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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgelsd (f08kc)

## Purpose

nag_lapack_dgelsd (f08kc) computes the minimum norm solution to a real linear least squares problem
 $minx b-Ax2 .$

## Syntax

[a, b, s, rank, info] = f08kc(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, s, rank, info] = nag_lapack_dgelsd(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgelsd (f08kc) uses the singular value decomposition (SVD) of $A$, where $A$ is a real $m$ 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $n$ coefficient matrix $A$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 contents of a are destroyed.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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 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: iwork, 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 complex analogue of this function is nag_lapack_zgelsd (f08kq).

## Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = -0.09 -1.56 -1.48 -1.09 0.08 -1.59 0.14 0.20 -0.43 0.84 0.55 -0.72 -0.46 0.29 0.89 0.77 -1.13 1.06 0.68 1.09 -0.71 2.11 0.14 1.24 1.29 0.51 -0.96 -1.27 1.74 0.34 and b= 7.4 4.3 -8.1 1.8 8.7 .$
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 f08kc_example

fprintf('f08kc example results\n\n');

% Least squares problem min ||b - Ax|| where A and b are:
a = [-0.09, -1.56, -1.48, -1.09,  0.08, -1.59;
0.14,  0.20, -0.43,  0.84,  0.55, -0.72;
-0.46,  0.29,  0.89,  0.77, -1.13,  1.06;
0.68,  1.09, -0.71,  2.11,  0.14,  1.24;
1.29,  0.51, -0.96, -1.27,  1.74,  0.34];
[m,n] = size(a);
b = [ 7.4;
4.3;
-8.1;
1.8;
8.7;
0.0];

% Treat singular values less than 0.01 as zero
rcond = 0.01;
[vr, x, s, rank, info] = f08kc( ...
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');

```
```f08kc example results

Least squares solution
1.5938   -0.1180   -3.1501    0.1554    2.5529   -1.6730

Tolerance used to estimate the rank of A
1.00e-02
Estimated rank of A
4

Singular values of A
3.9997    2.9962    2.0001    0.9988    0.0025

```