g01gc returns the probability associated with the lower tail of the noncentral ${\chi}^{2}$-distribution
.

# Syntax

C# |
---|

public static double g01gc( double x, double df, double rlamda, double tol, int maxit, out int ifail ) |

Visual Basic |
---|

Public Shared Function g01gc ( _ x As Double, _ df As Double, _ rlamda As Double, _ tol As Double, _ maxit As Integer, _ <OutAttribute> ByRef ifail As Integer _ ) As Double |

Visual C++ |
---|

public: static double g01gc( double x, double df, double rlamda, double tol, int maxit, [OutAttribute] int% ifail ) |

F# |
---|

static member g01gc : x : float * df : float * rlamda : float * tol : float * maxit : int * ifail : int byref -> float |

#### Parameters

- x
- Type: System..::..Double
*On entry*: the deviate from the noncentral ${\chi}^{2}$-distribution with $\nu $ degrees of freedom and noncentrality parameter $\lambda $.*Constraint*: ${\mathbf{x}}\ge 0.0$.

- df
- Type: System..::..Double
*On entry*: $\nu $, the degrees of freedom of the noncentral ${\chi}^{2}$-distribution.*Constraint*: ${\mathbf{df}}\ge 0.0$.

- rlamda
- Type: System..::..Double
*On entry*: $\lambda $, the noncentrality parameter of the noncentral ${\chi}^{2}$-distribution.*Constraint*: ${\mathbf{rlamda}}\ge 0.0$ if ${\mathbf{df}}>0.0$ or ${\mathbf{rlamda}}>0.0$ if ${\mathbf{df}}=0.0$.

- tol
- Type: System..::..Double

- maxit
- Type: System..::..Int32
*On entry*: the maximum number of iterations to be performed.*Constraint*: ${\mathbf{maxit}}\ge 1$.

- ifail
- Type: System..::..Int32%
*On exit*: ${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

#### Return Value

g01gc returns the probability associated with the lower tail of the noncentral ${\chi}^{2}$-distribution
.

# Description

The lower tail probability of the noncentral ${\chi}^{2}$-distribution with $\nu $ degrees of freedom and noncentrality parameter $\lambda $, $P\left(X\le x:\nu \text{;}\lambda \right)$, is defined by

where $P\left(X\le x:\nu +2j\text{;}0\right)$ is a central ${\chi}^{2}$-distribution with $\nu +2j$ degrees of freedom.

$$P\left(X\le x:\nu \text{;}\lambda \right)=\sum _{j=0}^{\infty}{e}^{-\lambda /2}\frac{{\left(\lambda /2\right)}^{j}}{j!}P\left(X\le x:\nu +2j\text{;}0\right)\text{,}$$ | (1) |

The value of $j$ at which the Poisson weight, ${e}^{-\lambda /2}\frac{{\left(\lambda /2\right)}^{j}}{j!}$, is greatest is determined and the summation (1) is made forward and backward from that value of $j$.

The recursive relationship:

is used during the summation in (1).

$$P\left(X\le x:a+2\text{;}0\right)=P\left(X\le x:a\text{;}0\right)-\frac{\left({x}^{a}/2\right){e}^{-x/2}}{\Gamma \left(a+1\right)}$$ | (2) |

# References

Abramowitz M and Stegun I A (1972)

*Handbook of Mathematical Functions*(3rd Edition) Dover Publications# Error Indicators and Warnings

**Note:**g01gc may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

- ${\mathbf{ifail}}=1$
On entry, ${\mathbf{df}}<0.0$, or ${\mathbf{rlamda}}<0.0$, or ${\mathbf{df}}=0.0$ and ${\mathbf{rlamda}}=0.0$, or ${\mathbf{x}}<0.0$, or ${\mathbf{maxit}}<1$.

- ${\mathbf{ifail}}=2$
- The initial value of the Poisson weight used in the summation (1) was too small to be calculated. The value of $P\left({\mathbf{x}}\le x:\nu \text{;}\lambda \right)$ is likely to be zero.

- ${\mathbf{ifail}}=3$
- The solution has failed to converge in maxit iterations.

- ${\mathbf{ifail}}=4$

- ${\mathbf{ifail}}=5$
- The calculations for the central ${\chi}^{2}$ probability has failed to converge. This is an unlikely error exit. A larger value of tol should be used.

# Accuracy

The summations described in [Description] are made until an upper bound on the truncation error relative to the current summation value is less than tol.

# Parallelism and Performance

None.

# Further Comments

The number of terms in (1) required for a given accuracy will depend on the following factors:

(i) | The rate at which the Poisson weights tend to zero. This will be slower for larger values of $\lambda $. |

(ii) | The rate at which the central ${\chi}^{2}$ probabilities tend to zero. This will be slower for larger values of $\nu $ and $x$. |

# Example

This example reads values from various noncentral ${\chi}^{2}$-distributions, calculates the lower tail probabilities and prints all these values until the end of data is reached.

Example program (C#): g01gce.cs