NAG Library Routine Document

G05RZF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G05RZF 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

SUBROUTINE G05RZF (

MODE, N, M, XMU, C, LDC, R, LR, STATE, X, LDX, IFAIL)

INTEGER	MODE, N, M, LDC, LR, STATE(*), LDX, IFAIL
REAL (KIND=nag_wp)	XMU(M), C(LDC,M), R(LR), X(LDX,M)

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.

G05RZF 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 routine 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 routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05RZF.

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 Parameters

1: MODE – INTEGERInput

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

$MODE = 0$: Set up reference vector only.
$MODE = 1$: Generate variates using reference vector set up in a prior call to G05RZF.
$MODE = 2$: Set up reference vector and generate variates.

Constraint:

MODE = 0

1

2

2: N – INTEGERInput

On entry:

n

, the number of random variates required.

Constraint:

N \geq 0

3: M – INTEGERInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

M > 0

4: XMU(M) – REAL (KIND=nag_wp) arrayInput

On entry:

a

, the vector of means of the distribution.

5: C(LDC,M) – REAL (KIND=nag_wp) arrayInput

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: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G05RZF is called.

Constraint:

LDC \geq M

7: R(LR) – REAL (KIND=nag_wp) arrayInput/Output

On entry: if

MODE = 1

, the reference vector as set up by G05RZF in a previous call with

MODE = 0

2

On exit: if

MODE = 0

2

, the reference vector that can be used in subsequent calls to G05RZF with

MODE = 1

8: LR – INTEGERInput

On entry: the dimension of the array R as declared in the (sub)program from which G05RZF is called. If

MODE = 1

, it must be the same as the value of LR specified in the prior call to G05RZF with

MODE = 0

2

Constraint:

LR \geq M \times (M + 1) + 1

9: STATE( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

10: X(LDX,M) – REAL (KIND=nag_wp) arrayOutput

On exit: the array of pseudorandom multivariate Normal vectors generated by the routine, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

11: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G05RZF is called.

Constraint:

LDX \geq N

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $MODE \neq 0$ , $1$ or $2$ .

$IFAIL = 2$: On entry, $N < 1$ .

$IFAIL = 3$: On entry, $M < 1$ .

$IFAIL = 5$: The covariance matrix C is not positive semidefinite to machine precision.

$IFAIL = 6$: On entry, $LDC < M$ .

$IFAIL = 7$: The reference vector R has been corrupted or M has changed since R was set up in a previous call to G05RZF with $MODE = 0$ or $2$ .

$IFAIL = 8$: On entry, $LR \leq M \times (M + 1)$ .

$IFAIL = 9$

On entry,

STATE vector was not initialized or has been corrupted.

$IFAIL = 11$: On entry, $LDX < N$ .

7 Accuracy

Not applicable.

8 Further Comments

The time taken by G05RZF 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.

9 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 G05RZF. All ten observations are generated by a single call to G05RZF with

MODE = 2

. The random number generator is initialized by G05KFF.

NAG Library Routine DocumentG05RZF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G05RZF