NAG Toolbox: nag_lapack_dgghrd (f08we)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

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

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

Do not compute $Q$.

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

The orthogonal matrix $Q$ is returned.

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

q must contain an orthogonal 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 orthogonal matrix $Z$.

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

Do not compute $Z$.

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

The orthogonal matrix $Z$ is returned.

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

z must contain an orthogonal matrix $Z_1$, and the product $Z_{1}Z$ 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_dggbal (f08wh). 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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_dormqr (f08ag).

6:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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_dgeqrf (f08ae).

7:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – double 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 an orthogonal matrix $Q_1$.

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

8:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double 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 an orthogonal 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 array z.

$n$, the order of the matrices $A$ and $B$.

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

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

a stores the upper Hessenberg matrix $H$.

2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – double 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 orthogonal matrix $Q$.

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

4:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double 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 orthogonal 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: f08we_example

function f08we_example


fprintf('f08we example results\n\n');

a = [ 1.0   1.0    1.0    1.0     1.0;
      2.0   4.0    8.0   16.0    32.0;
      3.0   9.0   27.0   81.0   243.0;
      4.0  16.0   64.0  256.0  1024.0;
      5.0  25.0  125.0  625.0  3125.0];
b = a';

%' Balance A and B
job = 'B';
[a, b, ilo, ihi, lscale, rscale, info] = ...
  f08wh( ...
	 job, a, b);

bbal = b(ilo:ihi,ilo:ihi);
abal = a(ilo:ihi,ilo:ihi);

% QR factorize balanced B
[QR, tau, info] = f08ae(bbal);

% Perform C = Q^T*A
side = 'Left';
trans = 'Transpose';
[c, info] = f08ag( ...
		   side, trans, QR, tau, abal);

% Generalized Hessenberg form (C,R) -> (H,T)
compq = 'No Q';
compz = 'No Z';
z = eye(4);
q = eye(4);
jlo = int64(1);
jhi = int64(ihi-ilo+1);
[H, T, ~, ~, info] = ...
  f08we( ...
	 compq, compz, jlo, jhi, c, QR, q, z);

% Find eigenvalues of generalized Hessenberg form
%    = eigenvalues of (A,B).
job = 'Eigenvalues';
[~, ~, alphar, alphai, beta, ~, ~, info] = ...
  f08xe( ...
	 job, compq, compz, jlo, jhi, H, T, q, z);

disp('Generalized eigenvalues of (A,B):');
w = complex(alphar+i*alphai);
disp(w./beta);

f08we example results

Generalized eigenvalues of (A,B):
  -2.4367 + 0.0000i
   0.6069 + 0.7948i
   0.6069 - 0.7948i
   1.0000 + 0.0000i
  -0.4104 + 0.0000i