naginterfaces.library.rand.matrix_​orthog

naginterfaces.library.rand.matrix_orthog(side, init, statecomm, a)[source]

matrix_orthog generates a random orthogonal matrix.

For full information please refer to the NAG Library document for g05px

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g05/g05pxf.html

Parameters
sidestr, length 1

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., premultiplied.

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

initstr, length 1

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

is initialized to the identity matrix.

is not initialized and the matrix must be supplied in .

statecommdict, RNG communication object, modified in place

RNG communication structure.

This argument must have been initialized by a prior call to init_repeat() or init_nonrepeat().

afloat, array-like, shape

If , must contain the matrix .

Returns
afloat, ndarray, shape

The matrix when or the matrix when .

Raises
NagValueError
(errno )

On entry, is not valid: .

(errno )

On entry, is not valid: .

(errno )

On entry, , .

Constraint: if , ; otherwise .

(errno )

On entry, , .

Constraint: if , ; otherwise .

(errno )

On entry, [‘state’] vector has been corrupted or not initialized.

(errno )

On entry, , .

Constraint: if , ; otherwise .

(errno )

On entry, , .

Constraint: if , ; otherwise .

Notes

matrix_orthog pre - or post-multiplies an 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 summarised 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 Theorem 3.3 in Stewart (1980).

One of the initialization functions init_repeat() (for a repeatable sequence if computed sequentially) or init_nonrepeat() (for a non-repeatable sequence) must be called prior to the first call to matrix_orthog.

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