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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgelss (f08ka)

## Purpose

nag_lapack_dgelss (f08ka) computes the minimum norm solution to a real linear least squares problem
 min ‖b − Ax‖2. x
$minx ‖b-Ax‖2 .$

## Syntax

[a, b, s, rank, info] = f08ka(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)
[a, b, s, rank, info] = nag_lapack_dgelss(a, b, rcond, 'm', m, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgelss (f08ka) 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, : $:$) – double 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, : $:$) – double 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

### Output Parameters

1:     a(lda, : $:$) – double 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, : $:$) – double 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 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: 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 complex analogue of this function is nag_lapack_zgelss (f08kn).

## Example

```function nag_lapack_dgelss_example
a = [-0.09, 0.14, -0.46, 0.68, 1.29;
-1.56, 0.2, 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];
rcond = 0.01;
[aOut, bOut, s, rank, info] = nag_lapack_dgelss(a, b, rcond)
```
```

aOut =

0.6554    0.0104   -0.4376   -0.5300    0.3130
0.1391   -0.1857    0.2942   -0.5255   -0.7638
0.5134    0.5066   -0.1540    0.5115   -0.4409
0.5064   -0.6588    0.3827    0.3792    0.1389
-0.1765   -0.5241   -0.7428    0.1934   -0.3239
0.5335   -0.1595   -0.2363   -0.3490   -0.6641

bOut =

0.6344
0.9699
-1.4403
3.3678
3.3992
-0.0035

s =

3.9997
2.9962
2.0001
0.9988
0.0025

rank =

4

info =

0

```
```function f08ka_example
a = [-0.09, 0.14, -0.46, 0.68, 1.29;
-1.56, 0.2, 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];
rcond = 0.01;
[aOut, bOut, s, rank, info] = f08ka(a, b, rcond)
```
```

aOut =

0.6554    0.0104   -0.4376   -0.5300    0.3130
0.1391   -0.1857    0.2942   -0.5255   -0.7638
0.5134    0.5066   -0.1540    0.5115   -0.4409
0.5064   -0.6588    0.3827    0.3792    0.1389
-0.1765   -0.5241   -0.7428    0.1934   -0.3239
0.5335   -0.1595   -0.2363   -0.3490   -0.6641

bOut =

0.6344
0.9699
-1.4403
3.3678
3.3992
-0.0035

s =

3.9997
2.9962
2.0001
0.9988
0.0025

rank =

4

info =

0

```