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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgelsd (f08kq)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgelsd (f08kq) computes the minimum norm solution to a complex linear least squares problem
minx b-Ax2 .  

Syntax

[a, b, s, rank, info] = f08kq(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p, 'lwork', lwork)
[a, b, s, rank, info] = nag_lapack_zgelsd(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p, 'lwork', lwork)

Description

nag_lapack_zgelsd (f08kq) uses the singular value decomposition (SVD) of A, where A is a complex m by n matrix which may be rank-deficient.
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.
The problem is solved in three steps:
1. reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a ‘bidiagonal least squares problem’ (BLS);
2. solve the BLS using a divide-and-conquer approach;
3. apply back all the Householder transformations to solve the original least squares problem.
The effective rank of A is determined by treating as zero those singular values which are less than rcond times the largest singular value.

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: – 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 coefficient matrix A.
2:     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 m by r right-hand side matrix B.
3:     rcond – double scalar
Used to determine the effective rank of A. Singular values sircond×s1 are treated as zero. If rcond<0, machine precision is used instead.

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.
4:     lwork int64int32nag_int scalar
Suggested value: for optimal performance, lwork should generally be larger than the required minimum. Consider increasing lwork by at least nb×minm,n , where nb  is the optimal block size.
Default: max1, 64 minm,n maxm+n+r,2r+r×nrhs_p  
The dimension of the array work.
The exact minimum amount of workspace needed depends on m, n and nrhs_p. As long as lwork is at least
max1,m+n+r,2r+r×nrhs_p ,  
where r=minm,n, the code will execute correctly.
Constraint: lwork must be at least max1,m+n+r,2r+r×nrhs_p.

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.
The contents of a are destroyed.
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 n by r solution matrix X. If mn and rank=n, the residual sum of squares for the solution in the ith column is given by the sum of squares of the modulus of elements n+1,,m in that column.
3:     s: – double array
The dimension of the array s will be max1,minm,n
The singular values of A in decreasing order.
4:     rank int64int32nag_int scalar
The effective rank of A, i.e., the number of singular values which are greater than rcond×s1.
5:     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: nrhs_p, 4: a, 5: lda, 6: b, 7: ldb, 8: s, 9: rcond, 10: rank, 11: work, 12: lwork, 13: rwork, 14: iwork, 15: 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
The algorithm for computing the SVD failed to converge; if info=i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero.

Accuracy

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

Further Comments

The real analogue of this function is nag_lapack_dgelsd (f08kc).

Example

This example solves the linear least squares problem
minx b-Ax2  
for the solution, x, of minimum norm, where
A = 0.47-0.34i -0.32-0.23i 0.35-0.60i 0.89+0.71i -0.19+0.06i -0.40+0.54i -0.05+0.20i -0.52-0.34i -0.45-0.45i 0.11-0.85i 0.60+0.01i -0.26-0.44i 0.87-0.11i -0.02-0.57i 1.44+0.80i 0.80-1.02i -0.43+0.17i -0.34-0.09i 1.14-0.78i 0.07+1.14i  
and
b = 2.15-0.20i -2.24+1.82i 4.45-4.28i 5.70-6.25i .  
A tolerance of 0.01 is used to determine the effective rank of A.
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 f08kq_example


fprintf('f08kq example results\n\n');

% Least squares problem min ||b - Ax|| where A and b are:
a = [ 0.47 - 0.34i, -0.32 - 0.23i,  0.35 - 0.60i,  0.89 + 0.71i, -0.19 + 0.06i;
     -0.40 + 0.54i, -0.05 + 0.20i, -0.52 - 0.34i, -0.45 - 0.45i,  0.11 - 0.85i;
      0.60 + 0.01i, -0.26 - 0.44i,  0.87 - 0.11i, -0.02 - 0.57i,  1.44 + 0.80i;
      0.80 - 1.02i, -0.43 + 0.17i, -0.34 - 0.09i,  1.14 - 0.78i,  0.07 + 1.14i];
[m,n] = size(a);
b = [ 2.15 - 0.20i;
     -2.24 + 1.82i;
      4.45 - 4.28i;
      5.70 - 6.25i;
      0    + 0i];

% Treat singular values less than 0.01 as zero
rcond = 0.01;
[vr, x, s, rank, info] = f08kq( ...
                                a, b, rcond);

disp('Least squares solution');
disp(x(1:n));
disp('Tolerance used to estimate the rank of A');
fprintf('%12.2e\n',rcond);
disp('Estimated rank of A');
fprintf('%5d\n\n',rank);
disp('Singular values of A');
disp(s');


f08kq example results

Least squares solution
   3.9747 - 1.8377i
  -0.9186 + 0.8253i
  -0.3105 + 0.1477i
   1.0050 + 0.8626i
  -0.2256 - 1.9425i

Tolerance used to estimate the rank of A
    1.00e-02
Estimated rank of A
    3

Singular values of A
    2.9979    1.9983    1.0044    0.0064


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Chapter Introduction
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