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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgels (f08an)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgels (f08an) solves linear least squares problems of the form
minx b-Ax2   or   minx b-AHx2 ,  
where A is an m by n complex matrix of full rank, using a QR or LQ factorization of A.

Syntax

[a, b, info] = f08an(trans, a, b, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, info] = nag_lapack_zgels(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='C' and mn: find the minimum norm solution of an undetermined system AHx=b.
4. If trans='C' and m<n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
minx b-AHx2 .  
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='C', the linear system involves AH.
Constraint: trans='N' or 'C'.
2:     alda: – complex 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: – complex 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='C'.

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: – complex 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_zgeqrf (f08as).
If m<n, a stores details of its LQ factorization, as returned by nag_lapack_zgelqf (f08av).
2:     bldb: – complex 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='C' 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

   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.
   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 83 n2 3m-n  if mn and 83 m2 3n-m  otherwise. Following the factorization the solution for a single vector x requires O minm2,n2  operations.
The real analogue of this function is nag_lapack_dgels (f08aa).

Example

This example solves the linear least squares problem
minx b-Ax2 ,  
where
A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i  
and
b = -2.09+1.93i 3.34-3.53i -4.94-2.04i 0.17+4.23i -5.19+3.63i 0.98+2.53i .  
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 f08an_example


fprintf('f08an example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i,  -0.91 + 2.06i, -0.05 + 0.41i;
     -0.98 + 1.98i, -1.20 + 0.19i,  -0.66 + 0.42i, -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,   0.63 - 0.17i, -1.11 + 0.60i;
     -0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.20i;
      0.83 + 0.51i,  0.20 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.20 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
b = [-2.09 + 1.93i;
      3.34 - 3.53i;
     -4.94 - 2.04i;
      0.17 + 4.23i;
     -5.19 + 3.63i;
      0.98 + 2.53i];
[m,n] = size(a);

% Solve the least squares problem min( norm2(b - Ax) ) for x
trans = 'No transpose';
[af, x, info] = f08an( ...
		       trans, a, b);

% Print Solution
fprintf('\nLeast Squares Solution:\n');
disp(transpose(x(1:n)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(x(n+1:m),2));


f08an example results


Least Squares Solution:
  -0.5044 - 1.2179i  -2.4281 + 2.8574i   1.4872 - 2.1955i   0.4537 + 2.6904i

Square root of the residual sum of squares
    0.0688


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Chapter Contents
Chapter Introduction
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