NAG Toolbox: nag_lapack_zgesdd (f08kr)

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 $V^{\mathrm{H}}$ 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 $V^{\mathrm{H}}$ 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 $V^{\mathrm{H}}$ are returned in the array vt. Otherwise, all columns of $U$ are returned in the array u and the first $m$ rows of $V^{\mathrm{H}}$ are overwritten in the array vt.

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

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

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

2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex 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 $V^{\mathrm{H}}$ (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:     $\mathrm{u}\left(\mathit{ldu},:\right)$ – complex 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 $m$ unitary 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:     $\mathrm{vt}\left(\mathit{ldvt},:\right)$ – complex 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 $n$ unitary matrix $V^{\mathrm{H}}$.

If $\mathbf{jobz}=\text{'S'}$, vt contains the first $\min \left(m,n\right)$ rows of $V^{\mathrm{H}}$ (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: rwork, 14: iwork, 15: 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_zgesdd (f08kr) did not converge, the updating process failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08kr_example

function f08kr_example


fprintf('f08kr example results\n\n');

% SVD of complex matrix A
a = [ 0.96 + 0.81i, -0.98 - 1.98i,  0.62 + 0.46i, ...
                    -0.37 - 0.38i,  0.83 - 0.51i,  1.08 + 0.28i;
     -0.03 - 0.96i, -1.20 - 0.19i,  1.01 - 0.02i, ...
                     0.19 + 0.54i,  0.20 - 0.01i,  0.20 + 0.12i;
     -0.91 - 2.06i, -0.66 - 0.42i,  0.63 + 0.17i, ...
                    -0.98 + 0.36i, -0.17 + 0.46i, -0.07 - 1.23i;
     -0.05 - 0.41i, -0.81 - 0.56i, -1.11 - 0.60i, ...
                     0.22 + 0.20i,  1.47 - 1.59i,  0.26 - 0.26i];
m = int64(size(a,1));
n = int64(size(a,2));
b = complex(ones(m,1));


jobz = 'Singular vector parts of U and VT';
[~, s, u, vt, info] = f08kr( ...
                             jobz, a);


disp('Singular values of A');
disp(s');

% Use SVD to compute minimum-norm solution: VS^(-1)U'b
y = u'*b;
y = y./s;
x = vt'*y;

disp('Minimum-norm solution:');
disp(x);
disp('Norm of Solution:');
disp(norm(x));
disp('Norm of Residual:');
fprintf('%11.1e\n',norm(b - a*x));

f08kr example results

Singular values of A
    3.9994    3.0003    1.9944    0.9995

Minimum-norm solution:
  -0.4024 + 0.3777i
  -0.2272 + 0.3626i
   0.1704 - 0.1532i
   0.2125 + 0.0781i
   0.2041 + 0.2236i
   0.2766 - 0.1517i

Norm of Solution:
    0.8846

Norm of Residual:
    1.1e-15