NAG Toolbox: nag_lapack_zstegr (f08jy)

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)

Indicates which eigenvalues should be returned.

$\mathbf{range}=\text{'A'}$

All eigenvalues will be found.

$\mathbf{range}=\text{'V'}$

All eigenvalues in the half-open interval $\left(\mathbf{vl},\mathbf{vu}\right]$ will be found.

$\mathbf{range}=\text{'I'}$

The ilth through iuth eigenvectors 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 $T$.

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

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

$\mathbf{e}\left(1:\mathbf{n}-1\right)$ contains the subdiagonal elements of the tridiagonal matrix $T$. $\mathbf{e}\left(\mathbf{n}\right)$ need not be set.

5:     $\mathrm{vl}$ – double scalar

6:     $\mathrm{vu}$ – double scalar

If $\mathbf{range}=\text{'V'}$, vl and vu contain the lower and upper bounds respectively 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 $\mathbf{range}=\text{'I'}$, il and iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.

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

Constraints:

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

Optional Input Parameters

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

Default: the dimension of the array d.

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

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

Output Parameters

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

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

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

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

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

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

The eigenvalues in ascending order.

5:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – 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 $\mathbf{jobz}=\text{'V'}$, then if $\mathbf{info}=\mathbf{0}$, the columns of z contain the orthonormal eigenvectors of the matrix $T$, 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.

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

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

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 $\mathbf{isuppz}\left(2\times i\right)$.

7:     $\mathrm{info}$ – 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: n, 4: d, 5: e, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: isuppz, 16: work, 17: lwork, 18: iwork, 19: liwork, 20: 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 $\mathbf{info}=\mathbf{1}$, the $\mathrm{dqds}$ algorithm failed to converge, if $\mathbf{info}=\mathbf{2}$, inverse iteration failed to converge.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08jy_example

function f08jy_example


fprintf('f08jy example results\n\n');

% Hermitian tridiagonal matrix A, stored as diagonal and of-diagonal
d = [1; 4; 9; 16];
e = [1; 2; 3; 0];

% Calculate all eigenvalues and eigenvectors
jobz = 'V';
range = 'A';
vl = 0;
vu = 0;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, ~, m, w, z, ~, info] = f08jy( ...
                                  jobz, range, d, e, vl, vu, il, iu);

% Normalize: largest elements are real
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

disp(' Eigenvalues of A:');
disp(w');
disp(' Corresponding eigenvectors:');
disp(z);

f08jy example results

 Eigenvalues of A:
    0.6476    3.5470    8.6578   17.1477

 Corresponding 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