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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgelsy (f08bn)

Purpose

nag_lapack_zgelsy (f08bn) computes the minimum norm solution to a complex linear least squares problem
 $minx b-Ax2$
using a complete orthogonal factorization of $A$. $A$ 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.

Syntax

[a, b, jpvt, rank, info] = f08bn(a, b, jpvt, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, jpvt, rank, info] = nag_lapack_zgelsy(a, b, jpvt, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

Description

The right-hand side vectors are stored as the columns of the $m$ by $r$ matrix $B$ and the solution vectors in the $n$ by $r$ matrix $X$.
nag_lapack_zgelsy (f08bn) first computes a $QR$ factorization with column pivoting
 $AP= Q R11 R12 0 R22 ,$
with ${R}_{11}$ defined as the largest leading sub-matrix whose estimated condition number is less than $1/{\mathbf{rcond}}$. The order of ${R}_{11}$, rank, is the effective rank of $A$.
Then, ${R}_{22}$ is considered to be negligible, and ${R}_{12}$ is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
 $AP= Q T11 0 0 0 Z .$
The minimum norm solution is then
 $X = PZH T11-1 Q1H b 0$
where ${Q}_{1}$ consists of the first rank columns of $Q$.

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:     $\mathrm{jpvt}\left(:\right)$int64int32nag_int array
The dimension of the array jpvt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{jpvt}}\left(i\right)\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.
4:     $\mathrm{rcond}$ – double scalar
Suggested value: if the condition number of a is not known then ${\mathbf{rcond}}=\sqrt{\left(\epsilon \right)/2}$ (where $\epsilon$ is machine precision, see nag_machine_precision (x02aj)) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective $\text{rank}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ that could be larger than its actual rank, leading to meaningless results.
Used to determine the effective rank of $A$, which is defined as the order of the largest leading triangular sub-matrix ${R}_{11}$ in the $QR$ factorization of $A$, whose estimated condition number is $\text{}<1/{\mathbf{rcond}}$.

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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores details of its complete orthogonal factorization.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ solution matrix $X$.
3:     $\mathrm{jpvt}\left(:\right)$int64int32nag_int array
The dimension of the array jpvt will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{jpvt}}\left(i\right)=k$, then the $i$th column of $AP$ was the $k$th column of $A$.
4:     $\mathrm{rank}$int64int32nag_int scalar
The effective rank of $A$, i.e., the order of the sub-matrix ${R}_{11}$. This is the same as the order of the sub-matrix ${T}_{11}$ in the complete orthogonal factorization of $A$.
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: jpvt, 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.

Accuracy

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

The real analogue of this function is nag_lapack_dgelsy (f08ba).

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 f08bn_example

fprintf('f08bn example results\n\n');

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);
jpvt = zeros(n,1,'int64');
rcond = 0.01;

[af, x, jpvt, rank, info] = f08bn( ...
a, b, jpvt, rcond);

disp('Least squares solution');
disp(x(1:n)');
disp('Tolerance used to estimate the rank of A');
disp(rcond);
disp('Estimated rank of A');
disp(rank);

```
```f08bn example results

Least squares solution
1.1669 + 3.3224i   1.3486 - 5.5027i   4.1764 - 2.3435i   0.6467 - 0.0107i

Tolerance used to estimate the rank of A
0.0100

Estimated rank of A
3

```