void	g05rzc (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)

The function may be called by the names: g05rzc, nag_rand_multivar_normal or nag_rand_matrix_multi_normal.

3 Description

When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function

f (x) = \sqrt{\frac{| C^{- 1} |}{{(2 π)}^{m}}} \exp (- \frac{1}{2} {(x - a)}^{T} 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^{T} = C

L

is not unique, if

C

is singular.

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 + L x

, 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 g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to 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: $order$ – Nag_OrderType Input

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

order = Nag_RowMajor

. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $mode$ – Nag_ModeRNG Input

On entry: a code for selecting the operation to be performed by the function.

$mode = Nag_InitializeReference$: Set up reference vector only.
$mode = Nag_GenerateFromReference$: Generate variates using reference vector set up in a prior call to g05rzc.
$mode = Nag_InitializeAndGenerate$: Set up reference vector and generate variates.

Constraint:

mode = Nag_InitializeReference

Nag_GenerateFromReference

Nag_InitializeAndGenerate

3: $n$ – Integer Input

On entry:

n

, the number of random variates required.

Constraint:

n \geq 0

4: $m$ – Integer Input

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

5: $xmu [m]$ – const double Input

On entry:

a

, the vector of means of the distribution.

6: $c [pdc \times m]$ – const double Input

Note: the

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

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: $pdc$ – Integer Input

On entry: the stride separating row or column elements (depending on the value of order) in the array c.

Constraint:

pdc \geq m

8: $r [lr]$ – double Communication Array

On entry: if

mode = Nag_GenerateFromReference

, the reference vector as set up by g05rzc in a previous call with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

On exit: if

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

, the reference vector that can be used in subsequent calls to g05rzc with

mode = Nag_GenerateFromReference

9: $lr$ – Integer Input

On entry: the dimension of the array r. If

mode = Nag_GenerateFromReference

, it must be the same as the value of lr specified in the prior call to g05rzc with

mode = Nag_InitializeReference

Nag_InitializeAndGenerate

Constraint:

lr \geq m \times (m + 1) + 1

10: $state [\dim]$ – Integer Communication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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: $x [\dim]$ – double Output

Note: where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the array of pseudorandom multivariate Normal vectors generated by the function.

Two possible storage orders are available. If

order = Nag_ColMajor

then

X (i, j)

holds the

j

th dimension for the

i

th variate. If

order = Nag_RowMajor

this ordering is reversed and

X (j, i)

holds the

j

th dimension for the

i

th variate.

12: $pdx$ – Integer Input

On entry: the stride used in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
otherwise $pdx \geq m$ .

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, lr is not large enough, $lr = ⟨ value ⟩$ : minimum length required $= ⟨ value ⟩$ .

On entry, $m = ⟨ value ⟩$ .
Constraint: $m > 0$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdc = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdc \geq m$ .

On entry, $pdx = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $pdx \geq m$ .

On entry, $pdx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $pdx \geq n$ .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
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 $m = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05rzc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g05rzc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

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.

9 Further Comments

The time taken by 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^{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.

10 Example

This example prints ten pseudorandom observations from a multivariate Normal distribution with means vector

[\begin{matrix} 1.0 \\ 2.0 \\ - 3.0 \\ 0.0 \end{matrix}]

and covariance matrix

[\begin{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 \end{matrix}],

generated by g05rzc. All ten observations are generated by a single call to g05rzc with

mode = Nag_InitializeAndGenerate

. The random number generator is initialized by g05kfc.

10.1 Program Text

Program Text (g05rzce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (g05rzce.r)

NAG Library Manual, Mark 30

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

G05 (Rand) Chapter Contents

G05 (Rand) Chapter Introduction

g05rz: FL CL CPP AD PY MB

NAG CL Interfaceg05rzc (multivar_​normal)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g05rzc (multivar_normal)