NAG Toolbox: nag_lapack_zheev (f08fn)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates whether eigenvectors are computed.

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

Only eigenvalues are computed.

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

Eigenvalues and eigenvectors are computed.

Constraint: $\mathbf{jobz}=\text{'N'}$ or $\text{'V'}$.

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

If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $A$ is stored.

If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $A$ is stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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 $n$ by $n$ Hermitian matrix $A$.

• If $\mathbf{uplo}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If $\mathbf{uplo}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

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

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

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

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

If $\mathbf{jobz}=\text{'V'}$, then a contains the orthonormal eigenvectors of the matrix $A$.

If $\mathbf{jobz}=\text{'N'}$, then on exit the lower triangle (if $\mathbf{uplo}=\text{'L'}$) or the upper triangle (if $\mathbf{uplo}=\text{'U'}$) of a, including the diagonal, is overwritten.

2:     $\mathrm{w}\left(\mathbf{n}\right)$ – double array

The eigenvalues in ascending order.

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: jobz, 2: uplo, 3: n, 4: a, 5: lda, 6: w, 7: work, 8: lwork, 9: rwork, 10: 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 $\mathbf{info}=i$, the algorithm failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08fn_example

function f08fn_example


fprintf('f08fn example results\n\n');

% upper triangular part of Hernitian matrix A
n = int64(4);
uplo = 'Upper';
a = [ 1,       2 - 1i,  3 - 1i,  4 - 1i;
      0 + 0i,  2 + 0i,  3 - 2i,  4 - 2i;
      0 + 0i,  0 + 0i,  3 + 0i,  4 - 3i;
      0 + 0i,  0 + 0i,  0 + 0i,  4 + 0i];

jobz = 'Vectors';
[z, w, info] = f08fn( ...
		      jobz, uplo, a);

disp('Eigenvalues');
disp(w);

% Normalize
for i = 1:n
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(1:n,i) = z(1:n,i)*conj(z(k,i))/abs(z(k,i));
end
[ifail] = x04da( ...
                 'General', ' ', z, 'Eigenvectors');

% Eigenvalue error bound
errbnd = x02aj*max(abs(w(1)),abs(w(end)));
% Eigenvector condition numbers
[rcondz, info] = f08fl( ...
		        'Eigenvectors', n, n, w);

% Eigenvector error bounds
zerrbd = errbnd./rcondz;

fprintf('\n');
disp('Error estimate for the eigenvalues');
fprintf('%12.1e\n',errbnd);
disp('Error estimates for the eigenvectors');
fprintf('%12.1e',zerrbd);
fprintf('\n');

f08fn example results

Eigenvalues
   -4.2443
   -0.6886
    1.1412
   13.7916

 Eigenvectors
          1       2       3       4
 1   0.4836  0.6470  0.0179  0.3809
     0.0000  0.0000 -0.4453 -0.0622

 2   0.2912 -0.4984  0.5706  0.4358
    -0.3618 -0.1130  0.0000 -0.0869

 3  -0.3163  0.2949 -0.1530  0.5241
    -0.3696  0.3165  0.5273  0.0000

 4  -0.4447 -0.2241 -0.2118  0.5719
     0.3406 -0.2878 -0.3598  0.2276

Error estimate for the eigenvalues
     1.5e-15
Error estimates for the eigenvectors
     4.3e-16     8.4e-16     8.4e-16     1.2e-16