NAG Toolbox: nag_lapack_zheevr (f08fr)

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.

For $\mathbf{range}=\text{'V'}$ or $\text{'I'}$ and $\mathbf{iu}-\mathbf{il}<\mathbf{n}-1$, nag_lapack_dstebz (f08jj) and nag_lapack_zstein (f08jx) are called.

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – 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.

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 $\epsilon {‖T‖}_1$ will be used in its place, where $T$ is the real tridiagonal matrix obtained by reducing $A$ to tridiagonal form. See Demmel and Kahan (1990).

If high relative accuracy is important, set abstol to $\mathbf{x02am}\left( \right)$, although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

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

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

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

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

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

4:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – complex 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 $\mathbf{jobz}=\text{'V'}$, 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 $\mathbf{jobz}=\text{'N'}$, z is not referenced.

5:     $\mathrm{isuppz}\left(:\right)$ – int64int32nag_int array

The dimension of the array isuppz will be $\max \left(1,2\times \mathbf{m}\right)$

The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The $i$th eigenvector is nonzero only in elements $\mathbf{isuppz}\left(2\times i-1\right)$ through $\mathbf{isuppz}\left(2\times i\right)$. Implemented only for $\mathbf{range}=\text{'A'}$ or $\text{'I'}$ and $\mathbf{iu}-\mathbf{il}=\mathbf{n}-1$.

6:     $\mathrm{info}$ – 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: range, 3: uplo, 4: n, 5: a, 6: lda, 7: vl, 8: vu, 9: il, 10: iu, 11: abstol, 12: m, 13: w, 14: z, 15: ldz, 16: isuppz, 17: work, 18: lwork, 19: rwork, 20: lrwork, 21: iwork, 22: liwork, 23: 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.

   $\mathbf{info}>0$

nag_lapack_zheevr (f08fr) failed to converge.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08fr_example

function f08fr_example


fprintf('f08fr example results\n\n');

% Upper triangular part of Hermitian matrix A
uplo  = 'Upper';
n = 4;
a = [ 1 + 0i,  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];

% Compute 2nd and 3rd eigenvalues and coresponding vectors
jobz  = 'Vectors';
range = 'I';
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[~, m, w, z, ~, info] = ...
  f08fr( ...
         jobz, range, uplo, a, vl, vu, il, iu, abstol);

% Normalize vectors, largest element is real and positive.
for i = 1:m
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

% Display
fprintf(' Eigenvalues numbered 2 to 3 are:\n');
disp(w(1:m));

ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
                 'General', ' ', z, 'Bracketed', 'F7.4', ...
                 'Corresponding eigenvectors', 'Integer', 'Integer', ...
                 ncols, indent);

f08fr example results

 Eigenvalues numbered 2 to 3 are:
   -0.6886
    1.1412

 Corresponding eigenvectors
                    1                 2
 1  ( 0.6470, 0.0000) ( 0.0179,-0.4453)
 2  (-0.4984,-0.1130) ( 0.5706, 0.0000)
 3  ( 0.2949, 0.3165) (-0.1530, 0.5273)
 4  (-0.2241,-0.2878) (-0.2118,-0.3598)