g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_multi_normal (g05lzc)

## 1  Purpose

nag_rngs_multi_normal (g05lzc) sets up a reference vector and generates a vector of pseudorandom numbers from a multivariate Normal distribution with mean vector $a$ and covariance matrix $C$.

## 2  Specification

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

## 3  Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
 $fx = C-1 2πn exp - 12 x-aT C-1 x-a$
where $n$ 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_rngs_multi_normal (g05lzc) decomposes $C$ to find such an $L$. It then stores $n$, $a$ 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_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_multi_normal (g05lzc).

## 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:     nIntegerInput
On entry: $n$, the number of dimensions of the distribution.
Constraint: ${\mathbf{n}}>0$.
4:     xmu[n]const doubleInput
On entry: $a$, the vector of means of the distribution.
5:     c[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array c must be at least ${\mathbf{pdc}}×{\mathbf{n}}$.
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.
6:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
7:     x[n]doubleOutput
On exit: the pseudorandom multivariate Normal vector generated by the function.
8:     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).
9:     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.
10:   r[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array r must be at least $\left(\left({\mathbf{n}}+1\right)\left({\mathbf{n}}+2\right)/2\right)$.
On entry: if ${\mathbf{mode}}=2$, the reference vector as set up by nag_rngs_multi_normal (g05lzc) 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_multi_normal (g05lzc) with ${\mathbf{mode}}=2$.
11:   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, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}>0$.
NE_INT_2
n is not the same as when r was set up in a previous call.
Previous value $\text{}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{pdc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdc}}\ge {\mathbf{n}}$.
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 $\left(n×\epsilon +\left(n+3\right)\epsilon /2\right)$ times the maximum element of $C$, where $\epsilon$ is the machine precision. Under normal circumstances, the above will be small compared to sampling error.

The time taken by nag_rngs_multi_normal (g05lzc) is of order ${n}^{3}$.
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 two pseudorandom observations from a bivariate Normal distribution with means vector
 $1.0 2.0$
and covariance matrix
 $2.0 1.0 1.0 3.0 ,$
generated by nag_rngs_multi_normal (g05lzc). The first observation is generated by a single call to nag_rngs_multi_normal (g05lzc) with ${\mathbf{mode}}=0$, and the second observation is generated using the same reference vector a call to nag_rngs_multi_normal (g05lzc) with ${\mathbf{mode}}=2$. The random number generator is initialized by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05lzce.c)

None.

### 9.3  Program Results

Program Results (g05lzce.r)