NAG Toolbox: nag_lapack_dsbev (f08ha)

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:     $\mathrm{kd}$ – int64int32nag_int scalar

If $\mathbf{uplo}=\text{'U'}$, the number of superdiagonals, $k_d$, of the matrix $A$.

If $\mathbf{uplo}=\text{'L'}$, the number of subdiagonals, $k_d$, of the matrix $A$.

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

4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array

The first dimension of the array ab must be at least $\mathbf{kd}+1$.

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

The upper or lower triangle of the $n$ by $n$ symmetric band matrix $A$.

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

Optional Input Parameters

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

Default: the second dimension of the array ab.

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

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

Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array

The first dimension of the array ab will be $\mathbf{kd}+1$.

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

ab stores values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix $T$ are returned in ab using the same storage format as described above.

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

The eigenvalues in ascending order.

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

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: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: 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 an intermediate tridiagonal form did not converge to zero.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08ha_example

function f08ha_example


fprintf('f08ha example results\n\n');

% Symmetric band matrix A, stored on symmetric banded format
uplo = 'U';
kd = int64(2);
n  = int64(5);
ab = [0, 0, 3, 4, 5;
      0, 2, 3, 4, 5;
      1, 2, 3, 4, 5];

% Calculate all eigenvalues and eigenvectors
jobz = 'Vectors';
[abf, w, z, info] = f08ha( ...
                           jobz, uplo, kd, ab);

% 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');

f08ha example results

Eigenvalues
   -3.2474
   -2.6633
    1.7511
    4.1599
   14.9997

Eigenvectors
    0.0394    0.6238    0.5635   -0.5165    0.1582
    0.5721   -0.2575   -0.3896   -0.5955    0.3161
   -0.4372   -0.5900    0.4008   -0.1470    0.5277
   -0.4424    0.4308   -0.5581    0.0470    0.5523
    0.5332    0.1039    0.2421    0.5956    0.5400

Error estimate for the eigenvalues
     1.7e-15
Error estimates for the eigenvectors
     2.9e-15     2.9e-15     6.9e-16     6.9e-16     1.5e-16