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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsyev (f08fa)

## Purpose

nag_lapack_dsyev (f08fa) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric matrix $A$.

## Syntax

[a, w, info] = f08fa(jobz, uplo, a, 'n', n)
[a, w, info] = nag_lapack_dsyev(jobz, uplo, a, 'n', n)

## Description

The symmetric 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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 $n$ symmetric 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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double 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

${\mathbf{info}}=-i$
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: 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 ${n}^{3}$.
The complex analogue of this function is nag_lapack_zheev (f08fn).

## Example

This example finds all the eigenvalues and eigenvectors of the symmetric matrix
 $A = 1 2 3 4 2 2 3 4 3 3 3 4 4 4 4 4 ,$
together with approximate error bounds for the computed eigenvalues and eigenvectors.
```function f08fa_example

fprintf('f08fa example results\n\n');

a = [1, 2, 3, 4;
0, 2, 3, 4;
0, 0, 3, 4;
0, 0, 0, 4];
n = int64(size(a,1));

% Eigenvalues and eogenvectors of upper triangular A
jobz = 'Vectors';
uplo = 'Upper';
[v, w, info] = f08fa( ...
jobz, uplo, a);

disp('Eigenvectors');
disp(v);

% 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');

```
```f08fa example results

Eigenvectors
0.7003   -0.5144    0.2767    0.4103
0.3592    0.4851   -0.6634    0.4422
-0.1569    0.5420    0.6504    0.5085
-0.5965   -0.4543   -0.2457    0.6144

Error estimate for the eigenvalues
1.4e-15
Error estimates for the eigenvectors
9.3e-16     6.5e-15     6.5e-15     1.1e-16
```