NAG Toolbox: nag_lapack_zstein (f08jx)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{d}\left(:\right)$ – double array

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

The diagonal elements of the tridiagonal matrix $T$.

2:     $\mathrm{e}\left(:\right)$ – double array

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

The off-diagonal elements of the tridiagonal matrix $T$.

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

$m$, the number of eigenvectors to be returned.

Constraint: $0\le \mathbf{m}\le \mathbf{n}$.

4:     $\mathrm{w}\left(:\right)$ – double array

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

The eigenvalues of the tridiagonal matrix $T$ stored in $\mathbf{w}\left(1\right)$ to $\mathbf{w}\left(m\right)$, as returned by nag_lapack_dstebz (f08jj) with $\mathbf{order}=\text{'B'}$. Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.

Constraint: if $\mathbf{iblock}\left(\mathit{i}\right)=\mathbf{iblock}\left(\mathit{i}+1\right)$, $\mathbf{w}\left(\mathit{i}\right)\le \mathbf{w}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}-1$.

5:     $\mathrm{iblock}\left(:\right)$ – int64int32nag_int array

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

The first $m$ elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by nag_lapack_dstebz (f08jj) with $\mathbf{order}=\text{'B'}$. If the eigenvalues were not computed by nag_lapack_dstebz (f08jj) with $\mathbf{order}=\text{'B'}$, set $\mathbf{iblock}\left(\mathit{i}\right)$ to $1$, for $\mathit{i}=1,2,\dots ,m$.

Constraint: $\mathbf{iblock}\left(\mathit{i}\right)\le \mathbf{iblock}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{m}-1$.

6:     $\mathrm{isplit}\left(:\right)$ – int64int32nag_int array

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

The points at which $T$ breaks up into sub-matrices, as returned by nag_lapack_dstebz (f08jj) with $\mathbf{order}=\text{'B'}$. If the eigenvalues were not computed by nag_lapack_dstebz (f08jj) with $\mathbf{order}=\text{'B'}$, set $\mathbf{isplit}\left(1\right)$ to n.

Optional Input Parameters

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

Default: the first dimension of the array d and the second dimension of the array d. (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:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex array

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

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

The $m$ eigenvectors, stored as columns of $Z$; the $i$th column corresponds to the $i$th specified eigenvalue, unless $\mathbf{info}>\mathbf{0}$ (in which case see Error Indicators and Warnings).

2:     $\mathrm{ifailv}\left(\mathbf{m}\right)$ – int64int32nag_int array

If $\mathbf{info}=i>0$, the first $i$ elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to $0$.

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

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: n, 2: d, 3: e, 4: m, 5: w, 6: iblock, 7: isplit, 8: z, 9: ldz, 10: work, 11: iwork, 12: ifailv, 13: 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.

W  $\mathbf{info}>0$

If $\mathbf{info}=i$, then $i$ eigenvectors (as indicated by the argument ifailv above) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08jx_example

function f08jx_example


fprintf('f08jx example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'L';
a = [-2.28 + 0.00i,  0.00 + 0i,     0    + 0i,     0    + 0i;
      1.78 + 2.03i, -1.12 + 0i,     0    + 0i,     0    + 0i;
      2.26 - 0.10i,  0.01 - 0.43i, -0.37 + 0i,     0    + 0i;
     -0.12 - 2.53i, -1.07 - 0.86i,  2.31 + 0.92i, -0.73 + 0i];

% Reduce A to tridiagonal form
[aq, d, e, tau, info] = f08fs( ...
                              uplo, a);

% Calculate two smallest eigenvalues
range = 'Indices';
order = 'Block';
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, nsplit, w, iblock, isplit, info] = ...
  f08jj( ...
         range, order, vl, vu, il, iu, abstol, d, e);

% Corresponding eigenvectors of T
[tz, ifailv, info] = f08jx( ...
                            d, e, m, w, iblock, isplit);

% TRansform to eigenvalues of A (by premultiplying by Q)
trans = 'No transpose';
side = 'Left';
[z, info] = f08fu( ...
                   side, uplo, trans, aq, tau, tz);

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

disp(' Selected eigenvalues of A:');
disp(w(1:m));
disp(' Corresponding eigenvectors:');
disp(z);

f08jx example results

 Selected eigenvalues of A:
   -6.0002
   -3.0030

 Corresponding eigenvectors:
   0.7299 + 0.0000i  -0.2120 + 0.1497i
  -0.1663 - 0.2061i   0.7307 + 0.0000i
  -0.4165 - 0.1417i  -0.3291 + 0.0479i
   0.1743 + 0.4162i   0.5200 + 0.1329i