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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$n$ by n$n$ symmetric tridiagonal matrix A$A$.

## 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$A$ using a combination of the QR$QR$ and QL$QL$ 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:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz = 'N'${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
jobz = 'V'${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: jobz = 'N'${\mathbf{jobz}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ diagonal elements of the tridiagonal matrix A$A$.
3:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix A$A$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array d.
n$n$, the order of the matrix.
Constraint: n0${\mathbf{n}}\ge 0$.

ldz work

### Output Parameters

1:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the eigenvalues in ascending order.
2:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The contents of e are destroyed.
3:     z(ldz, : $:$) – double array
The first dimension, ldz, of the array z will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldz1$\mathit{ldz}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then if ${\mathbf{INFO}}={\mathbf{0}}$, z contains the orthonormal eigenvectors of the matrix A$A$, with the i$i$th column of Z$Z$ holding the eigenvector associated with d(i)${\mathbf{d}}\left(i\right)$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the algorithm failed to converge; i$i$ off-diagonal elements of e did not converge to zero.

## Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A + E)$\left(A+E\right)$, where
 ‖E‖2 = O(ε) ‖A‖2 , $‖E‖2 = O(ε) ‖A‖2 ,$
and ε$\epsilon$ 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 n2${n}^{2}$ if jobz = 'N'${\mathbf{jobz}}=\text{'N'}$ and is proportional to n3${n}^{3}$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$.

## Example

```function nag_lapack_dstev_example
jobz = 'Vectors';
d = [1;
4;
9;
16];
e = [1;
2;
3];
[dOut, eOut, z, info] = nag_lapack_dstev(jobz, d, e)
```
```

dOut =

0.6476
3.5470
8.6578
17.1477

eOut =

0
0
0

z =

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

info =

0

```
```function f08ja_example
jobz = 'Vectors';
d = [1;
4;
9;
16];
e = [1;
2;
3];
[dOut, eOut, z, info] = f08ja(jobz, d, e)
```
```

dOut =

0.6476
3.5470
8.6578
17.1477

eOut =

0
0
0

z =

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

info =

0

```