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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_matop_real_gen_rq (f01qj)

## Purpose

nag_matop_real_gen_rq (f01qj) finds the RQ$RQ$ factorization of the real m$m$ by n$n$ (mn$m\le n$) matrix A$A$, so that A$A$ 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$m$ by n$n$ matrix A$A$ is factorized as
when  m < n, A = RPT when  m = n,
(R0)
 A = PT
where P$P$ is an n$n$ by n$n$ orthogonal matrix and R$R$ is an m$m$ by m$m$ upper triangular matrix.
P$P$ is given as a sequence of Householder transformation matrices
 P = Pm … P2P1, $P=Pm…P2P1,$
the (mk + 1$m-k+1$)th transformation matrix, Pk${P}_{k}$, being used to introduce zeros into the k$k$th row of A$A$. Pk${P}_{k}$ has the form
 Pk = I − ukukT, $Pk=I-ukukT,$
where
uk =
 wk ζk 0 zk
,
$uk= wk ζk 0 zk ,$
ζk${\zeta }_{k}$ is a scalar, wk${w}_{k}$ is an (k1)$\left(k-1\right)$ element vector and zk${z}_{k}$ is an (nm)$\left(n-m\right)$ element vector. uk${u}_{k}$ is chosen to annihilate the elements in the k$k$th row of A$A$.
The vector uk${u}_{k}$ is returned in the k$k$th element of zeta and in the k$k$th row of a, such that ζk${\zeta }_{k}$ is in zeta(k)${\mathbf{zeta}}\left(k\right)$, the elements of wk${w}_{k}$ are in a(k,1),,a(k,k1)${\mathbf{a}}\left(k,1\right),\dots ,{\mathbf{a}}\left(k,k-1\right)$ and the elements of zk${z}_{k}$ are in a(k,m + 1),,a(k,n)${\mathbf{a}}\left(k,m+1\right),\dots ,{\mathbf{a}}\left(k,n\right)$. The elements of R$R$ 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:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The leading m$m$ by n$n$ part of the array a must contain the matrix to be factorized.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
When m = 0${\mathbf{m}}=0$ then an immediate return is effected.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: nm${\mathbf{n}}\ge {\mathbf{m}}$.

lda

### Output Parameters

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

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=-1$
 On entry, m < 0${\mathbf{m}}<0$, or n < m${\mathbf{n}}<{\mathbf{m}}$, or lda < m$\mathit{lda}<{\mathbf{m}}$.

## Accuracy

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

The approximate number of floating point operations is given by 2 × m2(3nm) / 3$2×{m}^{2}\left(3n-m\right)/3$.
The first k$k$ rows of the orthogonal matrix PT${P}^{\mathrm{T}}$ can be obtained by calling nag_matop_real_gen_rq_formq (f01qk), which overwrites the k$k$ rows of PT${P}^{\mathrm{T}}$ on the first k$k$ rows of the array a. PT${P}^{\mathrm{T}}$ is obtained by the call:
```[a, ifail] = f01qk('Separate', m, k, a, zeta);
```

## Example

```function nag_matop_real_gen_rq_example
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];
[aOut, zeta, ifail] = nag_matop_real_gen_rq(a)
```
```

aOut =

-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

zeta =

1.0092
1.2981
1.2329

ifail =

0

```
```function f01qj_example
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];
[aOut, zeta, ifail] = f01qj(a)
```
```

aOut =

-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

zeta =

1.0092
1.2981
1.2329

ifail =

0

```