NAG Toolbox: nag_lapack_dsbevx (f08hb)

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:     $\mathrm{uplo}$ – 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 $\text{'L'}$.

4:     $\mathrm{kd}$ – 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: $\mathbf{kd}\ge 0$.

5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – 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 $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

6:     $\mathrm{vl}$ – double scalar

7:     $\mathrm{vu}$ – 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 $\mathbf{range}=\text{'V'}$, $\mathbf{vl}<\mathbf{vu}$.

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

9:     $\mathrm{iu}$ – 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 $\mathbf{n}>0$, $1\le \mathbf{il}\le \mathbf{iu}\le \mathbf{n}$.

10:   $\mathrm{abstol}$ – 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 $\epsilon {‖T‖}_1$ will be used in its place, where $T$ is the tridiagonal matrix obtained by reducing $A$ to tridiagonal form. 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 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:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – double array

The first dimension, $\mathit{ldq}$, of the array q will be

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

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

If $\mathbf{jobz}=\text{'V'}$, the $n$ by $n$ orthogonal matrix used in the reduction to tridiagonal form.

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

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.

   $\mathbf{info}=-i$

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

1: jobz, 2: range, 3: uplo, 4: n, 5: kd, 6: ab, 7: ldab, 8: q, 9: ldq, 10: vl, 11: vu, 12: il, 13: iu, 14: abstol, 15: m, 16: w, 17: z, 18: ldz, 19: work, 20: iwork, 21: jfail, 22: 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 failed to converge. Their indices are stored in array jfail. Please see abstol.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08hb_example

function f08hb_example


fprintf('f08hb example results\n\n');

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

% Eigenvalues in range [-3,3] and corresponding eigenvectors
jobz  = 'Vectors';
range = 'Values in range';
vl = -3;
vu = 3;
il = int64(0);
iu = int64(0);
abstol = 0;
[abf, q, m, w, z, jfail, info] = ...
f08hb( ...
       jobz, range, uplo, kd, ab, 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                            

fprintf('Number of eigenvalues in [-3,3] is %2d\n',m);
fprintf('\n Eigenvalues are:\n   ');
fprintf(' %7.4f',w(1:m));
fprintf('\n\n');

[ifail] = x04ca( ...
		 'General', ' ', z, 'Corresponding eigenvectors of A');

f08hb example results

Number of eigenvalues in [-3,3] is  2

 Eigenvalues are:
    -2.6633  1.7511

 Corresponding eigenvectors of A
          1       2
 1   0.6238  0.5635
 2  -0.2575 -0.3896
 3  -0.5900  0.4008
 4   0.4308 -0.5581
 5   0.1039  0.2421