One of the initialization functions g05kfc (for a repeatable sequence if computed sequentially) or g05kgc (for a non-repeatable sequence) must be called prior to the first call to 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_SideType Input

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_InitializeA Input

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: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraints:

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

4: $n$ – Integer Input

On entry:

n

, the number of columns of the matrix

A

Constraints:

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

5: $state [\dim]$ – Integer Communication 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: $a [m \times pda]$ – double Input/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: $pda$ – Integer Input

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 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

The maximum error in

U^{T} U

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g05pxc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.