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NAG Toolbox

NAG Toolbox: nag_lapack_dorgqr (f08af)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_lapack_dorgqr (f08af) generates all or part of the real orthogonal matrix Q from a QR factorization computed by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).

Syntax

[a, info] = f08af(a, tau, 'm', m, 'n', n, 'k', k)
[a, info] = nag_lapack_dorgqr(a, tau, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_dorgqr (f08af) is intended to be used after a call to nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf). which perform a QR 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 leading columns.
Usually Q is determined from the QR factorization of an m by p matrix A with m≥p. The whole of Q may be computed by: 
[a, info] = f08af(a, tau, 'k', p);
(note that the array a must have m columns) or its leading p columns by: 
[a, info] = f08af(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_dgeqrf (f08ae) followed by nag_lapack_dorgqr (f08af) can be used to orthogonalize the columns of A.
The information returned by the QR factorization functions also yields the QR factorization of the leading k columns of A, where k<p. The orthogonal matrix arising from this factorization can be computed by: 
[a, info] = f08af(a, tau, 'k', k);
or its leading k columns by: 
[a, info] = f08af(a(:,1:p), tau, 'k', k);

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).
2:     tau: – double array
The dimension of the array tau must be at least max1,k
Further details of the elementary reflectors, as returned by nag_lapack_dgeqrf (f08ae), nag_lapack_dgeqpf (f08be) or nag_lapack_dgeqp3 (f08bf).

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: the first dimension of the array a.
m, the order of the orthogonal matrix Q.
Constraint: m≥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≥n≥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≥k≥0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,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
E2 = Oε ,  
where ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 4mnk-2 m+n k2 + 43 k3 ; when n=k, the number is approximately 23 n2 3m-n .
The complex analogue of this function is nag_lapack_zungqr (f08at).

Example

This example forms the leading 4 columns of the orthogonal matrix Q from the QR factorization of the matrix A, where
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 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 .  
The columns of Q form an orthonormal basis for the space spanned by the columns of A.

Open in the MATLAB editor: f08af_example

function f08af_example


fprintf('f08af example results\n\n');

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,  0.30,  0.15, -2.13;
     -0.02,  1.03, -1.43,  0.50];

% Compute the QR Factorisation of A
[a, tau, info] = f08ae(a);

% Generate q
[q, info] = f08af(a, tau);

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



f08af example results

Orthogonal factor Q
   -0.1576    0.6744   -0.4571    0.4489
   -0.5335   -0.3861    0.2583    0.3898
    0.6358   -0.2928    0.0165    0.1930
   -0.5335   -0.1692   -0.0834   -0.2350
    0.0415   -0.1593    0.1475    0.7436
   -0.0055   -0.5064   -0.8339    0.0335
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Chapter Contents
Chapter Introduction
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