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NAG Toolbox: nag_lapack_zhpevx (f08gp)

Purpose

nag_lapack_zhpevx (f08gp) computes selected eigenvalues and, optionally, eigenvectors of a complex $n$ by $n$ Hermitian matrix $A$ in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[ap, m, w, z, jfail, info] = f08gp(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
[ap, m, w, z, jfail, info] = nag_lapack_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)

Description

The Hermitian matrix $A$ is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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:     $\mathrm{n}$int64int32nag_int scalar
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The upper or lower triangle of the $n$ by $n$ Hermitian matrix $A$, packed by columns.
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
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
 $abstol+ε maxa,b ,$
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 , not zero. If this function returns with ${\mathbf{info}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).

None.

Output Parameters

1:     $\mathrm{ap}\left(:\right)$ – complex array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of $A$.
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:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
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 ${\mathbf{jobz}}=\text{'V'}$, $\mathit{ldz}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldz}=1$.
The second dimension of the array z will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ 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.
5:     $\mathrm{jfail}\left(:\right)$int64int32nag_int array
The dimension of the array jfail will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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.
6:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The algorithm failed to converge; $_$ eigenvectors did not converge. Their indices are stored in array jfail.

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_lapack_dspevx (f08gb).

Example

This example finds the eigenvalues in the half-open interval $\left(-2,2\right]$, and the corresponding eigenvectors, of the Hermitian matrix
 $A = 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 .$
```function f08gp_example

fprintf('f08gp example results\n\n');

% Hermitian A stored in packed format.
uplo = 'U';
n = int64(4);
ap = [1;
2 - 1i;      2 + 0i;
3 - 1i;      3 - 2i;      3 + 0i;
4 - 1i;      4 - 2i;      4 - 3i;      4 + 0i];

% Eigenvalues in range [-2,2] of Hermitian A stored in packed format.
jobz  = 'Vectors';
range = 'Values in range';
vl = -2;
vu =  2;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, m, w, z, jfail, info] = ...
f08gp( ...
jobz, range, uplo, n, ap, vl, vu, il, iu, abstol);

% 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

fprintf('Number of eigenvalues in [-2,2] is %2d\n',m);
fprintf('\n Eigenvalues 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);

```
```f08gp example results

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

Eigenvalues 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)
```

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