Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zungqr (f08at)

## Purpose

nag_lapack_zungqr (f08at) generates all or part of the complex unitary matrix Q$Q$ from a QR$QR$ factorization computed by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).

## Syntax

[a, info] = f08at(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_zungqr(a, tau, 'm', m, 'n', n, 'k', k)

## Description

nag_lapack_zungqr (f08at) is intended to be used after a call to nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt), which perform a QR$QR$ factorization of a complex matrix A$A$. The unitary matrix Q$Q$ is represented as a product of elementary reflectors.
This function may be used to generate Q$Q$ explicitly as a square matrix, or to form only its leading columns.
Usually Q$Q$ is determined from the QR$QR$ factorization of an m$m$ by p$p$ matrix A$A$ with mp$m\ge p$. The whole of Q$Q$ may be computed by:
```[a, info] = f08at(a, tau);
```
(note that the array a must have m$m$ columns) or its leading p$p$ columns by:
```[a, info] = f08at(a(1:p,:), tau);
```
The columns of Q$Q$ returned by the last call form an orthonormal basis for the space spanned by the columns of A$A$; thus nag_lapack_zgeqrf (f08as) followed by nag_lapack_zungqr (f08at) can be used to orthogonalize the columns of A$A$.
The information returned by the QR$QR$ factorization functions also yields the QR$QR$ factorization of the leading k$k$ columns of A$A$, where k < p$k. The unitary matrix arising from this factorization can be computed by:
```[a, info] = f08at(a, tau);
```
or its leading k$k$ columns by:
```[a, info] = f08at(a(:,1:k), tau);
```

## References

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_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).
2:     tau( : $:$) – complex array
Note: the dimension of the array tau must be at least max (1,k)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{k}}\right)$.
Further details of the elementary reflectors, as returned by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the order of the unitary 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: mn0${\mathbf{m}}\ge {\mathbf{n}}\ge 0$.
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: nk0${\mathbf{n}}\ge {\mathbf{k}}\ge 0$.

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 = O(ε) , $‖E‖2 = O(ε) ,$
where ε$\epsilon$ is the machine precision.

The total number of real 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 n = k$n=k$, the number is approximately (8/3) n2 (3mn) $\frac{8}{3}{n}^{2}\left(3m-n\right)$.
The real analogue of this function is nag_lapack_dorgqr (f08af).

## Example

```function nag_lapack_zungqr_example
a = [-3.087005021051958,  -0.4884993674179767 - 1.141689105124614i, ...
0.3773560431732106 - 1.243729755480491i,  -0.8551654376967619 - 0.7073198731811355i;
-0.326978431123342 + 0.4238066080640211i, ...
1.516316047290931 + 0i,  1.373055096626977 - 0.8176293354211591i,  -0.2508627202628476 + 0.8203486040451443i;
0.1691724764304651 - 0.07980476733072399i, ...
-0.4537104861498296 - 0.006491499591352979i,  -2.17134536255717 + 0i,  -0.2272676203283603 - 0.2957314059070643i;
-0.1059736295130279 + 0.0726861860966964i, ...
-0.2734071741396468 + 0.0978078838704349i,  -0.291822737804996 + 0.4888081441553061i,  -2.353376106555421 + 0i;
0.1729396325459321 + 0.1606326404292985i, ...
-0.3236304714632618 + 0.1230007002199739i,  0.2727685061644792 + 0.04697693306903757i, ...
0.7054226886031557 + 0.2515080566109891i;
0.2698996744687472 - 0.01516708364852971i, ...
-0.1645935439354584 + 0.3389007203482612i,  0.5348395253617789 + 0.3988290677840221i, ...
0.2703069905230761 - 0.07268783264065712i];
tau = [ 1.310981029656006 - 0.2623902437722555i;
1.105103989110574 - 0.450362538745018i;
1.040251871615519 + 0.2121758107261096i;
1.18595901116611 + 0.2011836003307436i];
[aOut, info] = nag_lapack_zungqr(a, tau)
```
```

aOut =

-0.3110 + 0.2624i  -0.3175 + 0.4835i   0.4966 - 0.2997i  -0.0072 - 0.3718i
0.3175 - 0.6414i  -0.2062 + 0.1577i  -0.0793 - 0.3094i  -0.0282 - 0.1491i
-0.2008 + 0.1490i   0.4892 - 0.0900i   0.0357 - 0.0219i   0.5625 - 0.0710i
0.1199 - 0.1231i   0.2566 - 0.3055i   0.4489 - 0.2141i  -0.1651 + 0.1800i
-0.2689 - 0.1652i   0.1697 - 0.2491i  -0.0496 + 0.1158i  -0.4885 - 0.4540i
-0.3499 + 0.0907i  -0.0491 - 0.3133i  -0.1256 - 0.5300i   0.1039 + 0.0450i

info =

0

```
```function f08at_example
a = [-3.087005021051958,  -0.4884993674179767 - 1.141689105124614i, ...
0.3773560431732106 - 1.243729755480491i, ...
-0.8551654376967619 - 0.7073198731811355i;
-0.326978431123342 + 0.4238066080640211i, ...
1.516316047290931 + 0i,  1.373055096626977 - 0.8176293354211591i, ...
-0.2508627202628476 + 0.8203486040451443i;
0.1691724764304651 - 0.07980476733072399i, ...
-0.4537104861498296 - 0.006491499591352979i,  -2.17134536255717 + 0i, ...
-0.2272676203283603 - 0.2957314059070643i;
-0.1059736295130279 + 0.0726861860966964i, ...
-0.2734071741396468 + 0.0978078838704349i, ...
-0.291822737804996 + 0.4888081441553061i,  -2.353376106555421 + 0i;
0.1729396325459321 + 0.1606326404292985i, ...
-0.3236304714632618 + 0.1230007002199739i,  ...
0.2727685061644792 + 0.04697693306903757i, ...
0.7054226886031557 + 0.2515080566109891i;
0.2698996744687472 - 0.01516708364852971i, ...
-0.1645935439354584 + 0.3389007203482612i,  ...
0.5348395253617789 + 0.3988290677840221i, ...
0.2703069905230761 - 0.07268783264065712i];
tau = [ 1.310981029656006 - 0.2623902437722555i;
1.105103989110574 - 0.450362538745018i;
1.040251871615519 + 0.2121758107261096i;
1.18595901116611 + 0.2011836003307436i];
[aOut, info] = f08at(a, tau)
```
```

aOut =

-0.3110 + 0.2624i  -0.3175 + 0.4835i   0.4966 - 0.2997i  -0.0072 - 0.3718i
0.3175 - 0.6414i  -0.2062 + 0.1577i  -0.0793 - 0.3094i  -0.0282 - 0.1491i
-0.2008 + 0.1490i   0.4892 - 0.0900i   0.0357 - 0.0219i   0.5625 - 0.0710i
0.1199 - 0.1231i   0.2566 - 0.3055i   0.4489 - 0.2141i  -0.1651 + 0.1800i
-0.2689 - 0.1652i   0.1697 - 0.2491i  -0.0496 + 0.1158i  -0.4885 - 0.4540i
-0.3499 + 0.0907i  -0.0491 - 0.3133i  -0.1256 - 0.5300i   0.1039 + 0.0450i

info =

0

```