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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhbev (f08hn)

## Purpose

nag_lapack_zhbev (f08hn) computes all the eigenvalues and, optionally, all the eigenvectors of a complex $n$ by $n$ Hermitian band matrix $A$ of bandwidth $\left(2{k}_{d}+1\right)$.

## Syntax

[ab, w, z, info] = f08hn(jobz, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_zhbev(jobz, uplo, kd, ab, 'n', n)

## Description

The Hermitian band 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:     $\mathrm{kd}$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: ${\mathbf{kd}}\ge 0$.
4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of the $n$ by $n$ Hermitian 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 ${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$

### 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:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab will be ${\mathbf{kd}}+1$.
The second dimension of the array ab will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
The eigenvalues in ascending order.
3:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobz}}=\text{'V'}$, z contains the orthonormal eigenvectors of the matrix $A$, 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.
4:     $\mathrm{info}$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: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: rwork, 12: 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 ${\mathbf{info}}=i$, 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.

The total number of floating-point operations is proportional to ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$ and is proportional to ${k}_{d}{n}^{2}$ otherwise.
The real analogue of this function is nag_lapack_dsbev (f08ha).

## Example

This example finds all the eigenvalues and eigenvectors of the Hermitian band matrix
 $A = 1 2-i 3-i 0 0 2+i 2 3-2i 4-2i 0 3+i 3+2i 3 4-3i 5-3i 0 4+2i 4+3i 4 5-4i 0 0 5+3i 5+4i 5 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.
```function f08hn_example

fprintf('f08hn example results\n\n');

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

% All eigenvalues and eigenvectors of A
jobz = 'Vectors';
[~, w, z, info] = f08hn( ...
jobz, uplo, kd, ab);

% Normalize: largest elements are real
for i = 1:n
[~,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');
disp(w');
[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('\nError estimate for the eigenvalues\n');
fprintf('%12.1e\n',errbnd);
disp('Error estimates for the eigenvectors');
fprintf('%12.1e',zerrbd);
fprintf('\n');

```
```f08hn example results

Eigenvalues
-6.4185   -1.4094    1.4421    4.4856   16.9002

Eigenvectors
1       2       3       4       5
1  -0.2534 -0.4188 -0.2560  0.4767  0.1439
-0.0538  0.4797  0.3721 -0.2748  0.0000

2  -0.0662 -0.0122  0.5344  0.5524  0.3060
0.4301 -0.3529  0.0000  0.0000  0.0411

3   0.5274  0.4621 -0.4245  0.2076  0.4681
0.0000  0.0000  0.0915 -0.0660  0.2306

4   0.1061 -0.1642  0.4964 -0.1379  0.4098
-0.4981  0.3146 -0.1546  0.1026  0.3832

5  -0.4519 -0.0360 -0.1979 -0.2651  0.1819
0.0424 -0.3593 -0.1114 -0.4948  0.5136

Error estimate for the eigenvalues
1.9e-15
Error estimates for the eigenvectors
3.7e-16     6.6e-16     6.6e-16     6.2e-16
```