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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dstev (f08ja)

## Purpose

nag_lapack_dstev (f08ja) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix $A$.

## Syntax

[d, e, z, info] = f08ja(jobz, d, e, 'n', n)
[d, e, z, info] = nag_lapack_dstev(jobz, d, e, 'n', n)

## Description

nag_lapack_dstev (f08ja) computes all the eigenvalues and, optionally, all the eigenvectors of $A$ using a combination of the $QR$ and $QL$ algorithms, with an implicit shift.

## 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:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The $n$ diagonal elements of the tridiagonal matrix $A$.
3:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
The $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array d.
$n$, the order of the matrix.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{info}}={\mathbf{0}}$, the eigenvalues in ascending order.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
The contents of e are destroyed.
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 ${\mathbf{jobz}}=\text{'V'}$, then if ${\mathbf{info}}={\mathbf{0}}$, z contains the orthonormal eigenvectors of the matrix $A$, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{d}}\left(i\right)$.
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: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: 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 e 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 ${n}^{2}$ if ${\mathbf{jobz}}=\text{'N'}$ and is proportional to ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$.

## Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.
```function f08ja_example

fprintf('f08ja example results\n\n');

% Symmetric tridiagonal matrix A stored as main and sub-diagonals
n = int64(4);
d = [1;     4;     9;     16];
e = [1;     2;     3];

% All eigenvalues and eigenvectors of A
jobz = 'Vectors';
[w, ~, z, info] = f08ja( ...
jobz, d, e);

% 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');

```
```f08ja example results

Eigenvalues
0.6476
3.5470
8.6578
17.1477

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

Error estimate for the eigenvalues
1.9e-15
Error estimates for the eigenvectors
6.6e-16     6.6e-16     3.7e-16     2.2e-16
```