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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dorgrq (f08cj)

▸▿ 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_dorgrq (f08cj) generates all or part of the real

n

n

orthogonal matrix

Q

from an

R Q

factorization computed by nag_lapack_dgerqf (f08ch).

Syntax

[a, info] = f08cj(a, tau, 'm', m, 'n', n, 'k', k)

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

Description

nag_lapack_dorgrq (f08cj) is intended to be used following a call to nag_lapack_dgerqf (f08ch), which performs an

R Q

factorization of a real matrix

A

and represents the orthogonal matrix

Q

as a product of

k

elementary reflectors of order

n

This function may be used to generate

Q

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

Usually

Q

is determined from the

R Q

factorization of a

p

n

matrix

A

with

p \leq n

. The whole of

Q

may be computed by:

[a, info] = f08cj(a, tau);

(note that the matrix

A

must have at least

n

rows), or its trailing

p

rows as:

[a, info] = f08cj(a(1:p,:), tau, 'k', p);

The rows of

Q

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

A

; thus nag_lapack_dgerqf (f08ch) followed by nag_lapack_dorgrq (f08cj) can be used to orthogonalize the rows of

A

The information returned by nag_lapack_dgerqf (f08ch) also yields the

R Q

factorization of the trailing

k

rows of

A

, where

k < p

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

[a, info] = f08cj(a, tau, 'k', k);

or its leading

k

columns by:

[a, info] = f08cj(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: $a (lda :)$ – 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_dgerqf (f08ch).
2: $tau (:)$ – double array: The dimension of the array tau must be at least $\max (1, k)$

$tau (i)$ must contain the scalar factor of the elementary reflector $H_{i}$ , as returned by nag_lapack_dgerqf (f08ch).

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

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

and

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

m = k

this becomes

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

The complex analogue of this function is nag_lapack_zungrq (f08cw).

Example

This example generates the first four rows of the matrix

Q

of the

R Q

factorization of

A

as returned by nag_lapack_dgerqf (f08ch), where

A = (\begin{array}{r} - 0.57 & - 1.93 & 2.30 & - 1.93 & 0.15 & - 0.02 \\ - 1.28 & 1.08 & 0.24 & 0.64 & 0.30 & 1.03 \\ - 0.39 & - 0.31 & 0.40 & - 0.66 & 0.15 & - 1.43 \\ 0.25 & - 2.14 & - 0.35 & 0.08 & - 2.13 & 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: f08cj_example

function f08cj_example


fprintf('f08cj example results\n\n');

a = [-0.57  -1.93   2.30  -1.93   0.15  -0.02;
     -1.28   1.08   0.24   0.64   0.30   1.03;
     -0.39  -0.31   0.40  -0.66   0.15  -1.43;
      0.25  -2.14  -0.35   0.08  -2.13   0.50];

% Compute the RQ Factorisation of A
[rq, tau, info] = f08ch(a);

% Form Q
[Q, info] = f08cj(rq, tau);

disp('Orthogonal factor Q');
disp(Q);

f08cj example results

Orthogonal factor Q
   -0.0833    0.2972   -0.6404    0.4461   -0.2938   -0.4575
    0.9100   -0.1080   -0.2351   -0.1620    0.2022   -0.1946
   -0.2202   -0.2706    0.2220   -0.3866    0.0015   -0.8243
   -0.0809    0.6922    0.1132   -0.0259    0.6890   -0.1617

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Chapter Contents

Chapter Introduction

NAG Toolbox