NAG Toolbox: nag_lapack_dstev (f08ja)

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

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

The $n$ diagonal elements of the tridiagonal matrix $A$.

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

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

The $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.

Optional Input Parameters

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

Default: the dimension of the array d.

$n$, the order of the matrix.

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

Output Parameters

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

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

If $\mathbf{info}=\mathbf{0}$, the eigenvalues in ascending order.

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

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

The contents of e are destroyed.

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

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

• if $\mathbf{jobz}=\text{'V'}$, $\mathit{ldz}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldz}=1$.

The second dimension of the array z will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'V'}$, then if $\mathbf{info}=\mathbf{0}$, z contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{d}\left(i\right)$.

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

4:     $\mathrm{info}$ – 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: jobz, 2: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: 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.

   $\mathbf{info}>0$

If $\mathbf{info}=i$, the algorithm failed to converge; $i$ off-diagonal elements of e did not converge to zero.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ja_example

function f08ja_example


fprintf('f08ja example results\n\n');

% Symmetric tridiagonal matrix A stored as main and sub-diagonals
n = int64(4);
d = [1;     4;     9;     16];
e = [1;     2;     3];

% All eigenvalues and eigenvectors of A
jobz = 'Vectors';
[w, ~, z, info] = f08ja( ...
                         jobz, d, e);

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            

disp('Eigenvalues');
disp(w);
disp('Eigenvectors');
disp(z);

% 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;

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

f08ja example results

Eigenvalues
    0.6476
    3.5470
    8.6578
   17.1477

Eigenvectors
    0.9396    0.3388    0.0494    0.0034
   -0.3311    0.8628    0.3781    0.0545
    0.0853   -0.3648    0.8558    0.3568
   -0.0167    0.0879   -0.3497    0.9326

Error estimate for the eigenvalues
     1.9e-15
Error estimates for the eigenvectors
     6.6e-16     6.6e-16     3.7e-16     2.2e-16