g01 Chapter Contents
g01 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_prob_lin_chi_sq (g01jdc)

## 1  Purpose

nag_prob_lin_chi_sq (g01jdc) calculates the lower tail probability for a linear combination of (central) ${\chi }^{2}$ variables.

## 2  Specification

 #include #include
 void nag_prob_lin_chi_sq (Nag_LCCMethod method, Integer n, const double rlam[], double d, double c, double *prob, NagError *fail)

## 3  Description

Let ${u}_{1},{u}_{2},\dots ,{u}_{n}$ be independent Normal variables with mean zero and unit variance, so that ${u}_{1}^{2},{u}_{2}^{2},\dots ,{u}_{n}^{2}$ have independent ${\chi }^{2}$-distributions with unit degrees of freedom. nag_prob_lin_chi_sq (g01jdc) evaluates the probability that
 $λ1u12+λ2u22+⋯+λnun2
If $c=0.0$ this is equivalent to the probability that
 $λ1u12+λ2u22+⋯+λnun2 u12+u22+⋯+un2
Alternatively let
 $λi*=λi-d, ​ i= 1,2,…,n,$
then nag_prob_lin_chi_sq (g01jdc) returns the probability that
 $λ1*u12+λ2*u22+⋯+λn*un2
Two methods are available. One due to Pan (1964) (see Farebrother (1980)) makes use of series approximations. The other method due to Imhof (1961) reduces the problem to a one-dimensional integral. If $n\ge 6$ then a non-adaptive method is used to compute the value of the integral otherwise nag_1d_quad_gen_1 (d01sjc) is used.
Pan's procedure can only be used if the ${\lambda }_{i}^{*}$ are sufficiently distinct; nag_prob_lin_chi_sq (g01jdc) requires the ${\lambda }_{i}^{*}$ to be at least $1%$ distinct; see Section 9. If the ${\lambda }_{i}^{*}$ are at least $1%$ distinct and $n\le 60$, then Pan's procedure is recommended; otherwise Imhof's procedure is recommended.
Farebrother R W (1980) Algorithm AS 153. Pan's procedure for the tail probabilities of the Durbin–Watson statistic Appl. Statist. 29 224–227
Imhof J P (1961) Computing the distribution of quadratic forms in Normal variables Biometrika 48 419–426
Pan Jie–Jian (1964) Distributions of the noncircular serial correlation coefficients Shuxue Jinzhan 7 328–337

## 5  Arguments

1:    $\mathbf{method}$Nag_LCCMethodInput
On entry: indicates whether Pan's, Imhof's or an appropriately selected procedure is to be used.
${\mathbf{method}}=\mathrm{Nag_LCCPan}$
Pan's method is used.
${\mathbf{method}}=\mathrm{Nag_LCCImhof}$
Imhof's method is used.
${\mathbf{method}}=\mathrm{Nag_LCCDefault}$
Pan's method is used if ${\lambda }_{\mathit{i}}^{*}$, for $\mathit{i}=1,2,\dots ,n$ are at least $1%$ distinct and $n\le 60$; otherwise Imhof's method is used.
Constraint: ${\mathbf{method}}=\mathrm{Nag_LCCPan}$, $\mathrm{Nag_LCCImhof}$ or $\mathrm{Nag_LCCDefault}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of independent standard Normal variates, (central ${\chi }^{2}$ variates).
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{rlam}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the weights, ${\lambda }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of the central ${\chi }^{2}$ variables.
Constraint: ${\mathbf{rlam}}\left[\mathit{i}-1\right]\ne {\mathbf{d}}$ for at least one $\mathit{i}$. If ${\mathbf{method}}=\mathrm{Nag_LCCPan}$, then the ${\lambda }_{\mathit{i}}^{*}$ must be at least $1%$ distinct; see Section 9, for $\mathit{i}=1,2,\dots ,n$.
4:    $\mathbf{d}$doubleInput
On entry: $d$, the multiplier of the central ${\chi }^{2}$ variables.
Constraint: ${\mathbf{d}}\ge 0.0$.
5:    $\mathbf{c}$doubleInput
On entry: $c$, the value of the constant.
6:    $\mathbf{prob}$double *Output
On exit: the lower tail probability for the linear combination of central ${\chi }^{2}$ variables.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_REAL
On entry, ${\mathbf{d}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{d}}\ge 0.0$.
NE_REAL_ARRAY
On entry, all values of ${\mathbf{rlam}}={\mathbf{d}}$.
NE_REAL_ARRAY_ENUM
On entry, ${\mathbf{method}}=\mathrm{Nag_LCCPan}$ but two successive values of $\lambda *$ were not $1$ percent distinct.

## 7  Accuracy

On successful exit at least four decimal places of accuracy should be achieved.

## 8  Parallelism and Performance

nag_prob_lin_chi_sq (g01jdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Pan's procedure can only work if the ${\lambda }_{i}^{*}$ are sufficiently distinct. nag_prob_lin_chi_sq (g01jdc) uses the check $\left|{w}_{j}-{w}_{j-1}\right|\ge 0.01×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\left|{w}_{j}\right|,\left|{w}_{j-1}\right|\right)$, where the ${w}_{j}$ are the ordered nonzero values of ${\lambda }_{i}^{*}$.
For the situation when all the ${\lambda }_{i}$ are positive nag_prob_lin_non_central_chi_sq (g01jcc) may be used. If the probabilities required are for the Durbin–Watson test, then the bounds for the probabilities are given by nag_prob_durbin_watson (g01epc).

## 10  Example

For $n=10$, the choice of method, values of $c$ and $d$ and the ${\lambda }_{i}$ are input and the probabilities computed and printed.

### 10.1  Program Text

Program Text (g01jdce.c)

### 10.2  Program Data

Program Data (g01jdce.d)

### 10.3  Program Results

Program Results (g01jdce.r)