# NAG FL Interfaceg01fcf (inv_​cdf_​chisq)

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## 1Purpose

g01fcf returns the deviate associated with the given lower tail probability of the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Function g01fcf ( p, df,
 Real (Kind=nag_wp) :: g01fcf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p, df
#include <nag.h>
 double g01fcf_ (const double *p, const double *df, Integer *ifail)
The routine may be called by the names g01fcf or nagf_stat_inv_cdf_chisq.

## 3Description

The deviate, ${x}_{p}$, associated with the lower tail probability $p$ of the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is defined as the solution to
 $P(X≤xp:ν)=p=12ν/2Γ(ν/2) ∫0xpe-X/2Xv/2-1dX, 0≤xp<∞;ν>0.$
The required ${x}_{p}$ is found by using the relationship between a ${\chi }^{2}$-distribution and a gamma distribution, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.
For very large values of $\nu$, greater than ${10}^{5}$, Wilson and Hilferty's normal approximation to the ${\chi }^{2}$ is used; see Kendall and Stuart (1969).
Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the ${\chi }^{2}$ distribution Appl. Statist. 24 385–388
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth
Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

## 5Arguments

1: $\mathbf{p}$Real (Kind=nag_wp) Input
On entry: $p$, the lower tail probability from the required ${\chi }^{2}$-distribution.
Constraint: $0.0\le {\mathbf{p}}<1.0$.
2: $\mathbf{df}$Real (Kind=nag_wp) Input
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
3: $\mathbf{ifail}$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 $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. 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:
Note: in some cases g01fcf may return useful information.
If ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$, ${\mathbf{3}}$ or ${\mathbf{5}}$ on exit, then g01fcf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}<1.0$.
On entry, ${\mathbf{p}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{p}}\ge 0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{df}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df}}>0.0$.
${\mathbf{ifail}}=3$
The probability is too close to $0.0$ or $1.0$.
${\mathbf{ifail}}=4$
The algorithm has failed to converge in $⟨\mathit{\text{value}}⟩$ iterations. The result should be a reasonable approximation.
${\mathbf{ifail}}=5$
The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
${\mathbf{ifail}}=-99$
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

The results should be accurate to five significant digits for most argument values. Some accuracy is lost for $p$ close to $0.0$.

## 8Parallelism and Performance

g01fcf is not threaded in any implementation.

For higher accuracy the relationship described in Section 3 may be used and a direct call to g01fff made.

## 10Example

This example reads lower tail probabilities for several ${\chi }^{2}$-distributions, and calculates and prints the corresponding deviates until the end of data is reached.

### 10.1Program Text

Program Text (g01fcfe.f90)

### 10.2Program Data

Program Data (g01fcfe.d)

### 10.3Program Results

Program Results (g01fcfe.r)