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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgelss (f08kn)

## Purpose

nag_lapack_zgelss (f08kn) computes the minimum norm solution to a complex linear least squares problem
 $minx b-Ax2 .$

## Syntax

[a, b, s, rank, info] = f08kn(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, s, rank, info] = nag_lapack_zgelss(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgelss (f08kn) uses the singular value decomposition (SVD) of $A$, where $A$ is an $m$ 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 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $m$ by $n$ matrix $A$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $m$ by $r$ right-hand side matrix $B$.
3:     $\mathrm{rcond}$ – double scalar
Used to determine the effective rank of $A$. Singular values ${\mathbf{s}}\left(i\right)\le {\mathbf{rcond}}×{\mathbf{s}}\left(1\right)$ are treated as zero. If ${\mathbf{rcond}}<0$, machine precision is used instead.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the first dimension of the array a.
$m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathrm{n}$int64int32nag_int scalar
Default: the second dimension of the array a.
$n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathrm{nrhs_p}$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: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The first $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
b stores the $n$ by $r$ solution matrix $X$. If $m\ge n$ and ${\mathbf{rank}}=n$, the residual sum of squares for the solution in the $i$th column is given by the sum of squares of the modulus of elements $n+1,\dots ,m$ in that column.
3:     $\mathrm{s}\left(:\right)$ – double array
The dimension of the array s will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$
The singular values of $A$ in decreasing order.
4:     $\mathrm{rank}$int64int32nag_int scalar
The effective rank of $A$, i.e., the number of singular values which are greater than ${\mathbf{rcond}}×{\mathbf{s}}\left(1\right)$.
5:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, 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: 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.
${\mathbf{info}}>0$
The algorithm for computing the SVD failed to converge; if ${\mathbf{info}}=i$, $i$ 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.

The real analogue of this function is nag_lapack_dgelss (f08ka).

## Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i$
and
 $b = -1.08-2.59i -2.61-1.49i 3.13-3.61i 7.33-8.01i 9.12+7.63i .$
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 f08kn_example

fprintf('f08kn example results\n\n');

% Least Squares solution of Ax = b, where
a = [ 0.47 - 0.34i, -0.40 + 0.54i,  0.60 + 0.01i,  0.80 - 1.02i;
-0.32 - 0.23i, -0.05 + 0.20i, -0.26 - 0.44i, -0.43 + 0.17i;
0.35 - 0.60i, -0.52 - 0.34i,  0.87 - 0.11i, -0.34 - 0.09i;
0.89 + 0.71i, -0.45 - 0.45i, -0.02 - 0.57i,  1.14 - 0.78i;
-0.19 + 0.06i,  0.11 - 0.85i,  1.44 + 0.80i,  0.07 + 1.14i];
b = [-1.08 - 2.59i;
-2.61 - 1.49i;
3.13 - 3.61i;
7.33 - 8.01i;
9.12 + 7.63i];
[m,n] = size(a);

% treat singular values < 0.01 as zero
rcond = 0.01;
[~, x, s, rank, info] = f08kn( ...
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');

```
```f08kn example results

Least squares solution
1.1673 - 3.3222i
1.3480 + 5.5028i
4.1762 + 2.3434i
0.6465 + 0.0105i

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

```