NAG Toolbox: nag_lapack_ztrevc (f08qx)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates whether left and/or right eigenvectors are to be computed.

$\mathbf{job}=\text{'R'}$

Only right eigenvectors are computed.

$\mathbf{job}=\text{'L'}$

Only left eigenvectors are computed.

$\mathbf{job}=\text{'B'}$

Both left and right eigenvectors are computed.

Constraint: $\mathbf{job}=\text{'R'}$, $\text{'L'}$ or $\text{'B'}$.

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

Indicates how many eigenvectors are to be computed.

$\mathbf{howmny}=\text{'A'}$

All eigenvectors (as specified by job) are computed.

$\mathbf{howmny}=\text{'B'}$

All eigenvectors (as specified by job) are computed and then pre-multiplied by the matrix $Q$ (which is overwritten).

$\mathbf{howmny}=\text{'S'}$

Selected eigenvectors (as specified by job and select) are computed.

Constraint: $\mathbf{howmny}=\text{'A'}$, $\text{'B'}$ or $\text{'S'}$.

3:     $\mathrm{select}\left(:\right)$ – logical array

The dimension of the array select must be at least $\mathbf{n}$ if $\mathbf{howmny}=\text{'S'}$, and at least $1$ otherwise

Specifies which eigenvectors are to be computed if $\mathbf{howmny}=\text{'S'}$. To obtain the eigenvector corresponding to the eigenvalue ${\lambda }_j$, $\mathbf{select}\left(j\right)$ must be set true.

If $\mathbf{howmny}=\text{'A'}$ or $\text{'B'}$, select is not referenced.

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

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

The second dimension of the array t must be at least $\mathbf{n}$.

The $n$ by $n$ upper triangular matrix $T$, as returned by nag_lapack_zhseqr (f08ps).

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

The first dimension, $\mathit{ldvl}$, of the array vl must satisfy

• if $\mathbf{job}=\text{'L'}$ or $\text{'B'}$, $\mathit{ldvl}\ge \mathbf{n}$;
• if $\mathbf{job}=\text{'R'}$, $\mathit{ldvl}\ge 1$.

The second dimension of the array vl must be at least $\mathbf{mm}$ if $\mathbf{job}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{job}=\text{'R'}$.

If $\mathbf{howmny}=\text{'B'}$ and $\mathbf{job}=\text{'L'}$ or $\text{'B'}$, vl must contain an $n$ by $n$ matrix $Q$ (usually the matrix of Schur vectors returned by nag_lapack_zhseqr (f08ps)).

If $\mathbf{howmny}=\text{'A'}$ or $\text{'S'}$, vl need not be set.

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

The first dimension, $\mathit{ldvr}$, of the array vr must satisfy

• if $\mathbf{job}=\text{'R'}$ or $\text{'B'}$, $\mathit{ldvr}\ge \mathbf{n}$;
• if $\mathbf{job}=\text{'L'}$, $\mathit{ldvr}\ge 1$.

The second dimension of the array vr must be at least $\mathbf{mm}$ if $\mathbf{job}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{job}=\text{'L'}$.

If $\mathbf{howmny}=\text{'B'}$ and $\mathbf{job}=\text{'R'}$ or $\text{'B'}$, vr must contain an $n$ by $n$ matrix $Q$ (usually the matrix of Schur vectors returned by nag_lapack_zhseqr (f08ps)).

If $\mathbf{howmny}=\text{'A'}$ or $\text{'S'}$, vr need not be set.

7:     $\mathrm{mm}$ – int64int32nag_int scalar

The number of columns in the arrays vl and/or vr. The precise number of columns required, $\mathit{m}$, is $n$ if $\mathbf{howmny}=\text{'A'}$ or $\text{'B'}$; if $\mathbf{howmny}=\text{'S'}$, $\mathit{m}$ is the number of selected eigenvectors (see select), in which case $0\le \mathit{m}\le n$.

Constraints:

• if $\mathbf{howmny}=\text{'A'}$ or $\text{'B'}$, $\mathbf{mm}\ge \mathbf{n}$;
• otherwise $\mathbf{mm}\ge \mathit{m}$.

Optional Input Parameters

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

Default: the first dimension of the array t and the second dimension of the array t. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrix $T$.

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

Output Parameters

1:     $\mathrm{t}\left(\mathit{ldt},:\right)$ – complex array

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

The second dimension of the array t will be $\mathbf{n}$.

Is used as internal workspace prior to being restored and hence is unchanged.

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

The first dimension, $\mathit{ldvl}$, of the array vl will be

• if $\mathbf{job}=\text{'L'}$ or $\text{'B'}$, $\mathit{ldvl}=\mathbf{n}$;
• if $\mathbf{job}=\text{'R'}$, $\mathit{ldvl}=1$.

The second dimension of the array vl will be $\mathbf{mm}$ if $\mathbf{job}=\text{'L'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{job}=\text{'R'}$.

If $\mathbf{job}=\text{'L'}$ or $\text{'B'}$, vl contains the computed left eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.

If $\mathbf{job}=\text{'R'}$, vl is not referenced.

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

The first dimension, $\mathit{ldvr}$, of the array vr will be

• if $\mathbf{job}=\text{'R'}$ or $\text{'B'}$, $\mathit{ldvr}=\mathbf{n}$;
• if $\mathbf{job}=\text{'L'}$, $\mathit{ldvr}=1$.

The second dimension of the array vr will be $\mathbf{mm}$ if $\mathbf{job}=\text{'R'}$ or $\text{'B'}$ and at least $1$ if $\mathbf{job}=\text{'L'}$.

If $\mathbf{job}=\text{'R'}$ or $\text{'B'}$, vr contains the computed right eigenvectors (as specified by howmny and select). The eigenvectors are stored consecutively in the columns of the array, in the same order as their eigenvalues.

If $\mathbf{job}=\text{'L'}$, vr is not referenced.

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

$\mathit{m}$, the number of selected eigenvectors. If $\mathbf{howmny}=\text{'A'}$ or $\text{'B'}$, m is set to $n$.

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: job, 2: howmny, 3: select, 4: n, 5: t, 6: ldt, 7: vl, 8: ldvl, 9: vr, 10: ldvr, 11: mm, 12: m, 13: work, 14: rwork, 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.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08qx_example

function f08qx_example


fprintf('f08qx example results\n\n');

n = int64(4);
a = [ 1.50 - 2.75i,  0    + 0i,     0    + 0i,     0    + 0i;
     -8.06 - 1.24i, -2.50 - 0.50i,  0    + 0i,    -0.75 + 0.50i;
     -2.09 + 7.56i,  1.39 + 3.97i, -1.25 + 0.75i, -4.82 - 5.67i;
      6.18 + 9.79i, -0.92 - 0.62i,  0    + 0i,    -2.50 - 0.50i];

% Balance a
[a, ilo, ihi, scale, info] = f08nv( ...
				    'Both', a);

% Reduce a to upper Hessenberg form
[H, tau, info] = f08ns( ...
			ilo, ihi, a);

% Form Q explicitly, storing result in vr
[Q, info] = f08nt( ...
		   ilo, ihi, H, tau);

% Calculate the eigenvalues and Schur factorisation of A
[S, w, VR, info] = f08ps( ...
			  'Schur Form', 'Vectors', ilo, ihi, H, Q);

disp('Eigenvalues of A:');
disp(w);

% Calculate the eigenvectors of A
select = [false];
vl = [complex(0)];

[T, ~, Z, m, info] = ...
  f08qx( ...
	 'Right', 'Backtransform', select, H, vl, VR, n);

% Scale
[V, info] = f08nw( ...
		   'Both', 'Right', ilo, ihi, scale, Z);

% Normalize eigenvectors: largest element is real
for i = 1:m
  [~,k] = max(abs(real(V(:,i)))+abs(imag(V(:,i))));
  V(:,i) = V(:,i)*conj(V(k,i))/abs(V(k,i));
end

disp('Right eigenvectors of A:');
disp(V);

f08qx example results

Eigenvalues of A:
  -1.2500 + 0.7500i
  -1.5000 - 0.4975i
  -3.5000 - 0.5025i
   1.5000 - 2.7500i

Right eigenvectors of A:
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.1778 - 0.0372i
   0.0000 + 0.0000i   0.1902 - 0.0910i   0.1898 - 0.0908i   0.4382 - 0.1021i
   1.0000 + 0.0000i   0.7492 + 0.0000i   0.7479 + 0.0000i   0.8449 + 0.0000i
   0.0000 + 0.0000i  -0.2314 - 0.0337i  -0.2310 - 0.0336i   0.1950 - 0.3509i