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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsbev (f08ha)

## Purpose

nag_lapack_dsbev (f08ha) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric band matrix $A$ of bandwidth $\left(2{k}_{d}+1\right)$.

## Syntax

[ab, w, z, info] = f08ha(jobz, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab, 'n', n)

## Description

The symmetric band matrix $A$ is first reduced to tridiagonal form, using orthogonal 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:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double 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 $n$ symmetric 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:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double 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:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double 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: 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 complex analogue of this function is nag_lapack_zhbev (f08hn).

## Example

This example finds all the eigenvalues and eigenvectors of the symmetric band matrix
 $A = 1 2 3 0 0 2 2 3 4 0 3 3 3 4 5 0 4 4 4 5 0 0 5 5 5 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.
```function f08ha_example

fprintf('f08ha example results\n\n');

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

% Calculate all eigenvalues and eigenvectors
jobz = 'Vectors';
[abf, w, z, info] = f08ha( ...
jobz, uplo, kd, ab);

% Normalize eigenvectors: largest element positive
for j = 1:n
[~,k] = max(abs(z(:,j)));
if z(k,j) < 0;
z(:,j) = -z(:,j);
end
end

disp('Eigenvalues');
disp(w);
disp('Eigenvectors');
disp(z);

% 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;

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

```
```f08ha example results

Eigenvalues
-3.2474
-2.6633
1.7511
4.1599
14.9997

Eigenvectors
0.0394    0.6238    0.5635   -0.5165    0.1582
0.5721   -0.2575   -0.3896   -0.5955    0.3161
-0.4372   -0.5900    0.4008   -0.1470    0.5277
-0.4424    0.4308   -0.5581    0.0470    0.5523
0.5332    0.1039    0.2421    0.5956    0.5400

Error estimate for the eigenvalues
1.7e-15
Error estimates for the eigenvectors
2.9e-15     2.9e-15     6.9e-16     6.9e-16     1.5e-16
```