracy.

## 8Parallelism and Performance

g01ffc is not threaded in any implementation.

None.

## 10Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 10.1Program Text

Program Text (g01ffce.c)

### 10.2Program Data

Program Data (g01ffce.d)

### 10.3Program Results

Program Results (g01ffce.r)