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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgels (f08aa)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dgels (f08aa) solves linear least squares problems of the form
minx b-Ax2   or   minx b-ATx2 ,  
where A is an m by n real matrix of full rank, using a QR or LQ factorization of A.

Syntax

[a, b, info] = f08aa(trans, a, b, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, info] = nag_lapack_dgels(trans, a, b, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

Description

The following options are provided:
1. If trans='N' and mn: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minx b-Ax2 .  
2. If trans='N' and m<n: find the minimum norm solution of an underdetermined system Ax=b.
3. If trans='T' and mn: find the minimum norm solution of an undetermined system ATx=b.
4. If trans='T' and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minx b-ATx2 .  
Several right-hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the m by r right-hand side matrix B and the n by r solution matrix X.

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:     trans – string (length ≥ 1)
If trans='N', the linear system involves A.
If trans='T', the linear system involves AT.
Constraint: trans='N' or 'T'.
2:     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.
3:     bldb: – double array
The first dimension of the array b must be at least max1,m,n.
The second dimension of the array b must be at least max1,nrhs_p.
The matrix B of right-hand side vectors, stored in columns; b is m by r if trans='N', or n by r if trans='T'.

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.
3:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides, i.e., the number of columns of the matrices B and X.
Constraint: nrhs_p0.

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, a stores details of its QR factorization, as returned by nag_lapack_dgeqrf (f08ae).
If m<n, a stores details of its LQ factorization, as returned by nag_lapack_dgelqf (f08ah).
2:     bldb: – double array
The first dimension of the array b will be max1,m,n.
The second dimension of the array b will be max1,nrhs_p.
b stores the solution vectors, x, stored in columns:
  • if trans='N' and mn, or trans='T' and m<n, elements 1 to minm,n in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements minm,n+1 to maxm,n in that column;
  • otherwise, elements 1 to maxm,n in each column of b contain the minimum norm solution vectors.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: trans, 2: m, 3: n, 4: nrhs_p, 5: a, 6: lda, 7: b, 8: ldb, 9: work, 10: lwork, 11: 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.
W  info>0
If info=i, diagonal element i of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.

Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

Further Comments

The total number of floating-point operations required to factorize A is approximately 23 n2 3m-n  if mn and 23 m2 3n-m  otherwise. Following the factorization the solution for a single vector x requires O minm2,n2  operations.
The complex analogue of this function is nag_lapack_zgels (f08an).

Example

This example solves the linear least squares problem
minx b-Ax2 ,  
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   and   b= -2.67 -0.55 3.34 -0.77 0.48 4.10 .  
The square root of the residual sum of squares is also output.
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 f08aa_example


fprintf('f08aa example results\n\n');

% A and b
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];
b = [-2.67;
     -0.55;
      3.34;
     -0.77;
      0.48;
      4.1];
[m,n] = size(a);

% Find x which minimizes ||b-Ax||_2 
trans = 'No transpose';
[af, x, info] = f08aa(trans, a, b);

disp('Least squares solution');
disp(x(1:n)');
rnorm = norm(x(n+1:m));
fprintf('Square root of the residual sum of squares = %11.2e\n', rnorm);


f08aa example results

Least squares solution
    1.5339    1.8707   -1.5241    0.0392

Square root of the residual sum of squares =    2.22e-02

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