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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dstevr (f08jd)

## Purpose

nag_lapack_dstevr (f08jd) computes selected eigenvalues and, optionally, eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix $T$. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

## Syntax

[d, e, m, w, z, isuppz, info] = f08jd(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)
[d, e, m, w, z, isuppz, info] = nag_lapack_dstevr(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)

## Description

Whenever possible nag_lapack_dstevr (f08jd) computes the eigenspectrum using Relatively Robust Representations. nag_lapack_dstevr (f08jd) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ $LD{L}^{\mathrm{T}}$ representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the $i$th unreduced block of $T$:
 (a) compute $T-{\sigma }_{i}I={L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, such that ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ is a relatively robust representation, (b) compute the eigenvalues, ${\lambda }_{j}$, of ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$ to high relative accuracy by the dqds algorithm, (c) if there is a cluster of close eigenvalues, ‘choose’ ${\sigma }_{i}$ close to the cluster, and go to (a), (d) given the approximate eigenvalue ${\lambda }_{j}$ of ${L}_{i}{D}_{i}{L}_{i}^{\mathrm{T}}$, compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

## 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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new $\mathit{O}\left({n}^{2}\right)$ algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

## 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:     $\mathrm{range}$ – string (length ≥ 1)
If ${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If ${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If ${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
Constraint: ${\mathbf{range}}=\text{'A'}$, $\text{'V'}$ or $\text{'I'}$.
3:     $\mathrm{d}\left(:\right)$ – 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 $T$.
4:     $\mathrm{e}\left(:\right)$ – 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 $T$.
5:     $\mathrm{vl}$ – double scalar
6:     $\mathrm{vu}$ – double scalar
If ${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'I'}$, vl and vu are not referenced.
Constraint: if ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
7:     $\mathrm{il}$int64int32nag_int scalar
8:     $\mathrm{iu}$int64int32nag_int scalar
If ${\mathbf{range}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$;
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
9:     $\mathrm{abstol}$ – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $\left[a,b\right]$ of width less than or equal to
 $abstol+ε maxa,b ,$
where $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then $\epsilon {‖T‖}_{1}$ will be used in its place. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

### 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
3:     $\mathrm{m}$int64int32nag_int scalar
The total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\text{'A'}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\text{'I'}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
4:     $\mathrm{w}\left(:\right)$ – double array
The dimension of the array w will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The first m elements contain the selected eigenvalues in ascending order.
5:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobz}}=\text{'V'}$, the first m columns of $Z$ contain the orthonormal eigenvectors of the matrix $A$ corresponding to the selected eigenvalues, 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.
6:     $\mathrm{isuppz}\left(:\right)$int64int32nag_int array
The dimension of the array isuppz will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{m}}\right)$
The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The $i$th eigenvector is nonzero only in elements ${\mathbf{isuppz}}\left(2×i-1\right)$ through ${\mathbf{isuppz}}\left(2×i\right)$. Implemented only for ${\mathbf{range}}=\text{'A'}$ or $\text{'I'}$ and ${\mathbf{iu}}-{\mathbf{il}}={\mathbf{n}}-1$.
7:     $\mathrm{info}$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: range, 3: n, 4: d, 5: e, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: isuppz, 16: work, 17: lwork, 18: iwork, 19: liwork, 20: 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.
${\mathbf{info}}>0$
An internal error has occurred in this function. Please refer to info in nag_lapack_dstebz (f08jj).

## 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 ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{range}}=\text{'A'}$, otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.

## Example

This example finds the eigenvalues with indices in the range $\left[2,3\right]$, and the corresponding eigenvectors, of the symmetric tridiagonal matrix
 $T = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 .$
```function f08jd_example

fprintf('f08jd example results\n\n');

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

% Eigenvalues 2 and 3 and corresponding eigenvectors
jobz  = 'Vectors';
range = 'Indices';
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[~, ~, m, w, z, ~, info] = ...
f08jd(...
jobz, range, d, e, vl, vu, il, iu, abstol);

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

disp(' Eigenvalues of A in range:');
disp(w(1:m)');
disp(' Corresponding eigenvectors:');
disp(z);

```
```f08jd example results

Eigenvalues of A in range:
3.5470    8.6578

Corresponding eigenvectors:
0.3388    0.0494
0.8628    0.3781
-0.3648    0.8558
0.0879   -0.3497

```