# NAG CL Interfaceg05pxc (matrix_​orthog)

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## 1Purpose

g05pxc generates a random orthogonal matrix.

## 2Specification

 #include
 void g05pxc (Nag_SideType side, Nag_InitializeA init, Integer m, Integer n, Integer state[], double a[], Integer pda, NagError *fail)
The function may be called by the names: g05pxc, nag_rand_matrix_orthog or nag_rand_orthog_matrix.

## 3Description

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 $I{\sigma }^{2}$ 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 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.

## 4References

Stewart G W (1980) The efficient generation of random orthogonal matrices with an application to condition estimates SIAM J. Numer. Anal. 17 403–409

## 5Arguments

1: $\mathbf{side}$Nag_SideType Input
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_InitializeA Input
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: $\mathbf{m}$Integer Input
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: $\mathbf{n}$Integer Input
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: $\mathbf{state}\left[\mathit{dim}\right]$Integer Communication 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: $\mathbf{a}\left[{\mathbf{m}}×{\mathbf{pda}}\right]$double Input/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: $\mathbf{pda}$Integer Input
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 7 in the Introduction to the NAG Library CL Interface).

## 6Error 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 $⟨\mathit{\text{value}}⟩$ 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 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.

## 7Accuracy

The maximum error in ${U}^{\mathrm{T}}U$ should be a modest multiple of machine precision (see Chapter X02).

## 8Parallelism 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.

None.

## 10Example

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

### 10.1Program Text

Program Text (g05pxce.c)

None.

### 10.3Program Results

Program Results (g05pxce.r)