g05 Chapter Contents
g05 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_rand_orthog_matrix (g05pxc)

## 1  Purpose

nag_rand_orthog_matrix (g05pxc) generates a random orthogonal matrix.

## 2  Specification

 #include #include
 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 $m$ by $n$ 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 $I{\sigma }^{2}$ and dimensions $n,n-1,\dots ,2$; let ${H}_{j}=\mathrm{diag}\left({I}_{j-1},{H}_{j}^{*}\right)$, where ${I}_{j-1}$ is the identity matrix and ${H}_{j}^{*}$ is the Householder transformation that reduces ${x}_{j}$ to ${r}_{jj}{e}_{1}$, ${e}_{1}$ being the vector with first element one and the remaining elements zero and ${r}_{jj}$ being a scalar, and let $D=\mathrm{diag}\left(\mathrm{sign}\left({r}_{11}\right),\mathrm{sign}\left({r}_{22}\right),\dots ,\mathrm{sign}\left({r}_{nn}\right)\right)$. 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:    $\mathbf{side}$Nag_SideTypeInput
On entry: indicates whether the matrix $A$ is multiplied on the left or right by the random orthogonal matrix $U$.
${\mathbf{side}}=\mathrm{Nag_LeftSide}$
The matrix $A$ is multiplied on the left, i.e., premultiplied.
${\mathbf{side}}=\mathrm{Nag_RightSide}$
The matrix $A$ is multiplied on the right, i.e., post-multiplied.
Constraint: ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or $\mathrm{Nag_RightSide}$.
2:    $\mathbf{init}$Nag_InitializeAInput
On entry: indicates whether or not a should be initialized to the identity matrix.
${\mathbf{init}}=\mathrm{Nag_InitializeI}$
a is initialized to the identity matrix.
${\mathbf{init}}=\mathrm{Nag_InputA}$
a is not initialized and the matrix $A$ must be supplied in a.
Constraint: ${\mathbf{init}}=\mathrm{Nag_InitializeI}$ or $\mathrm{Nag_InputA}$.
3:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraints:
• if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{m}}>1$;
• otherwise ${\mathbf{m}}\ge 1$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraints:
• if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{n}}>1$;
• otherwise ${\mathbf{n}}\ge 1$.
5:    $\mathbf{state}\left[\mathit{dim}\right]$IntegerCommunication Array
Note: the dimension, $\mathit{dim}$, 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:    $\mathbf{a}\left[{\mathbf{m}}×{\mathbf{pda}}\right]$doubleInput/Output
On entry: if ${\mathbf{init}}=\mathrm{Nag_InputA}$, a must contain the matrix $A$, with the $\left(i,j\right)$th element of $A$ stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$.
On exit: the matrix $UA$ when ${\mathbf{side}}=\mathrm{Nag_LeftSide}$ or the matrix $AU$ when ${\mathbf{side}}=\mathrm{Nag_RightSide}$.
7:    $\mathbf{pda}$IntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
8:    $\mathbf{fail}$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.
See Section 3.2.1.2 in the Essential Introduction for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT
On entry, ${\mathbf{side}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_LeftSide}$, ${\mathbf{m}}>1$;
otherwise ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{side}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{side}}=\mathrm{Nag_RightSide}$, ${\mathbf{n}}>1$;
otherwise ${\mathbf{n}}\ge 1$.
NE_INT
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge {\mathbf{n}}$.
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 3.6.6 in the Essential Introduction 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 3.6.5 in the Essential Introduction for further information.

## 7  Accuracy

The maximum error in ${U}^{\mathrm{T}}U$ should be a modest multiple of machine precision (see Chapter x02).

## 8  Parallelism and Performance

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

None.

## 10  Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeatable (g05kfc), a $4$ by $4$ orthogonal matrix is generated using the ${\mathbf{init}}=\mathrm{Nag_InitializeI}$ option and the result printed.

### 10.1  Program Text

Program Text (g05pxce.c)

None.

### 10.3  Program Results

Program Results (g05pxce.r)