NAG Toolbox: nag_lapack_dstevx (f08jb)

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

If $\mathbf{range}=\text{'A'}$, all eigenvalues will be found.

If $\mathbf{range}=\text{'V'}$, all eigenvalues in the half-open interval $\left(\mathbf{vl},\mathbf{vu}\right]$ will be found.

If $\mathbf{range}=\text{'I'}$, the ilth to iuth eigenvalues will be found.

Constraint: $\mathbf{range}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.

3:     $\mathrm{d}\left(:\right)$ – 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 $A$.

4:     $\mathrm{e}\left(:\right)$ – 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 $A$.

5:     $\mathrm{vl}$ – double scalar

6:     $\mathrm{vu}$ – double scalar

If $\mathbf{range}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.

If $\mathbf{range}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.

Constraint: if $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

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

8:     $\mathrm{iu}$ – int64int32nag_int scalar

If $\mathbf{range}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

If $\mathbf{range}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.

Constraints:

• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}=0$, $\mathbf{il}=1$ and $\mathbf{iu}=0$;
• if $\mathbf{range}=\text{'I'}$ and $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

9:     $\mathrm{abstol}$ – double scalar

The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to

$$\mathbf{abstol}+\epsilon \max \left(\left|a\right|,\left|b\right|\right)\text{,}$$

where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖A‖}_1$ will be used in its place. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold $2\times \mathbf{x02am}\left( \right)$, not zero. If this function returns with $\mathbf{info}>\mathbf{0}$, indicating that some eigenvectors did not converge, try setting abstol to $2\times \mathbf{x02am}\left( \right)$. See Demmel and Kahan (1990).

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

May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

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

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

May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

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

The total number of eigenvalues found. $0\le \mathbf{m}\le \mathbf{n}$.

If $\mathbf{range}=\text{'A'}$, $\mathbf{m}=\mathbf{n}$.

If $\mathbf{range}=\text{'I'}$, $\mathbf{m}=\mathbf{iu}-\mathbf{il}+1$.

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

The first m elements contain the selected eigenvalues in ascending order.

5:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – 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 $\max \left(1,\mathbf{m}\right)$ if $\mathbf{jobz}=\text{'V'}$ and $1$ otherwise.

If $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, with the $i$th column of $Z$ holding the eigenvector associated with $\mathbf{w}\left(i\right)$;
• if an eigenvector fails to converge ($\mathbf{info}>\mathbf{0}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.

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

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

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

If $\mathbf{jobz}=\text{'V'}$, then

• if $\mathbf{info}=\mathbf{0}$, the first m elements of jfail are zero;
• if $\mathbf{info}>\mathbf{0}$, jfail contains the indices of the eigenvectors that failed to converge.

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

7:     $\mathrm{info}$ – 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.

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W  $\mathbf{info}>0$

The algorithm failed to converge; $\_$ eigenvectors did not converge. Their indices are stored in array jfail.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08jb_example

function f08jb_example


fprintf('f08jb example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
d = [1;     4;     9;     16];
e = [1;     2;     3];

% Eigenvalues in range [0,5] and corresponding eigenvectors
jobz  = 'Vectors';
range = 'Values in range';
vl = 0;
vu = 5;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, ~, m, w, z, jfail, info] = ...
f08jb( ...
       jobz, range, d, e, vl, vu, il, iu, abstol);

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

disp(' Eigenvalues of A in range:');
disp(w(1:m));
disp(' Corresponding eigenvectors:');
disp(z);

f08jb example results

 Eigenvalues of A in range:
    0.6476
    3.5470

 Corresponding eigenvectors:
    0.9396    0.3388
   -0.3311    0.8628
    0.0853   -0.3648
   -0.0167    0.0879