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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgelsd (f08kq)

## Purpose

nag_lapack_zgelsd (f08kq) 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] = 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$A$, where A$A$ is a complex 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 problem is solved in three steps:
1. reduce the coefficient matrix A$A$ 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$A$ 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$n$ coefficient 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${\mathbf{nrhs}}\ge 0$.
4:     lwork – int64int32nag_int scalar
The dimension of the array work as declared in the (sub)program from which nag_lapack_zgelsd (f08kq) is called.
The exact minimum amount of workspace needed depends on m, n and nrhs_p. As long as lwork is at least
 max (1,m + n + r,2r + r × nrhs) , $max(1,m+n+r,2r+r×nrhs) ,$
where r = min (m,n)$r=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$, the code will execute correctly.
Suggested value: for optimal performance, lwork should generally be larger than the required minimum. Consider increasing lwork by at least nb × min (m,n) $\mathit{nb}×\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$, where nb $\mathit{nb}$ is the optimal block size.
Default: max (1, 64 min (m,n) max (m + n + r,2r + r × nrhs) ) $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,64\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}+{\mathbf{n}}+\mathit{r},2\mathit{r}+\mathit{r}×{\mathbf{nrhs}}\right)\right)$
Constraint: lwork must be at least max (1,m + n + r,2r + r × nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}+{\mathbf{n}}+r,2r+r×{\mathbf{nrhs}}\right)$.

### Input Parameters Omitted from the MATLAB Interface

lda ldb work rwork iwork

### 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 contents of a are destroyed.
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: 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${\mathbf{INFO}}>0$
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_dgelsd (f08kc).

## Example

```function nag_lapack_zgelsd_example
a = [ 0.47 - 0.34i,  -0.32 - 0.23i,  0.35 - 0.6i,  0.89 + 0.71i,  -0.19 + 0.06i;
-0.4 + 0.54i,  -0.05 + 0.2i,  -0.52 - 0.34i, ...
-0.45 - 0.45i,  0.11 - 0.85i;
0.6 + 0.01i,  -0.26 - 0.44i,  0.87 - 0.11i,  -0.02 - 0.57i,  1.44 + 0.8i;
0.8 - 1.02i,  -0.43 + 0.17i,  -0.34 - 0.09i, ...
1.14 - 0.78i,  0.07 + 1.14i];
b = [ 2.15 - 0.2i;
-2.24 + 1.82i;
4.45 - 4.28i;
5.7 - 6.25i;
0 + 0i];
rcond = 0.01;
[aOut, bOut, s, rank, info] = nag_lapack_zgelsd(a, b, rcond)
```
```

aOut =

-1.5199 + 0.0000i  -0.1371 - 0.1390i   0.2210 - 0.2638i   0.3753 + 0.4209i  -0.0978 + 0.0134i
-1.9194 + 0.0000i  -1.4211 + 0.0000i  -0.2911 + 0.4160i   0.0463 - 0.4369i  -0.2470 + 0.5898i
-0.0459 - 0.0067i   1.9985 + 0.0000i   0.9249 + 0.0000i   0.3343 + 0.4472i  -0.1840 + 0.1355i
-0.2659 + 0.5880i   0.1704 + 0.3519i  -1.0610 + 0.0000i   0.0192 + 0.0000i   0.3393 - 0.6232i

bOut =

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

s =

2.9979
1.9983
1.0044
0.0064

rank =

3

info =

0

```
```function f08kq_example
a = [ 0.47 - 0.34i,  -0.32 - 0.23i,  0.35 - 0.6i,  0.89 + 0.71i,  -0.19 + 0.06i;
-0.4 + 0.54i,  -0.05 + 0.2i,  -0.52 - 0.34i, ...
-0.45 - 0.45i,  0.11 - 0.85i;
0.6 + 0.01i,  -0.26 - 0.44i,  0.87 - 0.11i,  -0.02 - 0.57i,  1.44 + 0.8i;
0.8 - 1.02i,  -0.43 + 0.17i,  -0.34 - 0.09i, ...
1.14 - 0.78i,  0.07 + 1.14i];
b = [ 2.15 - 0.2i;
-2.24 + 1.82i;
4.45 - 4.28i;
5.7 - 6.25i;
0 + 0i];
rcond = 0.01;
[aOut, bOut, s, rank, info] = f08kq(a, b, rcond)
```
```

aOut =

-1.5199 + 0.0000i  -0.1371 - 0.1390i   0.2210 - 0.2638i   0.3753 + 0.4209i  -0.0978 + 0.0134i
-1.9194 + 0.0000i  -1.4211 + 0.0000i  -0.2911 + 0.4160i   0.0463 - 0.4369i  -0.2470 + 0.5898i
-0.0459 - 0.0067i   1.9985 + 0.0000i   0.9249 + 0.0000i   0.3343 + 0.4472i  -0.1840 + 0.1355i
-0.2659 + 0.5880i   0.1704 + 0.3519i  -1.0610 + 0.0000i   0.0192 + 0.0000i   0.3393 - 0.6232i

bOut =

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

s =

2.9979
1.9983
1.0044
0.0064

rank =

3

info =

0

```