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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zheev (f08fn)

## Purpose

nag_lapack_zheev (f08fn) computes all the eigenvalues and, optionally, all the eigenvectors of a complex $n$ by $n$ Hermitian matrix $A$.

## Syntax

[a, w, info] = f08fn(jobz, uplo, a, 'n', n)
[a, w, info] = nag_lapack_zheev(jobz, uplo, a, 'n', n)

## Description

The Hermitian matrix $A$ is first reduced to real tridiagonal form, using unitary similarity transformations, and then the $QR$ algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.

## 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
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:     $\mathrm{uplo}$ – 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 $\text{'L'}$.
3:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{jobz}}=\text{'V'}$, then a contains the orthonormal eigenvectors of the matrix $A$.
If ${\mathbf{jobz}}=\text{'N'}$, then on exit 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:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
The eigenvalues in ascending order.
3:     $\mathrm{info}$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: uplo, 3: n, 4: a, 5: lda, 6: w, 7: work, 8: lwork, 9: rwork, 10: 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 ${\mathbf{info}}=i$, the algorithm failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

## 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.

Each eigenvector is normalized so that the element of largest absolute value is real.
The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_lapack_dsyev (f08fa).

## Example

This example finds all the eigenvalues and eigenvectors of the Hermitian matrix
 $A = 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.
```function f08fn_example

fprintf('f08fn example results\n\n');

% upper triangular part of Hernitian matrix A
n = int64(4);
uplo = 'Upper';
a = [ 1,       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];

jobz = 'Vectors';
[z, w, info] = f08fn( ...
jobz, uplo, a);

disp('Eigenvalues');
disp(w);

% Normalize
for i = 1:n
[~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
z(1:n,i) = z(1:n,i)*conj(z(k,i))/abs(z(k,i));
end
[ifail] = x04da( ...
'General', ' ', z, 'Eigenvectors');

% Eigenvalue error bound
errbnd = x02aj*max(abs(w(1)),abs(w(end)));
% Eigenvector condition numbers
[rcondz, info] = f08fl( ...
'Eigenvectors', n, n, w);

% Eigenvector error bounds
zerrbd = errbnd./rcondz;

fprintf('\n');
disp('Error estimate for the eigenvalues');
fprintf('%12.1e\n',errbnd);
disp('Error estimates for the eigenvectors');
fprintf('%12.1e',zerrbd);
fprintf('\n');

```
```f08fn example results

Eigenvalues
-4.2443
-0.6886
1.1412
13.7916

Eigenvectors
1       2       3       4
1   0.4836  0.6470  0.0179  0.3809
0.0000  0.0000 -0.4453 -0.0622

2   0.2912 -0.4984  0.5706  0.4358
-0.3618 -0.1130  0.0000 -0.0869

3  -0.3163  0.2949 -0.1530  0.5241
-0.3696  0.3165  0.5273  0.0000

4  -0.4447 -0.2241 -0.2118  0.5719
0.3406 -0.2878 -0.3598  0.2276

Error estimate for the eigenvalues
1.5e-15
Error estimates for the eigenvectors
4.3e-16     8.4e-16     8.4e-16     1.2e-16
```