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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zungrq (f08cw)

Purpose

nag_lapack_zungrq (f08cw) generates all or part of the complex n$n$ by n$n$ unitary matrix Q$Q$ from an RQ$RQ$ factorization computed by nag_lapack_zgerqf (f08cv).

Syntax

[a, info] = f08cw(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_zungrq(a, tau, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_zungrq (f08cw) is intended to be used following a call to nag_lapack_zgerqf (f08cv), which performs an RQ$RQ$ factorization of a complex matrix A$A$ and represents the unitary matrix Q$Q$ as a product of k$k$ elementary reflectors of order n$n$.
This function may be used to generate Q$Q$ explicitly as a square matrix, or to form only its trailing rows.
Usually Q$Q$ is determined from the RQ$RQ$ factorization of a p$p$ by n$n$ matrix A$A$ with pn$p\le n$. The whole of Q$Q$ may be computed by:
```[a, info] = f08cw(a, tau);
```
(note that the matrix A$A$ must have at least n$n$ rows), or its trailing p$p$ rows as:
```[a, info] = f08cw(a(1:p,:), tau, 'k', p);
```
The rows of Q$Q$ returned by the last call form an orthonormal basis for the space spanned by the rows of A$A$; thus nag_lapack_zgerqf (f08cv) followed by nag_lapack_zungrq (f08cw) can be used to orthogonalize the rows of A$A$.
The information returned by nag_lapack_zgerqf (f08cv) also yields the RQ$RQ$ factorization of the trailing k$k$ rows of A$A$, where k < p$k. The unitary matrix arising from this factorization can be computed by:
```[a, info] = f08cw(a, tau, 'k', k);
```
or its leading k$k$ columns by:
```[a, info] = f08cw(a(1:k,:), tau, 'k', k);
```

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     a(lda, : $:$) – complex 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgerqf (f08cv).
2:     tau( : $:$) – complex array
Note: the dimension of the array tau must be at least max (1,k)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
tau(i)${\mathbf{tau}}\left(i\right)$ must contain the scalar factor of the elementary reflector Hi${H}_{i}$, as returned by nag_lapack_zgerqf (f08cv).

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 Q$Q$.
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 Q$Q$.
Constraint: nm${\mathbf{n}}\ge {\mathbf{m}}$.
3:     k – int64int32nag_int scalar
Default: The dimension of the array tau.
k$k$, the number of elementary reflectors whose product defines the matrix Q$Q$.
Constraint: mk0${\mathbf{m}}\ge {\mathbf{k}}\ge 0$.

lda work lwork

Output Parameters

1:     a(lda, : $:$) – complex 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 n$n$ matrix Q$Q$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: k, 4: a, 5: lda, 6: tau, 7: work, 8: lwork, 9: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed matrix Q$Q$ differs from an exactly unitary matrix by a matrix E$E$ such that
 ‖E‖2 = Oε $‖E‖2 = O⁡ε$
and ε$\epsilon$ is the machine precision.

The total number of floating point operations is approximately 16mnk8(m + n)k2 + (16/3)k3$16mnk-8\left(m+n\right){k}^{2}+\frac{16}{3}{k}^{3}$; when m = k$m=k$ this becomes (8/3)m2(3nm)$\frac{8}{3}{m}^{2}\left(3n-m\right)$.
The real analogue of this function is nag_lapack_dorgrq (f08cj).

Example

```function nag_lapack_zungrq_example
a = [ -0.1732849729587456 - 0.353512300439208i, ...
-0.2462313258920653 + 0.2360567043400598i,  -1.835343413987657 + 0i, ...
-1.000908232487384 + 1.32039822375845i,  0.4156798554945925 + 1.306842647249796i, ...
-0.9547699266414058 + 0.7897042065311224i;
0.09114631819652763 + 0.2475043484243413i, ...
-0.4802181323902914 - 0.1420168005324323i,  0.4821546631632406 - 0.09601786537269767i, ...
-1.575493318887248 + 0i,  -1.101370732269467 - 0.05966135311459431i, ...
-0.3055741168299267 + 0.2217392484654139i;
-0.3728300824744412 + 0.4958059927232391i, ...
-0.1765491709076989 + 0.1573286421732319i,  0.21466822640392 + 0.1362481009678133i, ...
-0.2486033881873816 - 0.20517929204053i, ...
-2.991745620440206 + 0i,  -0.01276691481133434 + 0.08423270423682098i;
-0.004843574322500398 + 0.1359553259506703i, ...
-0.2500132710791688 + 0.2066154899636792i,  -0.3473326302453916 + 0.2282035599626739i, ...
0.06655374884663712 - 0.0718369909305264i, ...
0.5272403281091034 + 0.480309771000454i,  -2.764959312539698 + 0i];
tau = [ 1.56428134034193 + 0.1176469112365368i;
1.251296240416963 + 0.1907664156695332i;
1.137210468449182 + 0.3464883871917274i;
1.094033933454587 - 0.09403393345458752i];
[aOut, info] = nag_lapack_zungrq(a, tau)
```
```

aOut =

Columns 1 through 5

0.2810 + 0.5020i   0.2707 - 0.3296i  -0.2864 - 0.0094i   0.2262 - 0.3854i   0.0341 - 0.0760i
-0.2051 - 0.1092i   0.5711 + 0.0432i  -0.5416 + 0.0454i  -0.3387 + 0.2228i   0.0098 - 0.0712i
0.3083 - 0.6874i   0.2251 - 0.1313i  -0.2062 + 0.0691i   0.3259 + 0.1178i   0.0753 + 0.1412i
0.0181 - 0.1483i   0.2930 - 0.2025i   0.4015 - 0.2170i  -0.0796 + 0.0723i  -0.5317 - 0.5751i

Column 6

-0.3936 - 0.2083i
-0.1296 + 0.3691i
0.0264 - 0.4134i
-0.0940 - 0.0940i

info =

0

```
```function f08cw_example
a = [ -0.1732849729587456 - 0.353512300439208i, ...
-0.2462313258920653 + 0.2360567043400598i,  -1.835343413987657 + 0i, ...
-1.000908232487384 + 1.32039822375845i,  0.4156798554945925 + 1.306842647249796i, ...
-0.9547699266414058 + 0.7897042065311224i;
0.09114631819652763 + 0.2475043484243413i, ...
-0.4802181323902914 - 0.1420168005324323i,  0.4821546631632406 - 0.09601786537269767i, ...
-1.575493318887248 + 0i,  -1.101370732269467 - 0.05966135311459431i, ...
-0.3055741168299267 + 0.2217392484654139i;
-0.3728300824744412 + 0.4958059927232391i, ...
-0.1765491709076989 + 0.1573286421732319i,  0.21466822640392 + 0.1362481009678133i, ...
-0.2486033881873816 - 0.20517929204053i, ...
-2.991745620440206 + 0i,  -0.01276691481133434 + 0.08423270423682098i;
-0.004843574322500398 + 0.1359553259506703i, ...
-0.2500132710791688 + 0.2066154899636792i,  -0.3473326302453916 + 0.2282035599626739i, ...
0.06655374884663712 - 0.0718369909305264i, ...
0.5272403281091034 + 0.480309771000454i,  -2.764959312539698 + 0i];
tau = [ 1.56428134034193 + 0.1176469112365368i;
1.251296240416963 + 0.1907664156695332i;
1.137210468449182 + 0.3464883871917274i;
1.094033933454587 - 0.09403393345458752i];
[aOut, info] = f08cw(a, tau)
```
```

aOut =

Columns 1 through 5

0.2810 + 0.5020i   0.2707 - 0.3296i  -0.2864 - 0.0094i   0.2262 - 0.3854i   0.0341 - 0.0760i
-0.2051 - 0.1092i   0.5711 + 0.0432i  -0.5416 + 0.0454i  -0.3387 + 0.2228i   0.0098 - 0.0712i
0.3083 - 0.6874i   0.2251 - 0.1313i  -0.2062 + 0.0691i   0.3259 + 0.1178i   0.0753 + 0.1412i
0.0181 - 0.1483i   0.2930 - 0.2025i   0.4015 - 0.2170i  -0.0796 + 0.0723i  -0.5317 - 0.5751i

Column 6

-0.3936 - 0.2083i
-0.1296 + 0.3691i
0.0264 - 0.4134i
-0.0940 - 0.0940i

info =

0

```