g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_matrix_multi_students_t (g05lxc)

## 1  Purpose

nag_rngs_matrix_multi_students_t (g05lxc) sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Student's $t$ distribution with $\nu$ degrees of freedom, mean vector $a$ and covariance matrix $\frac{\nu }{\nu -2}C$.

## 2  Specification

 #include #include
 void nag_rngs_matrix_multi_students_t (Nag_OrderType order, Integer mode, Integer df, Integer m, const double xmu[], const double c[], Integer pdc, Integer n, double x[], Integer pdx, Integer igen, Integer iseed[], double r[], Integer lr, NagError *fail)

## 3  Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
 $fx = Γ ν+m 2 πv m/2 Γ ν/2 C 12 1 + x-aT C-1 x-a ν -ν+m 2$
where $m$ is the number of dimensions, $\nu$ is the degrees of freedom, $a$ is the vector of means, $x$ is the vector of positions and $\frac{\nu }{\nu -2}C$ is the covariance matrix.
The function returns the value
 $x = a + vs z$
where $z$ is generated by nag_rgsn_matrix_multi_normal (g05lyc) from a Normal distribution with mean zero and covariance matrix $C$ and $s$ is generated by nag_rngs_chi_sq (g05lcc) from a ${\chi }^{2}$-distribution with $\nu$ degrees of freedom.
One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_matrix_multi_students_t (g05lxc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     modeIntegerInput
On entry: selects the operation to be performed.
${\mathbf{mode}}=0$
Initialize and generate random numbers.
${\mathbf{mode}}=1$
Initialize only (i.e., set up reference vector).
${\mathbf{mode}}=2$
Generate random numbers using previously set up reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
3:     dfIntegerInput
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 3$.
4:     mIntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
5:     xmu[m]const doubleInput
On entry: $a$, the vector of means of the distribution.
6:     c[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array c must be at least ${\mathbf{pdc}}×{\mathbf{m}}$.
The $\left(i,j\right)$th element of the matrix $C$ is stored in
• ${\mathbf{c}}\left[\left(j-1\right)×{\mathbf{pdc}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{c}}\left[\left(i-1\right)×{\mathbf{pdc}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: matrix which, along with df, defines the covariance of the distribution. Only the upper triangle need be set.
Constraint: c must be positive semidefinite to machine precision.
7:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
8:     nIntegerInput
On entry: $n$, the number of random variates required.
Constraint: ${\mathbf{n}}\ge 1$.
9:     x[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: the array of pseudorandom multivariate Student's $t$ vectors generated by the function, with ${\mathbf{X}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
10:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
11:   igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
12:   iseed[$4$]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
13:   r[lr]doubleInput/Output
On entry: if ${\mathbf{mode}}=2$, the reference vector as set up by nag_rngs_matrix_multi_students_t (g05lxc) in a previous call with ${\mathbf{mode}}=0$ or $1$.
On exit: if ${\mathbf{mode}}=0$ or $1$, the reference vector that can be used in subsequent calls to nag_rngs_matrix_multi_students_t (g05lxc) with ${\mathbf{mode}}=2$.
14:   lrIntegerInput
On entry: the dimension of the array r. If ${\mathbf{mode}}=2$, it must be the same as the value of lr specified in the prior call to nag_rngs_matrix_multi_students_t (g05lxc) with ${\mathbf{mode}}=0$ or $1$.
Constraint: ${\mathbf{lr}}>{\mathbf{m}}×\left({\mathbf{m}}+1\right)+1$.
15:   failNagError *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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{df}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{df}}>2$.
On entry, invalid value for igen. Ensure igen has not changed since random number generator was initialized.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}\ge 0$ and ${\mathbf{mode}}\le 2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
m is not the same as when r was set up in a previous call. Previous value $=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{lr}}=〈\mathit{\text{value}}〉$, $\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}+1\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lr}}\ge {\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}+1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
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.
NE_POS_DEF
The covariance matrix c is not positive semidefinite to machine precision.

## 7  Accuracy

The maximum absolute error in $L{L}^{\mathrm{T}}$, and hence in the covariance matrix of the resulting vectors, is less than $\left(m\epsilon +\left(m+3\right)\epsilon /2\right)$ times the maximum element of $C$, where $\epsilon$ is the machine precision. Under normal circumstances, the above will be small compared to sampling error.

## 8  Further Comments

The time taken by nag_rngs_matrix_multi_students_t (g05lxc) is of order $n{m}^{3}$.
It is recommended that the diagonal elements of $C$ should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is $C+E=L{L}^{\mathrm{T}}$, where $E$ is a diagonal matrix with small positive diagonal elements. This ensures that, even when $C$ is singular, or nearly singular, the Cholesky factor $L$ corresponds to a positive definite covariance matrix that agrees with $C$ within machine precision.

## 9  Example

This example prints ten pseudorandom observations from a multivariate Student's $t$-distribution ten degrees of freedom, means vector
 $1.0 2.0 -3.0 0.0$
and c matrix
 $1.69 0.39 -1.86 0.07 0.39 98.01 -7.07 -0.71 -1.86 -7.07 11.56 0.03 0.07 -0.71 0.03 0.01 ,$
generated by nag_rngs_matrix_multi_students_t (g05lxc). All ten observations are generated by a single call to nag_rngs_matrix_multi_students_t (g05lxc) with ${\mathbf{mode}}=0$. The random number generator is initialized by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05lxce.c)

None.

### 9.3  Program Results

Program Results (g05lxce.r)