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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dorgql (f08cf)

▸▿ 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_dorgql (f08cf) generates all or part of the real

m

m

orthogonal matrix

Q

from a

Q L

factorization computed by nag_lapack_dgeqlf (f08ce).

Syntax

[a, info] = f08cf(a, tau, 'm', m, 'n', n, 'k', k)

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

Description

nag_lapack_dorgql (f08cf) is intended to be used after a call to nag_lapack_dgeqlf (f08ce), which performs a

Q L

factorization of a real matrix

A

. The orthogonal 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] = f08cf(a, tau);

(note that the array a must have at least

m

columns) or its trailing

p

columns by:

[a, info] = f08cf(a(:,1:p), tau);

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_dgeqlf (f08ce) followed by nag_lapack_dorgql (f08cf) can be used to orthogonalize the columns of

A

The information returned by nag_lapack_dgeqlf (f08ce) also yields the

Q L

factorization of the trailing

k

columns of

A

, where

k < p

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

[a, info] = f08cf(a, tau, 'k', k);

or its trailing

k

columns by:

[a, info] = f08cf(a(:,1:p), 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: $a (lda :)$ – double 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_dgeqlf (f08ce).
2: $tau (:)$ – double 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_dgeqlf (f08ce).

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: $a (lda :)$ – double 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 orthogonal matrix by a matrix

E

such that

{‖E‖}_{2} = O (ε),

where

ε

is the machine precision.

Further Comments

The total number of floating-point operations is approximately

4 m n k - 2 (m + n) k^{2} + \frac{4}{3} k^{3}

; when

n = k

, the number is approximately

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

The complex analogue of this function is nag_lapack_zungql (f08ct).

Example

This example generates the first four columns of the matrix

Q

of the

Q L

factorization of

A

as returned by nag_lapack_dgeqlf (f08ce), where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \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: f08cf_example

function f08cf_example


fprintf('f08cf example results\n\n');

% Form Q from QL factorization of A
m = 6;
n = 4;
a = [-0.57, -1.28, -0.39,  0.25;
     -1.93,  1.08, -0.31, -2.14;
      2.30,  0.24,  0.40, -0.35;
     -1.93,  0.64, -0.66,  0.08;
      0.15,  0.3,   0.15, -2.13;
     -0.02,  1.03, -1.43,  0.5];

% Compute the QL factorization of A
[ql, tau, info] = f08ce(a);

% Form Q
[Q, info] = f08cf(ql, tau);
% Print Q
[ifail] = x04ca( ...
		 'General', ' ', Q, 'Orthogonal Q from QL of A');

f08cf example results

 Orthogonal Q from QL of A
          1       2       3       4
 1  -0.0833  0.9100 -0.2202 -0.0809
 2   0.2972 -0.1080 -0.2706  0.6922
 3  -0.6404 -0.2351  0.2220  0.1132
 4   0.4461 -0.1620 -0.3866 -0.0259
 5  -0.2938  0.2022  0.0015  0.6890
 6  -0.4575 -0.1946 -0.8243 -0.1617

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Chapter Contents

Chapter Introduction

NAG Toolbox