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_dgelqf (f08ah)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgelqf (f08ah) computes the LQ factorization of a real m by n matrix.

Syntax

[a, tau, info] = f08ah(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dgelqf(a, 'm', m, 'n', n)

Description

nag_lapack_dgelqf (f08ah) forms the LQ factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = L 0 Q  
where L is an m by m lower triangular matrix and Q is an n by n orthogonal matrix. It is sometimes more convenient to write the factorization as
A = L 0 Q1 Q2  
which reduces to
A = LQ1 ,  
where Q1 consists of the first m rows of Q, and Q2 the remaining n-m rows.
If m>n, L is trapezoidal, and the factorization can be written
A = L1 L2 Q  
where L1 is lower triangular and L2 is rectangular.
The LQ factorization of A is essentially the same as the QR factorization of AT, since
A = L 0 QAT= QT LT 0 .  
The matrix Q is not formed explicitly but is represented as a product of minm,n elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Further Comments).
Note also that for any k<m, the information returned in the first k rows of the array a represents an LQ factorization of the first k rows of the original matrix A.

References

None.

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.
The m by n matrix A.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
Constraint: n0.

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.
If mn, the elements above the diagonal store details of the orthogonal matrix Q and the lower triangle stores the corresponding elements of the m by m lower triangular matrix L.
If m>n, the strictly upper triangular part stores details of the orthogonal matrix Q and the remaining elements store the corresponding elements of the m by n lower trapezoidal matrix L.
2:     tau: – double array
The dimension of the array tau will be max1,minm,n
Further details of the orthogonal matrix Q.
3:     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: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: 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 factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2 ,  
and ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 23 m2 3n-m  if mn or 23 n2 3m-n  if m>n.
To form the orthogonal matrix Q nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dorglq (f08aj):
[a, info] = f08aj(a, tau, 'k', min(m,n));
but note that the first dimension of the array a, specified by the argument lda, must be at least n, which may be larger than was required by nag_lapack_dgelqf (f08ah).
When mn, it is often only the first m rows of Q that are required, and they may be formed by the call:
[a, info] = f08aj(a, tau, 'k', m);
To apply Q to an arbitrary real rectangular matrix C, nag_lapack_dgelqf (f08ah) may be followed by a call to nag_lapack_dormlq (f08ak). For example,
[c, info] = f08ak('Left', 'Transpose', a, tau, c, 'k', min(m, n));
forms the matrix product C=QTC, where C is m by p.
The complex analogue of this function is nag_lapack_zgelqf (f08av).

Example

This example finds the minimum norm solutions of the under-determined systems of linear equations
Ax1= b1   and   Ax2= b2  
where b1 and b2 are the columns of the matrix B,
A = -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50   and   B= -2.87 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 .  
function f08ah_example


fprintf('f08ah example results\n\n');

a = [-5.42, 3.28, -3.68, 0.27, 2.06, 0.46;
     -1.65, -3.4, -3.2, -1.03, -4.06, -0.01;
     -0.37, 2.35, 1.9, 4.31, -1.76, 1.13;
     -3.15, -0.11, 1.99, -2.7, 0.26, 4.5];
b = [-2.87, -5.23;
      1.63,  0.29;
     -3.52,  4.76;
      0.45, -8.41;
      0,     0;
      0,     0];
% Compute the LQ factorization of a
[a, tau, info] = f08ah(a);

% solve l*y=b
l = tril(a(:, 1:4));
b(1:4,:) = inv(l)*b(1:4,:);

% Compute minimum-norm solution x = (q^t)*b
[x, info] = f08ak( ...
		   'Left', 'Transpose', a, tau, b);

mtitle = 'Minimum-norm solution(s)';
[ifail] = x04ca( ...
                 'General', ' ', x, mtitle);


f08ah example results

 Minimum-norm solution(s)
             1          2
 1      0.2371     0.7383
 2     -0.4575     0.0158
 3     -0.0085    -0.0161
 4     -0.5192     1.0768
 5      0.0239    -0.6436
 6     -0.0543    -0.6613

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