NAG Toolbox: nag_lapack_zggsvp (f08vs)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

If $\mathbf{jobu}=\text{'U'}$, the unitary matrix $U$ is computed.

If $\mathbf{jobu}=\text{'N'}$, $U$ is not computed.

Constraint: $\mathbf{jobu}=\text{'U'}$ or $\text{'N'}$.

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

If $\mathbf{jobv}=\text{'V'}$, the unitary matrix $V$ is computed.

If $\mathbf{jobv}=\text{'N'}$, $V$ is not computed.

Constraint: $\mathbf{jobv}=\text{'V'}$ or $\text{'N'}$.

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

If $\mathbf{jobq}=\text{'Q'}$, the unitary matrix $Q$ is computed.

If $\mathbf{jobq}=\text{'N'}$, $Q$ is not computed.

Constraint: $\mathbf{jobq}=\text{'Q'}$ or $\text{'N'}$.

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

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

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

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

The $p$ by $n$ matrix $B$.

6:     $\mathrm{tola}$ – double scalar

7:     $\mathrm{tolb}$ – double scalar

tola and tolb are the thresholds to determine the effective numerical rank of matrix $B$ and a subblock of $A$. Generally, they are set to

$$\begin{array}{c}\mathbf{tola}=\max \left(\mathbf{m},\mathbf{n}\right)‖A‖\epsilon \text{,}\\ \mathbf{tolb}=\max \left(\mathbf{p},\mathbf{n}\right)‖B‖\epsilon \text{,}\end{array}$$

where $\epsilon$ is the machine precision.

The size of tola and tolb may affect the size of backward errors of the decomposition.

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:     $\mathrm{p}$ – int64int32nag_int scalar

Default: the first dimension of the array b.

$p$, the number of rows of the matrix $B$.

Constraint: $\mathbf{p}\ge 0$.

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

Default: the second dimension of the array a.

$n$, the number of columns of the matrices $A$ and $B$.

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

Contains the triangular (or trapezoidal) matrix described in Description.

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

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

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

Contains the triangular matrix described in Description.

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

4:     $\mathrm{l}$ – int64int32nag_int scalar

k and l specify the dimension of the subblocks $k$ and $l$ as described in Description; $\left(k+l\right)$ is the effective numerical rank of ${\left(\begin{array}{cc}{\mathbf{a}}^{\mathrm{T}}& {\mathbf{b}}^{\mathrm{T}}\end{array}\right)}^{\mathrm{T}}$.

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

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

• if $\mathbf{jobu}=\text{'U'}$, $\mathit{ldu}=\max \left(1,\mathbf{m}\right)$;
• otherwise $\mathit{ldu}=1$.

The second dimension of the array u will be $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobu}=\text{'U'}$ and $1$ otherwise.

If $\mathbf{jobu}=\text{'U'}$, u contains the unitary matrix $U$.

If $\mathbf{jobu}=\text{'N'}$, u is not referenced.

6:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – complex array

The first dimension, $\mathit{ldv}$, of the array v will be

• if $\mathbf{jobv}=\text{'V'}$, $\mathit{ldv}=\max \left(1,\mathbf{p}\right)$;
• otherwise $\mathit{ldv}=1$.

The second dimension of the array v will be $\max \left(1,\mathbf{p}\right)$ if $\mathbf{jobv}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobv}=\text{'V'}$, v contains the unitary matrix $V$.

If $\mathbf{jobv}=\text{'N'}$, v is not referenced.

7:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – complex array

The first dimension, $\mathit{ldq}$, of the array q will be

• if $\mathbf{jobq}=\text{'Q'}$, $\mathit{ldq}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldq}=1$.

The second dimension of the array q will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobq}=\text{'Q'}$ and $1$ otherwise.

If $\mathbf{jobq}=\text{'Q'}$, q contains the unitary matrix $Q$.

If $\mathbf{jobq}=\text{'N'}$, q is not referenced.

8:     $\mathrm{info}$ – 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: jobu, 2: jobv, 3: jobq, 4: m, 5: p, 6: n, 7: a, 8: lda, 9: b, 10: ldb, 11: tola, 12: tolb, 13: k, 14: l, 15: u, 16: ldu, 17: v, 18: ldv, 19: q, 20: ldq, 21: iwork, 22: rwork, 23: tau, 24: work, 25: 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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08vs_example

function f08vs_example


fprintf('f08vs example results\n\n');

m = 6;
n = 4;
a = [ 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i;
     -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i;
      0.62-0.46i  1.01+0.02i  0.63-0.17i -1.11+0.60i;
      0.37+0.38i  0.19-0.54i -0.98-0.36i  0.22-0.20i;
      0.83+0.51i  0.20+0.01i -0.17-0.46i  1.47+1.59i;
      1.08-0.28i  0.20-0.12i -0.07+1.23i  0.26+0.26i];
p = 2;
b = complex([ 1 0 -1  0;
              0 1  0 -1]); 

% Reduce A and B to upper triangular form S = U^H A Q, T = V^H B Q
tola = max(m,n)*norm(a,1)*x02aj;
tolb = max(p,n)*norm(a,1)*x02aj;
[S, T, k, l, U, V, Q, info] = ...
  f08vs( ...
	 'U', 'V', 'Q', a, b, tola, tolb);

% Compute singular values
[S, T, alpha, beta, U, V, Q, ncycle, info] = ...
  f08ys( ...
	 'U', 'V', 'Q', k, l, S, T, tola, tolb, U, V, Q);

fprintf('Number of infinite generalized singular values = %3d\n',k);
fprintf('Number of   finite generalized singular values = %3d\n',l);
fprintf('Effective rank of the matrix pair  (A^T B^T)^T = %3d\n',k+l);
fprintf('Number of cycles of the Kogbetliantz method    = %3d\n\n',ncycle);
disp('Finite generalized singular values');
disp(alpha(k+1:k+l)./beta(k+1:k+l));

f08vs example results

Number of infinite generalized singular values =   2
Number of   finite generalized singular values =   2
Effective rank of the matrix pair  (A^T B^T)^T =   4
Number of cycles of the Kogbetliantz method    =   2

Finite generalized singular values
    2.0720
    1.1058