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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zungql (f08ct)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zungql (f08ct) generates all or part of the complex

m

m

unitary matrix

Q

from a

Q L

factorization computed by nag_lapack_zgeqlf (f08cs).

Syntax

[a, info] = f08ct(a, tau, 'm', m, 'n', n, 'k', k)

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

Description

nag_lapack_zungql (f08ct) is intended to be used after a call to nag_lapack_zgeqlf (f08cs), which performs a

Q L

factorization of a complex matrix

A

. The unitary matrix

Q

is represented as a product of elementary reflectors.

This function may be used to generate

Q

explicitly as a square matrix, or to form only its trailing columns.

Usually

Q

is determined from the

Q L

factorization of an

m

p

matrix

A

with

m \geq p

. The whole of

Q

may be computed by:

[a, info] = f08ct(a, tau, 'k', p);

(note that the array a must have at least

m

columns) or its trailing

p

columns by:

[a, info] = f08ct(a(:,1:p), tau, 'k', p);

The columns of

Q

returned by the last call form an orthonormal basis for the space spanned by the columns of

A

; thus nag_lapack_zgeqlf (f08cs) followed by nag_lapack_zungql (f08ct) can be used to orthogonalize the columns of

A

The information returned by nag_lapack_zgeqlf (f08cs) also yields the

Q L

factorization of the trailing

k

columns of

A

, where

k < p

. The unitary matrix arising from this factorization can be computed by:

[a, info] = f08ct(a, tau);

or its trailing

k

columns by:

[a, info] = f08ct(a(:,1:k), tau);

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 $\max (1, m)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgeqlf (f08cs).
2: $tau (:)$ – complex array: The dimension of the array tau must be at least $\max (1, k)$

Further details of the elementary reflectors, as returned by nag_lapack_zgeqlf (f08cs).

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the first dimension of the array a.
$m$ , the number of rows of the matrix $Q$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $Q$ .

Constraint: $m \geq n \geq 0$ .
3: $k$ – int64int32nag_int scalar: Default: the dimension of the array tau.
$k$ , the number of elementary reflectors whose product defines the matrix $Q$ .

Constraint: $n \geq k \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, m)$ .
The second dimension of the array a will be $\max (1, n)$ .

The $m$ by $n$ matrix $Q$ .
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ 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

differs from an exactly unitary matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

Further Comments

The total number of real floating-point operations is approximately

16 m n k - 8 (m + n) k^{2} + \frac{16}{3} k^{3}

; when

n = k

, the number is approximately

\frac{8}{3} n^{2} (3 m - n)

The real analogue of this function is nag_lapack_dorgql (f08cf).

Example

This example generates the first four columns of the matrix

Q

of the

Q L

factorization of

A

as returned by nag_lapack_zgeqlf (f08cs), where

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ - 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array}) .

Note that the block size (NB) of

64

assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

Open in the MATLAB editor: f08ct_example

function f08ct_example


fprintf('f08ct example results\n\n');

% Form Q from QL factorization of A
m = 6;
n = 4;
a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i, -0.05 + 0.41i;
     -0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i, -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i, -1.11 + 0.60i;
     -0.37 + 0.38i,  0.19 - 0.54i, -0.98 - 0.36i,  0.22 - 0.20i;
      0.83 + 0.51i,  0.20 + 0.01i, -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i,  0.26 + 0.26i];

% Compute the QL factorization of A
[ql, tau, info] = f08cs(a);

% Form Q
[Q, info] = f08ct(ql, tau);

%Print Q
ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
		 'General', ' ', Q, 'Bracketed', 'F7.4', ...
                 'Unitary Q from QL of A', 'Integer', 'Integer', ...
		 ncols, indent);

f08ct example results

 Unitary Q from QL of A
                    1                 2                 3                 4
 1  ( 0.2810, 0.5020) (-0.2051,-0.1092) ( 0.3083,-0.6874) ( 0.0181,-0.1483)
 2  ( 0.2707,-0.3296) ( 0.5711, 0.0432) ( 0.2251,-0.1313) ( 0.2930,-0.2025)
 3  (-0.2864,-0.0094) (-0.5416, 0.0454) (-0.2062, 0.0691) ( 0.4015,-0.2170)
 4  ( 0.2262,-0.3854) (-0.3387, 0.2228) ( 0.3259, 0.1178) (-0.0796, 0.0723)
 5  ( 0.0341,-0.0760) ( 0.0098,-0.0712) ( 0.0753, 0.1412) (-0.5317,-0.5751)
 6  (-0.3936,-0.2083) (-0.1296, 0.3691) ( 0.0264,-0.4134) (-0.0940,-0.0940)

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Chapter Contents

Chapter Introduction

NAG Toolbox