NAG Toolbox: nag_lapack_dgelss (f08ka)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array a must be at least $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

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 $\max \left(1,\mathbf{m},\mathbf{n}\right)$.

The second dimension of the array b must be at least $\max \left(1,\mathbf{nrhs\_p}\right)$.

The $m$ by $r$ right-hand side matrix $B$.

3:     $\mathrm{rcond}$ – double scalar

Used to determine the effective rank of $A$. Singular values $\mathbf{s}\left(i\right)\le \mathbf{rcond}\times \mathbf{s}\left(1\right)$ 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 $\max \left(1,\mathbf{m}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The first $\min \left(m,n\right)$ rows of $A$ are overwritten with its right singular vectors, stored row-wise.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array

The first dimension of the array b will be $\max \left(1,\mathbf{m},\mathbf{n}\right)$.

The second dimension of the array b will be $\max \left(1,\mathbf{nrhs\_p}\right)$.

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 $\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$

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 $\mathbf{rcond}\times \mathbf{s}\left(1\right)$.

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: 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

Further Comments

Example

Open in the MATLAB editor: f08ka_example

function f08ka_example


fprintf('f08ka example results\n\n');

% Least squares problem min ||b - Ax|| where A and b are:
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);

% Treat singular values less than 0.01 as zero
rcond = 0.01;
[vr, x, s, rank, info] = f08ka( ...
                                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');

f08ka example results

Least squares solution
    0.6344    0.9699   -1.4403    3.3678    3.3992

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