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# NAG Toolbox: nag_matop_real_gen_rq (f01qj)

## Purpose

nag_matop_real_gen_rq (f01qj) finds the $RQ$ factorization of the real $m$ by $n$ ($m\le n$) matrix $A$, so that $A$ is reduced to upper triangular form by means of orthogonal transformations from the right.

## Syntax

[a, zeta, ifail] = f01qj(a, 'm', m, 'n', n)
[a, zeta, ifail] = nag_matop_real_gen_rq(a, 'm', m, 'n', n)

## Description

The $m$ by $n$ matrix $A$ is factorized as
where $P$ is an $n$ by $n$ orthogonal matrix and $R$ is an $m$ by $m$ upper triangular matrix.
$P$ is given as a sequence of Householder transformation matrices
 $P=Pm…P2P1,$
the ($m-k+1$)th transformation matrix, ${P}_{k}$, being used to introduce zeros into the $k$th row of $A$. ${P}_{k}$ has the form
 $Pk=I-ukukT,$
where
 $uk= wk ζk 0 zk ,$
${\zeta }_{k}$ is a scalar, ${w}_{k}$ is an $\left(k-1\right)$ element vector and ${z}_{k}$ is an $\left(n-m\right)$ element vector. ${u}_{k}$ is chosen to annihilate the elements in the $k$th row of $A$.
The vector ${u}_{k}$ is returned in the $k$th element of zeta and in the $k$th row of a, such that ${\zeta }_{k}$ is in ${\mathbf{zeta}}\left(k\right)$, the elements of ${w}_{k}$ are in ${\mathbf{a}}\left(k,1\right),\dots ,{\mathbf{a}}\left(k,k-1\right)$ and the elements of ${z}_{k}$ are in ${\mathbf{a}}\left(k,m+1\right),\dots ,{\mathbf{a}}\left(k,n\right)$. The elements of $R$ are returned in the upper triangular part of a.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The leading $m$ by $n$ part of the array a must contain the matrix to be factorized.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
When ${\mathbf{m}}=0$ then an immediate return is effected.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $m$ upper triangular part of a will contain the upper triangular matrix $R$, and the $m$ by $m$ strictly lower triangular part of a and the $m$ by $\left(n-m\right)$ rectangular part of a to the right of the upper triangular part will contain details of the factorization as described in Description.
2:     $\mathrm{zeta}\left({\mathbf{m}}\right)$ – double array
${\mathbf{zeta}}\left(k\right)$ contains the scalar ${\zeta }_{k}$ for the $\left(m-k+1\right)$th transformation. If ${P}_{k}=I$ then ${\mathbf{zeta}}\left(k\right)=0.0$, otherwise ${\mathbf{zeta}}\left(k\right)$ contains ${\zeta }_{k}$ as described in Description and ${\zeta }_{k}$ is always in the range $\left(1.0,\sqrt{2.0}\right)$.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=-1$
 On entry, ${\mathbf{m}}<0$, or ${\mathbf{n}}<{\mathbf{m}}$, or $\mathit{lda}<{\mathbf{m}}$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The computed factors $R$ and $P$ satisfy the relation
 $R 0 PT=A+E,$
where
 $E≤cε A,$
$\epsilon$ is the machine precision (see nag_machine_precision (x02aj)), $c$ is a modest function of $m$ and $n$, and $‖.‖$ denotes the spectral (two) norm.

The approximate number of floating-point operations is given by $2×{m}^{2}\left(3n-m\right)/3$.
The first $k$ rows of the orthogonal matrix ${P}^{\mathrm{T}}$ can be obtained by calling nag_matop_real_gen_rq_formq (f01qk), which overwrites the $k$ rows of ${P}^{\mathrm{T}}$ on the first $k$ rows of the array a. ${P}^{\mathrm{T}}$ is obtained by the call:
```[a, ifail] = f01qk('Separate', m, k, a, zeta);
```

## Example

This example obtains the $RQ$ factorization of the $3$ by $5$ matrix
 $A= 2.0 2.0 1.6 2.0 1.2 2.5 2.5 -0.4 -0.5 -0.3 2.5 2.5 2.8 0.5 -2.9 .$
```function f01qj_example

fprintf('f01qj example results\n\n');

a = [2,   2,    1.6,  2,    1.2;
2.5, 2.5, -0.4, -0.5, -0.3;
2.5, 2.5,  2.8,  0.5, -2.9];

[RQ, zeta, ifail] = f01qj(a);

disp('RQ Factorization of A');
disp('Vector zeta');
disp(zeta');
disp('Matrix A after factorization (R in left-hand upper triangle');
disp(RQ);

```
```f01qj example results

RQ Factorization of A
Vector zeta
1.0092    1.2981    1.2329

Matrix A after factorization (R in left-hand upper triangle
-3.1446   -1.0705   -2.2283    0.6333    0.7619
0.5277   -2.8345   -2.2283   -0.1662    0.0945
0.3766    0.3766   -5.3852    0.0753   -0.4368

```

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