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g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rand_matrix_multi_normal (g05rzc)

## 1  Purpose

nag_rand_matrix_multi_normal (g05rzc) sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Normal distribution with mean vector $a$ and covariance matrix $C$.

## 2  Specification

 #include #include
 void nag_rand_matrix_multi_normal (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer m, const double xmu[], const double c[], Integer pdc, double r[], Integer lr, Integer state[], double x[], Integer pdx, NagError *fail)

## 3  Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
 $fx = C-1 2πm exp - 12 x-aT C-1 x-a$
where $m$ is the number of dimensions, $C$ is the covariance matrix, $a$ is the vector of means and $x$ is the vector of positions.
Covariance matrices are symmetric and positive semidefinite. Given such a matrix $C$, there exists a lower triangular matrix $L$ such that $L{L}^{\mathrm{T}}=C$. $L$ is not unique, if $C$ is singular.
nag_rand_matrix_multi_normal (g05rzc) decomposes $C$ to find such an $L$. It then stores $m$, $a$ and $L$ in the reference vector $r$ which is used to generate a vector $x$ of independent standard Normal pseudorandom numbers. It then returns the vector $a+Lx$, which has the required multivariate Normal distribution.
It should be noted that this function will work with a singular covariance matrix $C$, provided $C$ is positive semidefinite, despite the fact that the above formula for the probability density function is not valid in that case. Wilkinson (1965) should be consulted if further information is required.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_matrix_multi_normal (g05rzc).

## 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:     modeNag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$
Set up reference vector only.
${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$
Generate variates using reference vector set up in a prior call to nag_rand_matrix_multi_normal (g05rzc).
${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$
Set up reference vector and generate variates.
Constraint: ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$, $\mathrm{Nag_GenerateFromReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
3:     nIntegerInput
On entry: $n$, the number of random variates required.
Constraint: ${\mathbf{n}}\ge 0$.
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: the covariance matrix 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:     r[lr]doubleInput/Output
On entry: if ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, the reference vector as set up by nag_rand_matrix_multi_normal (g05rzc) in a previous call with ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
On exit: if ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$, the reference vector that can be used in subsequent calls to nag_rand_matrix_multi_normal (g05rzc) with ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$.
9:     lrIntegerInput
On entry: the dimension of the array r. If ${\mathbf{mode}}=\mathrm{Nag_GenerateFromReference}$, it must be the same as the value of lr specified in the prior call to nag_rand_matrix_multi_normal (g05rzc) with ${\mathbf{mode}}=\mathrm{Nag_InitializeReference}$ or $\mathrm{Nag_InitializeAndGenerate}$.
Constraint: ${\mathbf{lr}}\ge {\mathbf{m}}×\left({\mathbf{m}}+1\right)+1$.
10:   state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
11:   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 Normal vectors generated by the function, with ${\mathbf{X}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
12:   pdxIntegerInput
On entry: the stride used in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
13:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, lr is not large enough, ${\mathbf{lr}}=〈\mathit{\text{value}}〉$: minimum length required $\text{}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
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_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_POS_DEF
On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
NE_PREV_CALL
m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

Not applicable.

The time taken by nag_rand_matrix_multi_normal (g05rzc) 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 Normal distribution with means vector
 $1.0 2.0 -3.0 0.0$
and covariance 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_rand_matrix_multi_normal (g05rzc). All ten observations are generated by a single call to nag_rand_matrix_multi_normal (g05rzc) with ${\mathbf{mode}}=\mathrm{Nag_InitializeAndGenerate}$. The random number generator is initialized by nag_rand_init_repeatable (g05kfc).

### 9.1  Program Text

Program Text (g05rzce.c)

None.

### 9.3  Program Results

Program Results (g05rzce.r)