Computes a random orthogonal matrix

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NAG[g05qac] NAG[nag_rngs_orthog_matrix] - Computes a random orthogonal matrix

Calling Sequence

g05qac(side, init, m, n, a, igen, iseed, 'fail'=fail)

nag_rngs_orthog_matrix(. . .)

Parameters

side - String;

On entry: indicates whether the matrix is multiplied on the left or right by the random orthogonal matrix .

The matrix is multiplied on the left, i.e., pre-multiplied.

The matrix is multiplied on the right, i.e., post-multiplied.

Constraint: "Nag_LeftSide" or "Nag_RightSide". .

init - String;

On entry: indicates whether or not a should be initialized to the identity matrix.

a is initialized to the identity matrix.

a is not initialized and the matrix must be supplied in a.

Constraint: "Nag_InitializeI" or "Nag_InputA". .

m - integer;

On entry: , the number of rows of the matrix .

Constraints:

if , ;

otherwise.

n - integer;

On entry: , the number of columns of the matrix .

Constraints:

if , ;

otherwise.

a - Matrix(1..dim1, 1..n, datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: if , a must contain the matrix .

On exit: the matrix when or the matrix when .

igen - integer;

On entry: must contain the identification number for the generator to be used to return a pseudo-random number and should remain unchanged following initialization by a prior call to one of the functions g05kbc (nag_rngs_init_repeatable) or g05kcc (nag_rngs_init_nonrepeatable).

iseed - Vector(1.., datatype=integer[kernelopts('wordsize')/8]);

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.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_rngs_orthog_matrix (g05qac) generates a random orthogonal matrix.

Description

nag_rngs_orthog_matrix (g05qac) pre- or post-multiplies an by matrix by a random orthogonal matrix , overwriting . The matrix may optionally be initialized to the identity matrix before multiplying by , hence is returned. is generated using the method of Stewart (1980). The algorithm can be summarized as follows.

Let follow independent multinormal distributions with zero mean and variance and dimensions ; let , where is the identity matrix and is the Householder transformation that reduces to , being the vector with first element one and the remaining elements zero and being a scalar, and let . Then the product is a random orthogonal matrix distributed according to the Haar measure over the set of orthogonal matrices of . See Stewart (1980), Theorem 3.3.

One of the initialization functions g05kbc (nag_rngs_init_repeatable) (for a repeatable sequence if computed sequentially) or g05kcc (nag_rngs_init_nonrepeatable) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_orthog_matrix (g05qac).

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_ENUM_INT"

On entry, , . Constraint: if , .

"NE_INT"

On entry, . Constraint: .

"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 consult NAG for assistance.

"NE_ORTHOGONAL_MATRIX"

Orthogonal matrix of dimension 1 requested.

	Accuracy
	The maximum error in should be a modest multiple of machine precision (see Chapter x02).

	Further Comments
	g05qbc (nag_rngs_corr_matrix) computes a random correlation matrix from a random orthogonal matrix.

Examples

side := "Nag_RightSide":
init := "Nag_InitializeI":
m := 4:
n := 4:
igen := 1:
a := Matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]], datatype=float[8]):
iseed := Vector([1762543, 9324783, 423446, 742355], datatype=integer[kernelopts('wordsize')/8]):
NAG:-g05qac(side, init, m, n, a, igen, iseed):

iseed;

	See Also
	Stewart G W (1980) The efficient generation of random orthogonal matrices with an application to condition estimates SIAM J. Numer. Anal. 17 403–409 g05 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.