NAG Toolbox: nag_lapack_dgesdd (f08kd)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Specifies options for computing all or part of the matrix $U$.

$\mathbf{jobz}=\text{'A'}$

All $m$ columns of $U$ and all $n$ rows of $V^{\mathrm{T}}$ are returned in the arrays u and vt.

$\mathbf{jobz}=\text{'S'}$

The first $\min \left(m,n\right)$ columns of $U$ and the first $\min \left(m,n\right)$ rows of $V^{\mathrm{T}}$ are returned in the arrays u and vt.

$\mathbf{jobz}=\text{'O'}$

If $\mathbf{m}\ge \mathbf{n}$, the first $n$ columns of $U$ are overwritten on the array a and all rows of $V^{\mathrm{T}}$ are returned in the array vt. Otherwise, all columns of $U$ are returned in the array u and the first $m$ rows of $V^{\mathrm{T}}$ are overwritten in the array vt.

$\mathbf{jobz}=\text{'N'}$

No columns of $U$ or rows of $V^{\mathrm{T}}$ are computed.

Constraint: $\mathbf{jobz}=\text{'A'}$, $\text{'S'}$, $\text{'O'}$ or $\text{'N'}$.

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 $A$.

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: $\mathbf{n}\ge 0$.

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 $\mathbf{jobz}=\text{'O'}$, a is overwritten with the first $n$ columns of $U$ (the left singular vectors, stored column-wise) if $\mathbf{m}\ge \mathbf{n}$; a is overwritten with the first $m$ rows of $V^{\mathrm{T}}$ (the right singular vectors, stored row-wise) otherwise.

If $\mathbf{jobz}\ne \text{'O'}$, the contents of a are destroyed.

2:     $\mathrm{s}\left(\min \left(\mathbf{m},\mathbf{n}\right)\right)$ – double array

The singular values of $A$, sorted so that $\mathbf{s}\left(i\right)\ge \mathbf{s}\left(i+1\right)$.

3:     $\mathrm{u}\left(\mathit{ldu},:\right)$ – double array

The first dimension, $\mathit{ldu}$, of the array u will be

• if $\mathbf{jobz}=\text{'S'}$ or $\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, $\mathit{ldu}=\max \left(1,\mathbf{m}\right)$;
• otherwise $\mathit{ldu}=1$.

The second dimension of the array u will be $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, $\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$ if $\mathbf{jobz}=\text{'S'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, u contains the $m$ by $m$ orthogonal matrix $U$.

If $\mathbf{jobz}=\text{'S'}$, u contains the first $\min \left(m,n\right)$ columns of $U$ (the left singular vectors, stored column-wise).

If $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$, or $\mathbf{jobz}=\text{'N'}$, u is not referenced.

4:     $\mathrm{vt}\left(\mathit{ldvt},:\right)$ – double array

The first dimension, $\mathit{ldvt}$, of the array vt will be

• if $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$, $\mathit{ldvt}=\max \left(1,\mathbf{n}\right)$;
• if $\mathbf{jobz}=\text{'S'}$, $\mathit{ldvt}=\max \left(1,\min \left(\mathbf{m},\mathbf{n}\right)\right)$;
• otherwise $\mathit{ldvt}=1$.

The second dimension of the array vt will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'A'}$ or $\text{'S'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'A'}$ or $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}\ge \mathbf{n}$, vt contains the $n$ by $n$ orthogonal matrix $V^{\mathrm{T}}$.

If $\mathbf{jobz}=\text{'S'}$, vt contains the first $\min \left(m,n\right)$ rows of $V^{\mathrm{T}}$ (the right singular vectors, stored row-wise).

If $\mathbf{jobz}=\text{'O'}$ and $\mathbf{m}<\mathbf{n}$, or $\mathbf{jobz}=\text{'N'}$, vt is not referenced.

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: jobz, 2: m, 3: n, 4: a, 5: lda, 6: s, 7: u, 8: ldu, 9: vt, 10: ldvt, 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.

   $\mathbf{info}>0$

nag_lapack_dgesdd (f08kd) did not converge, the updating process failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08kd_example

function f08kd_example


fprintf('f08kd example results\n\n');

a = [ 2.27,  0.28, -0.48, 1.07, -2.35,  0.62;
     -1.54, -1.67, -3.09, 1.22,  2.93, -7.39;
      1.15,  0.94,  0.99, 0.79, -1.45,  1.03;
     -1.94, -0.78, -0.21, 0.63,  2.30, -2.57];
m = int64(size(a,1));
n = int64(size(a,2));

jobz = 'Singular vector parts of U and VT';
[~, s, u, vt, info] = f08kd( ...
                             jobz, a);

disp('Singular values of A');
disp(s');
disp('Left singular vectors');
disp(u);
disp('Right singular vectors (by row)');
disp(vt);


% Singular values error bound
serrbd = x02aj*s(1);
% Reciprocal condition numbers for singular vectors
[rcondu, info] = f08fl( ...
		        'Left', m, n, s);
[rcondv, info] = f08fl( ...
		        'Right', m, n, s);

% Sigular vector error bounds
uerrbd = serrbd./rcondu;
verrbd = serrbd./rcondv;

disp('Error estimate for the singular values');
fprintf('%12.1e\n',serrbd);
disp('Error estimates for the left singular vectors');
fprintf('%12.1e',uerrbd);
fprintf('\n');
disp('Error estimates for the right singular vectors');
fprintf('%12.1e',verrbd);
fprintf('\n');

f08kd example results

Singular values of A
    9.9966    3.6831    1.3569    0.5000

Left singular vectors
   -0.1921    0.8030   -0.0041    0.5642
    0.8794    0.3926    0.0752   -0.2587
   -0.2140    0.2980   -0.7827   -0.5027
    0.3795   -0.3351   -0.6178    0.6017

Right singular vectors (by row)
   -0.2774   -0.2020   -0.2918    0.0938    0.4213   -0.7816
    0.6003    0.0301   -0.3348    0.3699   -0.5266   -0.3353
    0.1277   -0.2805   -0.6453   -0.6781   -0.0413    0.1645
   -0.1323   -0.7034   -0.1906    0.5399    0.0575    0.3957

Error estimate for the singular values
     1.1e-15
Error estimates for the left singular vectors
     1.8e-16     4.8e-16     1.3e-15     1.3e-15
Error estimates for the right singular vectors
     1.8e-16     4.8e-16     1.3e-15     2.2e-15