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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
 min ‖b − Ax‖2. x
$minx ‖b-Ax‖2 .$

## 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$A$, where A$A$ is an m$m$ by n$n$ matrix which may be rank-deficient.
Several right-hand side vectors b$b$ and solution vectors x$x$ can be handled in a single call; they are stored as the columns of the m$m$ by r$r$ right-hand side matrix B$B$ and the n$n$ by r$r$ solution matrix X$X$.
The effective rank of A$A$ 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:     a(lda, : $:$) – complex array
The first dimension of the array a must be at least max (1,m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The m$m$ by n$n$ matrix A$A$.
2:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,m,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The m$m$ by r$r$ right-hand side matrix B$B$.
3:     rcond – double scalar
Used to determine the effective rank of A$A$. Singular values s(i)rcond × s(1)${\mathbf{s}}\left(i\right)\le {\mathbf{rcond}}×{\mathbf{s}}\left(1\right)$ are treated as zero. If rcond < 0${\mathbf{rcond}}<0$, machine precision is used instead.

### Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
m$m$, the number of rows of the matrix A$A$.
Constraint: m0${\mathbf{m}}\ge 0$.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
n$n$, the number of columns of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
3:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrices B$B$ and X$X$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork rwork

### Output Parameters

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

## Error Indicators and Warnings

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

```function nag_lapack_zgelss_example
a = [ 0.47 - 0.34i,  -0.4 + 0.54i,  0.6 + 0.01i,  0.8 - 1.02i;
-0.32 - 0.23i,  -0.05 + 0.2i,  -0.26 - 0.44i,  -0.43 + 0.17i;
0.35 - 0.6i,  -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 + 0.8i,  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];
rcond = 0.01;
[aOut, bOut, s, rank, info] = nag_lapack_zgelss(a, b, rcond)
```
```

aOut =

-0.3352 + 0.0000i   0.3721 - 0.1084i  -0.0616 + 0.4811i  -0.4465 + 0.5503i
0.5389 + 0.0000i   0.0529 - 0.2020i   0.0597 + 0.7312i   0.0289 - 0.3564i
-0.3736 + 0.0000i   0.4361 + 0.5487i   0.3416 + 0.2185i   0.3581 - 0.2766i
-0.6765 + 0.0000i  -0.3830 - 0.4102i  -0.1106 + 0.2234i   0.0465 - 0.4038i
-0.0978 + 0.0134i  -0.2470 + 0.5898i  -0.1840 + 0.1355i   0.3393 - 0.6232i

bOut =

1.1673 - 3.3222i
1.3480 + 5.5028i
4.1762 + 2.3434i
0.6465 + 0.0105i
-0.0749 + 0.1345i

s =

2.9979
1.9983
1.0044
0.0064

rank =

3

info =

0

```
```function f08kn_example
a = [ 0.47 - 0.34i,  -0.4 + 0.54i,  0.6 + 0.01i,  0.8 - 1.02i;
-0.32 - 0.23i,  -0.05 + 0.2i,  -0.26 - 0.44i,  -0.43 + 0.17i;
0.35 - 0.6i,  -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 + 0.8i,  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];
rcond = 0.01;
[aOut, bOut, s, rank, info] = f08kn(a, b, rcond)
```
```

aOut =

-0.3352 + 0.0000i   0.3721 - 0.1084i  -0.0616 + 0.4811i  -0.4465 + 0.5503i
0.5389 + 0.0000i   0.0529 - 0.2020i   0.0597 + 0.7312i   0.0289 - 0.3564i
-0.3736 + 0.0000i   0.4361 + 0.5487i   0.3416 + 0.2185i   0.3581 - 0.2766i
-0.6765 + 0.0000i  -0.3830 - 0.4102i  -0.1106 + 0.2234i   0.0465 - 0.4038i
-0.0978 + 0.0134i  -0.2470 + 0.5898i  -0.1840 + 0.1355i   0.3393 - 0.6232i

bOut =

1.1673 - 3.3222i
1.3480 + 5.5028i
4.1762 + 2.3434i
0.6465 + 0.0105i
-0.0749 + 0.1345i

s =

2.9979
1.9983
1.0044
0.0064

rank =

3

info =

0

```