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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgelsy (f08ba)

## Purpose

nag_lapack_dgelsy (f08ba) computes the minimum norm solution to a real 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] = f08ba(a, b, jpvt, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, jpvt, rank, info] = nag_lapack_dgelsy(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_dgelsy (f08ba) 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
 $X = PZT T11-1 Q1T b 0$
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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $n$ 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:     $\mathrm{jpvt}\left(:\right)$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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
a stores details of its complete orthogonal factorization.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
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: 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 complex analogue of this function is nag_lapack_zgelsy (f08bn).

## Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and b= 7.4 4.2 -8.3 1.8 8.6 2.1 .$
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 f08ba_example

fprintf('f08ba example results\n\n');

a = [-0.09,  0.14, -0.46,  0.68,  1.29;
-1.56,  0.20,  0.29,  1.09,  0.51;
-1.48, -0.43,  0.89, -0.71, -0.96;
-1.09,  0.84,  0.77,  2.11, -1.27;
0.08,  0.55, -1.13,  0.14,  1.74;
-1.59, -0.72,  1.06,  1.24,  0.34];
b = [ 7.4;
4.2;
-8.3;
1.8;
8.6;
2.1];
[m,n] = size(a);

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

[af, x, jpvt, rank, info] = f08ba( ...
a, b, jpvt, rcond);

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

f08ba example results

Least squares solution
0.6344    0.9699   -1.4402    3.3678    3.3992   -0.0035

Tolerance used to estimate the rank of A
0.0100

Estimated rank of A
4