hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dorgql (f08cf)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dorgql (f08cf) generates all or part of the real m by m orthogonal matrix Q from a QL 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 QL 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 QL factorization of an m by p matrix A with mp. 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 QL 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:     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_dgeqlf (f08ce).
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_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: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix Q.
Constraint: mn0.
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: nk0.

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_zungql (f08ct).

Example

This example generates the first four columns of the matrix Q of the QL factorization of A as returned by nag_lapack_dgeqlf (f08ce), 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 .  
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.
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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015