One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_matrix_orthog (g05px).

References

Stewart G W (1980) The efficient generation of random orthogonal matrices with an application to condition estimates SIAM J. Numer. Anal. 17 403–409

Parameters

Compulsory Input Parameters

1: $side$ – string (length ≥ 1)

Indicates whether the matrix

A

is multiplied on the left or right by the random orthogonal matrix

U

$side ='L'$: The matrix $A$ is multiplied on the left, i.e., premultiplied.
$side ='R'$: The matrix $A$ is multiplied on the right, i.e., post-multiplied.

Constraint:

side ='L'

'R'

2: $init$ – string (length ≥ 1)

Indicates whether or not a should be initialized to the identity matrix.

$init ='I'$: a is initialized to the identity matrix.
$init ='N'$: a is not initialized and the matrix $A$ must be supplied in a.

Constraint:

init ='I'

'N'

3: $state (:)$ – int64int32nag_int array

Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).

Contains information on the selected base generator and its current state.

4: $a (lda, n)$ – double array

lda, the first dimension of the array, must satisfy the constraint

lda \geq m

init ='N'

, a must contain the matrix

A

Optional Input Parameters

1: $m$ – int64int32nag_int scalar

Default: the first dimension of the array a.

m

, the number of rows of the matrix

A

Constraints:

if $side ='L'$ , $m > 1$ ;
otherwise $m \geq 1$ .

2: $n$ – int64int32nag_int scalar

Default: the second dimension of the array a.

n

, the number of columns of the matrix

A

Constraints:

if $side ='R'$ , $n > 1$ ;
otherwise $n \geq 1$ .

Output Parameters

1: $state (:)$ – int64int32nag_int array: Contains updated information on the state of the generator.
2: $a (lda, n)$ – double array: The matrix $U A$ when $side ='L'$ or the matrix $A U$ when $side ='R'$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: On entry, side is not valid.

$ifail = 2$: On entry, init is not valid.

$ifail = 3$: Constraint: if $side ='L'$ , $m > 1$ ; otherwise $m \geq 1$ .

$ifail = 4$: Constraint: if $side ='R'$ , $n > 1$ ; otherwise $n \geq 1$ .

$ifail = 5$: On entry, state vector has been corrupted or not initialized.

$ifail = 7$: Constraint: $lda \geq m$ .

$ifail = 8$: Constraint: if $side ='L'$ , $m > 1$ ; otherwise $m \geq 1$ .

Constraint: if $side ='R'$ , $n > 1$ ; otherwise $n \geq 1$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The maximum error in

U^{T} U

should be a modest multiple of machine precision (see Chapter X02).

Further Comments

None.

Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeat (g05kf), a

4

4

orthogonal matrix is generated using the

init ='I'

option and the result printed.

Open in the MATLAB editor: g05px_example

function g05px_example


fprintf('g05px example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Control parameters
side = 'Right';
init = 'Initialize';

% Generate the random orthogonal matrix
a = zeros(4, 4);
[state, a, ifail] = g05px( ...
                           side, init, state, a);

disp('Random orthogonal matrix');
disp(a);

g05px example results

Random orthogonal matrix
    0.1756    0.7401   -0.3067   -0.5722
    0.6593   -0.5781   -0.2191   -0.4279
    0.6680    0.3172    0.6077    0.2895
   -0.2971   -0.1323    0.6990   -0.6369

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox