NAG Library Function Document

nag_rand_orthog_matrix (g05pxc)

+− 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

1 Purpose

nag_rand_orthog_matrix (g05pxc) generates a random orthogonal matrix.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_orthog_matrix (Nag_SideType side, Nag_InitializeA init, Integer m, Integer n, Integer state[], double a[], Integer pda, NagError *fail)

3 Description

nag_rand_orthog_matrix (g05pxc) pre- or post-multiplies an

m

n

matrix

A

by a random orthogonal matrix

U

, overwriting

A

. The matrix

A

may optionally be initialized to the identity matrix before multiplying by

U

, hence

U

is returned.

U

is generated using the method of Stewart (1980). The algorithm can be summarised as follows.

Let

x_{1}, x_{2}, \dots, x_{n - 1}

follow independent multinormal distributions with zero mean and variance

I σ^{2}

and dimensions

n, n - 1, \dots, 2

; let

H_{j} = diag (I_{j - 1}, H_{j}^{*})

, where

I_{j - 1}

is the identity matrix and

H_{j}^{*}

is the Householder transformation that reduces

x_{j}

r_{j j} e_{1}

e_{1}

being the vector with first element one and the remaining elements zero and

r_{j j}

being a scalar, and let

D = diag (sign (r_{11}), sign (r_{22}), \dots, sign (r_{n n}))

. Then the product

U = D H_{1} H_{2} \dots H_{n - 1}

is a random orthogonal matrix distributed according to the Haar measure over the set of orthogonal matrices of

n

. See Theorem 3.3 in Stewart (1980).

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_orthog_matrix (g05pxc).

4 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

5 Arguments

1: side – Nag_SideTypeInput

On entry: indicates whether the matrix

A

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

U

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

Constraint:

side = Nag_LeftSide

Nag_RightSide

2: init – Nag_InitializeAInput

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

$init = Nag_InitializeI$: a is initialized to the identity matrix.
$init = Nag_InputA$: a is not initialized and the matrix $A$ must be supplied in a.

Constraint:

init = Nag_InitializeI

Nag_InputA

3: m – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraints:

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

4: n – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraints:

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

5: state[ $\dim$ ] – IntegerCommunication 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.

6: a[ $m \times pda$ ] – doubleInput/Output

On entry: if

init = Nag_InputA

, a must contain the matrix

A

, with the

(i, j)

th element of

A

stored in

a [(i - 1) \times pda + j - 1]

On exit: the matrix

U A

when

side = Nag_LeftSide

or the matrix

A U

when

side = Nag_RightSide

7: pda – IntegerInput

On entry: the stride separating matrix column elements in the array a.

Constraint:

pda \geq n

8: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_ENUM_INT: On entry, $side = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: if $side = Nag_LeftSide$ , $m > 1$ ;
otherwise $m \geq 1$ .
On entry, $side = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: if $side = Nag_RightSide$ , $n > 1$ ;
otherwise $n \geq 1$ .
NE_INT: On entry, $pda = ⟨value⟩$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pda \geq n$ .
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.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.

7 Accuracy

The maximum error in

U^{T} U

should be a modest multiple of machine precision (see Chapter x02).

8 Parallelism and Performance

nag_rand_orthog_matrix (g05pxc) is not threaded by NAG in any implementation.

nag_rand_orthog_matrix (g05pxc) 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 Users' Note for your implementation for any additional implementation-specific information.