NAG Toolbox: nag_lapack_zggsvd (f08vn)

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 $B$.

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 matrices $A$ and $B$.

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

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

Default: the first dimension of the array b.

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

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

Output Parameters

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

2:     $\mathrm{l}$ – 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 $\left(\begin{array}{c}A\\ B\end{array}\right)$.

3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 matrix $R$, or part of $R$. See Description for details.

4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $R$ if $m-k-l<0$. See Description for details.

5:     $\mathrm{alpha}\left(\mathbf{n}\right)$ – double array

See the description of beta.

6:     $\mathrm{beta}\left(\mathbf{n}\right)$ – double array

alpha and beta contain the generalized singular value pairs of $A$ and $B$, ${\alpha }_i$ and ${\beta }_i$;

• $\mathbf{alpha}\left(1:\mathbf{k}\right)=1$,
• $\mathbf{beta}\left(1:\mathbf{k}\right)=0$,

and if $m-k-l\ge 0$,

• $\mathbf{alpha}\left(\mathbf{k}+1:\mathbf{k}+\mathbf{l}\right)=C$,
• $\mathbf{beta}\left(\mathbf{k}+1:\mathbf{k}+\mathbf{l}\right)=S$,

or if $m-k-l<0$,

• $\mathbf{alpha}\left(\mathbf{k}+1:\mathbf{m}\right)=C$,
• $\mathbf{alpha}\left(\mathbf{m}+1:\mathbf{k}+\mathbf{l}\right)=0$,
• $\mathbf{beta}\left(\mathbf{k}+1:\mathbf{m}\right)=S$,
• $\mathbf{beta}\left(\mathbf{m}+1:\mathbf{k}+\mathbf{l}\right)=1$, and
• $\mathbf{alpha}\left(\mathbf{k}+\mathbf{l}+1:\mathbf{n}\right)=0$,
• $\mathbf{beta}\left(\mathbf{k}+\mathbf{l}+1:\mathbf{n}\right)=0$.

The notation $\mathbf{alpha}\left(\mathbf{k}:\mathbf{n}\right)$ above refers to consecutive elements $\mathbf{alpha}\left(\mathit{i}\right)$, for $\mathit{i}=\mathbf{k},\dots ,\mathbf{n}$.

7:     $\mathrm{u}\left(\mathit{ldu},:\right)$ – 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 $m$ by $m$ unitary matrix $U$.

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

8:     $\mathrm{v}\left(\mathit{ldv},:\right)$ – 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 $p$ by $p$ unitary matrix $V$.

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

9:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – 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 $n$ by $n$ unitary matrix $Q$.

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

10:   $\mathrm{iwork}\left(\mathbf{n}\right)$ – int64int32nag_int array

Stores the sorting information. More precisely, the following loop will sort alpha

 for i=k+1:min(m,k+l)
 % swap alpha(i) and alpha(iwork(i))
 end 

such that $\mathbf{alpha}\left(1\right)\ge \mathbf{alpha}\left(2\right)\ge \cdots \ge \mathbf{alpha}\left(\mathbf{n}\right)$.

11:   $\mathrm{info}$ – 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: n, 6: p, 7: k, 8: l, 9: a, 10: lda, 11: b, 12: ldb, 13: alpha, 14: beta, 15: u, 16: ldu, 17: v, 18: ldv, 19: q, 20: ldq, 21: work, 22: rwork, 23: iwork, 24: 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.

   $\mathbf{info}=1$

If $\mathbf{info}=1$, the Jacobi-type procedure failed to converge.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08vn_example

function f08vn_example


fprintf('f08vn example results\n\n');

% Generalized SVD of (A, B), where
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];
b = complex([ 1,  0, -1,  0;
              0,  1,  0, -1]);

% Compute the GSVD of (a, b)
  % (a,b) = (U*D1,v*D2)*(0 R)*(Q^H), m>=n
jobu = 'U';
jobv = 'V';
jobq = 'Q';
[k, l, DR, b, alpha, beta, U, V, Q, iwork, info] = ...
  f08vn( ...
	 jobu, jobv, jobq, a, b);

irank = k + l;
fprintf('\nInfinite generalized singular values = %5d\n', k);
fprintf('  Finite generalized singular values = %5d\n', l);
fprintf('  Numerical rank of (a'' b'')''         = %5d\n', irank);

fprintf('\nFinite generalized singular values\n');
w = alpha(k+1:irank)./beta(k+1:irank);
disp(transpose(w));

fprintf('Orthogonal matrix u\n');
disp(U);

fprintf('Orthogonal matrix v\n');
disp(V);

fprintf('Orthogonal matrix q\n');
disp(Q);

fprintf('Non singular upper triangular matrix R\n');
R = DR(1:n,n-(irank-1):n);
disp(R);

% Estimate the reciprocal condition number of R
[rcond, info] = f07tu( ...
		       'Infinity-norm','Upper','Non-unit', R);

fprintf('Estimate of reciprocal condition number for R\n%11.1e\n', rcond);

% If irank = n, compute error bound for generalized singular values
if (irank==n)
  serrbd = x02aj/rcond;
  fprintf('\nError bound for the generalized singular values\n%11.1e\n', ...
          serrbd);
else
  fprintf('\n(A^H B^H)^H is not of full rank\n');
end

f08vn example results


Infinite generalized singular values =     2
  Finite generalized singular values =     2
  Numerical rank of (a' b')'         =     4

Finite generalized singular values
    2.0720    1.1058

Orthogonal matrix u
  -0.0130 - 0.3259i  -0.1404 - 0.2617i   0.2518 - 0.7979i  -0.0510 - 0.2175i  -0.0459 + 0.0001i  -0.0528 - 0.2249i
   0.4276 - 0.6258i   0.0863 - 0.0382i  -0.3219 + 0.1611i   0.1198 + 0.1632i  -0.0803 - 0.4360i  -0.0381 - 0.2191i
  -0.3259 + 0.1643i   0.3816 - 0.1822i   0.1323 - 0.0146i  -0.5067 + 0.1862i   0.0597 - 0.5897i  -0.1385 - 0.0909i
   0.1591 - 0.0052i  -0.2821 + 0.1973i   0.2160 + 0.1881i  -0.4016 + 0.2679i  -0.0464 + 0.3086i  -0.3735 - 0.5515i
  -0.1721 - 0.0130i  -0.5094 - 0.5032i   0.0365 + 0.2032i   0.1927 + 0.1557i   0.5784 - 0.1244i  -0.0188 - 0.0557i
  -0.2634 - 0.2477i  -0.1086 + 0.2847i   0.1091 - 0.1271i  -0.0882 + 0.5617i   0.0158 + 0.0471i   0.6501 + 0.0049i

Orthogonal matrix v
   0.9893 - 0.0000i  -0.1146 + 0.0903i
  -0.1146 - 0.0903i  -0.9893 - 0.0000i

Orthogonal matrix q
   0.7071 + 0.0000i   0.0000 + 0.0000i   0.6995 - 0.0000i   0.0810 - 0.0638i
   0.0000 + 0.0000i   0.7071 + 0.0000i  -0.0810 - 0.0638i   0.6995 + 0.0000i
   0.7071 + 0.0000i   0.0000 + 0.0000i  -0.6995 + 0.0000i  -0.0810 + 0.0638i
   0.0000 + 0.0000i   0.7071 + 0.0000i   0.0810 + 0.0638i  -0.6995 - 0.0000i

Non singular upper triangular matrix R
  -2.7118 + 0.0000i  -1.4390 - 1.0315i  -0.0769 + 1.3613i  -0.2814 - 0.0324i
   0.0000 + 0.0000i  -1.8583 + 0.0000i  -1.0760 + 0.0310i   1.3292 + 0.3677i
   0.0000 + 0.0000i   0.0000 + 0.0000i   3.2537 + 0.0000i  -0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i  -2.1084 + 0.0000i

Estimate of reciprocal condition number for R
    1.3e-01

Error bound for the generalized singular values
    8.3e-16