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# NAG Toolbox: nag_lapack_dstevd (f08jc)

## Purpose

nag_lapack_dstevd (f08jc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the $QL$ or $QR$ algorithm.

## Syntax

[d, e, z, info] = f08jc(job, d, e, 'n', n)
[d, e, z, info] = nag_lapack_dstevd(job, d, e, 'n', n)

## Description

nag_lapack_dstevd (f08jc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix $T$. In other words, it can compute the spectral factorization of $T$ as
 $T=ZΛZT,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues ${\lambda }_{i}$, and $Z$ is the orthogonal matrix whose columns are the eigenvectors ${z}_{i}$. Thus
 $Tzi=λizi, i=1,2,…,n.$

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{job}$ – string (length ≥ 1)
Indicates whether eigenvectors are computed.
${\mathbf{job}}=\text{'N'}$
Only eigenvalues are computed.
${\mathbf{job}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\text{'N'}$ 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 $T$.
3:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The $n-1$ off-diagonal elements of the tridiagonal matrix $T$. The $n$th element of this array is used as workspace.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array d and the second dimension of the array d. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The eigenvalues of the matrix $T$ in ascending order.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
e is overwritten with intermediate results.
3:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double array
The first dimension, $\mathit{ldz}$, of the array z will be
• if ${\mathbf{job}}=\text{'V'}$, $\mathit{ldz}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\text{'N'}$, $\mathit{ldz}=1$.
The second dimension of the array z will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{job}}=\text{'V'}$ and at least $1$ if ${\mathbf{job}}=\text{'N'}$.
If ${\mathbf{job}}=\text{'V'}$, z stores the orthogonal matrix $Z$ which contains the eigenvectors of $T$.
If ${\mathbf{job}}=\text{'N'}$, 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: job, 2: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: lwork, 9: iwork, 10: liwork, 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 ${\mathbf{info}}=i$ and ${\mathbf{job}}=\text{'N'}$, the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if ${\mathbf{info}}=i$ and ${\mathbf{job}}=\text{'V'}$, then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i/\left({\mathbf{n}}+1\right)$ through .

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix $\left(T+E\right)$, where
 $E2 = Oε T2 ,$
and $\epsilon$ is the machine precision.
If ${\lambda }_{i}$ is an exact eigenvalue and ${\stackrel{~}{\lambda }}_{i}$ is the corresponding computed value, then
 $λ~i - λi ≤ c n ε T2 ,$
where $c\left(n\right)$ is a modestly increasing function of $n$.
If ${z}_{i}$ is the corresponding exact eigenvector, and ${\stackrel{~}{z}}_{i}$ is the corresponding computed eigenvector, then the angle $\theta \left({\stackrel{~}{z}}_{i},{z}_{i}\right)$ between them is bounded as follows:
 $θ z~i,zi ≤ c n ε T2 min i≠j λi - λj .$
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

There is no complex analogue of this function.

## Example

This example computes all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix $T$, where
 $T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .$
```function f08jc_example

fprintf('f08jc example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
n = 4;
d = [1;     4;     9;     16];
e = [1;     2;     3;     0];

% All eigenvalues and eigenvectors of A
job = 'Vectors';
[w, ~, z, info] = f08jc( ...
job, 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);

```
```f08jc 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

```

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