NAG Toolbox: nag_lapack_zgghrd (f08ws)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Specifies the form of the computed unitary matrix $Q$.

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

Do not compute $Q$.

$\mathbf{compq}=\text{'I'}$

The unitary matrix $Q$ is returned.

$\mathbf{compq}=\text{'V'}$

q must contain a unitary matrix $Q_1$, and the product $Q_{1}Q$ is returned.

Constraint: $\mathbf{compq}=\text{'N'}$, $\text{'I'}$ or $\text{'V'}$.

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

Specifies the form of the computed unitary matrix $Z$.

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

Do not compute $Z$.

$\mathbf{compz}=\text{'V'}$

z must contain a unitary matrix $Z_1$, and the product $Z_{1}Z$ is returned.

$\mathbf{compz}=\text{'I'}$

The unitary matrix $Z$ is returned.

Constraint: $\mathbf{compz}=\text{'N'}$, $\text{'V'}$ or $\text{'I'}$.

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

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

$i_{\mathrm{lo}}$ and $i_{\mathrm{hi}}$ as determined by a previous call to nag_lapack_zggbal (f08wv). Otherwise, they should be set to $1$ and $n$, respectively.

Constraints:

• if $\mathbf{n}>0$, $1\le \mathbf{ilo}\le \mathbf{ihi}\le \mathbf{n}$;
• if $\mathbf{n}=0$, $\mathbf{ilo}=1$ and $\mathbf{ihi}=0$.

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

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

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

The matrix $A$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $A$ returned by nag_lapack_zunmqr (f08au).

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

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

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

The upper triangular matrix $B$ of the matrix pair $\left(A,B\right)$. Usually, this is the matrix $B$ returned by the $QR$ factorization function nag_lapack_zgeqrf (f08as).

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

The first dimension, $\mathit{ldq}$, of the array q must satisfy

• if $\mathbf{compq}=\text{'I'}$ or $\text{'V'}$, $\mathit{ldq}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{compq}=\text{'N'}$, $\mathit{ldq}\ge 1$.

The second dimension of the array q must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{compq}=\text{'I'}$ or $\text{'V'}$ and at least $1$ if $\mathbf{compq}=\text{'N'}$.

If $\mathbf{compq}=\text{'V'}$, q must contain a unitary matrix $Q_1$.

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

8:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex array

The first dimension, $\mathit{ldz}$, of the array z must satisfy

• if $\mathbf{compz}=\text{'V'}$ or $\text{'I'}$, $\mathit{ldz}\ge \max \left(1,\mathbf{n}\right)$;
• if $\mathbf{compz}=\text{'N'}$, $\mathit{ldz}\ge 1$.

The second dimension of the array z must be at least $\max \left(1,\mathbf{n}\right)$ if $\mathbf{compz}=\text{'V'}$ or $\text{'I'}$ and at least $1$ if $\mathbf{compz}=\text{'N'}$.

If $\mathbf{compz}=\text{'V'}$, z must contain a unitary matrix $Z_1$.

If $\mathbf{compz}=\text{'N'}$, z is not referenced.

Optional Input Parameters

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

Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b.

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

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

a stores the upper Hessenberg matrix $H$.

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

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

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

b stores the upper triangular matrix $T$.

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

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

• if $\mathbf{compq}=\text{'I'}$ or $\text{'V'}$, $\mathit{ldq}=\max \left(1,\mathbf{n}\right)$;
• if $\mathbf{compq}=\text{'N'}$, $\mathit{ldq}=1$.

The second dimension of the array q will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{compq}=\text{'I'}$ or $\text{'V'}$ and at least $1$ if $\mathbf{compq}=\text{'N'}$.

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

Iif $\mathbf{compq}=\text{'V'}$, q stores $Q_{1}Q$.

4:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex array

The first dimension, $\mathit{ldz}$, of the array z will be

• if $\mathbf{compz}=\text{'V'}$ or $\text{'I'}$, $\mathit{ldz}=\max \left(1,\mathbf{n}\right)$;
• if $\mathbf{compz}=\text{'N'}$, $\mathit{ldz}=1$.

The second dimension of the array z will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{compz}=\text{'V'}$ or $\text{'I'}$ and at least $1$ if $\mathbf{compz}=\text{'N'}$.

If $\mathbf{compz}=\text{'I'}$, z contains the unitary matrix $Z$.

If $\mathbf{compz}=\text{'V'}$, z stores $Z_{1}Z$.

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: compq, 2: compz, 3: n, 4: ilo, 5: ihi, 6: a, 7: lda, 8: b, 9: ldb, 10: q, 11: ldq, 12: z, 13: ldz, 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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ws_example

function f08ws_example


fprintf('f08ws example results\n\n');

% Generalized eigenvalues of matrix pair (A,B) , where
a = [ 1.0+3.0i  1.0+4.0i  1.0+5.0i   1.0+6.0i;
      2.0+2.0i  4.0+3.0i  8.0+4.0i  16.0+5.0i;
      3.0+1.0i  9.0+2.0i 27.0+3.0i  81.0+4.0i;
      4.0+0.0i 16.0+1.0i 64.0+2.0i 256.0+3.0i];
b = [ 1.0+0.0i  2.0+1.0i  3.0+2.0i   4.0+3.0i;
      1.0+1.0i  4.0+2.0i  9.0+3.0i  16.0+4.0i;
      1.0+2.0i  8.0+3.0i 27.0+4.0i  64.0+5.0i;
      1.0+3.0i 16.0+4.0i 81.0+5.0i 256.0+6.0i];

% Balance matrix pair
job = 'Balance';
[a, b, ilo, ihi, lscale, rscale, info] = ...
  f08wv(job, a, b);
bbal = b(ilo:ihi,ilo:ihi);
abal = a(ilo:ihi,ilo:ihi);

% QR factorize balanced B
[QR, tau, info] = f08as(bbal);

% Perform C = Q^H*A
side = 'Left';
trans = 'Conjugate transpose';
[c, info] = f08au( ...
		   side, trans, QR, tau, abal);

% Generalized Hessenberg form (C,R) -> (H,T)
compq = 'No Q';
compz = 'No Z';
z = complex(eye(4));
q = complex(eye(4));
jlo = int64(1);
jhi = int64(ihi-ilo+1);
[H, T, ~, ~, info] = ...
  f08ws( ...
	 compq, compz, jlo, jhi, c, QR, q, z);

% Find eigenvalues of generalized Hessenberg form
%    = eigenvalues of (A,B).
job = 'Eigenvalues';
[~, ~, alpha, beta, ~, ~, info] = ...
  f08xs( ...
	 job, compq, compz, jlo, jhi, H, T, q, z);

disp('Generalized eigenvalues of (A,B):');
disp(sort(alpha./beta));

f08ws example results

Generalized eigenvalues of (A,B):
   0.6744 - 0.0499i
   0.4580 - 0.8426i
   0.4934 + 0.9102i
  -0.6354 + 1.6529i