# NAG FL Interfaceg01ecf (prob_​chisq)

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## 1Purpose

g01ecf returns the lower or upper tail probability for the ${\chi }^{2}$-distribution with real degrees of freedom.

## 2Specification

Fortran Interface
 Function g01ecf ( tail, x, df,
 Real (Kind=nag_wp) :: g01ecf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x, df Character (1), Intent (In) :: tail
C Header Interface
#include <nag.h>
 double g01ecf_ (const char *tail, const double *x, const double *df, Integer *ifail, const Charlen length_tail)
The routine may be called by the names g01ecf or nagf_stat_prob_chisq.

## 3Description

The lower tail probability for the ${\chi }^{2}$-distribution with $\nu$ degrees of freedom, $P\left(X\le x:\nu \right)$ is defined by:
 $P(X≤x:ν)=12ν/2Γ(ν/2) ∫0.0xXν/2-1e-X/2dX, x≥0,ν>0.$
To calculate $P\left(X\le x:\nu \right)$ a transformation of a gamma distribution is employed, i.e., a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom is equal to a gamma distribution with scale parameter $2$ and shape parameter $\nu /2$.
NIST Digital Library of Mathematical Functions
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## 5Arguments

1: $\mathbf{tail}$Character(1) Input
On entry: indicates whether the upper or lower tail probability is required.
${\mathbf{tail}}=\text{'L'}$
The lower tail probability is returned, i.e., $P\left(X\le x:\nu \right)$.
${\mathbf{tail}}=\text{'U'}$
The upper tail probability is returned, i.e., $P\left(X\ge x:\nu \right)$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
2: $\mathbf{x}$Real (Kind=nag_wp) Input
On entry: $x$, the value of the ${\chi }^{2}$ variate with $\nu$ degrees of freedom.
Constraint: ${\mathbf{x}}\ge 0.0$.
3: $\mathbf{df}$Real (Kind=nag_wp) Input
On entry: $\nu$, the degrees of freedom of the ${\chi }^{2}$-distribution.
Constraint: ${\mathbf{df}}>0.0$.
4: $\mathbf{ifail}$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 g01ecf may return useful information.
If ${\mathbf{ifail}}={\mathbf{1}}$, ${\mathbf{2}}$ or ${\mathbf{3}}$ on exit, then g01ecf returns $0.0$.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{tail}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{tail}}=\text{'L'}$ or $\text{'U'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{x}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{x}}\ge 0.0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df}}>0.0$.
${\mathbf{ifail}}=4$
The series used to calculate the gamma probabilities has failed to converge. The result returned should represent an approximation to the solution.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
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

A relative accuracy of five significant figures is obtained in most cases.

## 8Parallelism and Performance

g01ecf is not threaded in any implementation.

## 9Further Comments

For higher accuracy the transformation described in Section 3 may be used with a direct call to s14baf.

## 10Example

Values from various ${\chi }^{2}$-distributions are read, the lower tail probabilities calculated, and all these values printed out, until the end of data is reached.

### 10.1Program Text

Program Text (g01ecfe.f90)

### 10.2Program Data

Program Data (g01ecfe.d)

### 10.3Program Results

Program Results (g01ecfe.r)