NAG Toolbox: nag_lapack_dgels (f08aa)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{trans}$ – string (length ≥ 1)

If $\mathbf{trans}=\text{'N'}$, the linear system involves $A$.

If $\mathbf{trans}=\text{'T'}$, the linear system involves $A^{\mathrm{T}}$.

Constraint: $\mathbf{trans}=\text{'N'}$ or $\text{'T'}$.

2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $A$.

3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 matrix $B$ of right-hand side vectors, stored in columns; b is $m$ by $r$ if $\mathbf{trans}=\text{'N'}$, or $n$ by $r$ if $\mathbf{trans}=\text{'T'}$.

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

If $\mathbf{m}\ge \mathbf{n}$, a stores details of its $QR$ factorization, as returned by nag_lapack_dgeqrf (f08ae).

If $\mathbf{m}<\mathbf{n}$, a stores details of its $LQ$ factorization, as returned by nag_lapack_dgelqf (f08ah).

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 solution vectors, $x$, stored in columns:

• if $\mathbf{trans}=\text{'N'}$ and $m\ge n$, or $\mathbf{trans}=\text{'T'}$ and $m<n$, elements $1$ to $\min \left(m,n\right)$ in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $\left(\min \left(m,n\right)+1\right)$ to $\max \left(m,n\right)$ in that column;
• otherwise, elements $1$ to $\max \left(m,n\right)$ in each column of b contain the minimum norm solution vectors.

3:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: trans, 2: m, 3: n, 4: nrhs_p, 5: a, 6: lda, 7: b, 8: ldb, 9: work, 10: lwork, 11: 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.

W  $\mathbf{info}>0$

If $\mathbf{info}=i$, diagonal element $i$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08aa_example

function f08aa_example


fprintf('f08aa example results\n\n');

% A and b
a = [-0.57, -1.28, -0.39,  0.25;
     -1.93,  1.08, -0.31, -2.14;
      2.30,  0.24,  0.40, -0.35;
     -1.93,  0.64, -0.66,  0.08;
      0.15,  0.30,  0.15, -2.13;
     -0.02,  1.03, -1.43,  0.50];
b = [-2.67;
     -0.55;
      3.34;
     -0.77;
      0.48;
      4.1];
[m,n] = size(a);

% Find x which minimizes ||b-Ax||_2 
trans = 'No transpose';
[af, x, info] = f08aa(trans, a, b);

disp('Least squares solution');
disp(x(1:n)');
rnorm = norm(x(n+1:m));
fprintf('Square root of the residual sum of squares = %11.2e\n', rnorm);

f08aa example results

Least squares solution
    1.5339    1.8707   -1.5241    0.0392

Square root of the residual sum of squares =    2.22e-02