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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dstegr (f08jl)

## Purpose

nag_lapack_dstegr (f08jl) computes all the eigenvalues and, optionally, all the eigenvectors of a real $n$ by $n$ symmetric tridiagonal matrix.

## Syntax

[d, e, m, w, z, isuppz, info] = f08jl(jobz, range, d, e, vl, vu, il, iu, 'n', n)
[d, e, m, w, z, isuppz, info] = nag_lapack_dstegr(jobz, range, d, e, vl, vu, il, iu, 'n', n)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 23: abstol was made an optional input parameter

## Description

nag_lapack_dstegr (f08jl) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix $T$. That is, the function computes the spectral factorization of $T$ given by
 $T = ZΛZT ,$
where $\Lambda$ is a diagonal matrix whose diagonal elements are the eigenvalues, ${\lambda }_{i}$, of $T$ and $Z$ is an orthogonal matrix whose columns are the eigenvectors, ${z}_{i}$, of $T$. Thus
 $Tzi= λi zi , i = 1,2,…,n .$
The function may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix $A$ which has been reduced to tridiagonal form $T$:
 $A =QTQT, where ​Q​ is orthogonal =QZΛQZT.$
In this case, the matrix $Q$ must be explicitly applied to the output matrix $Z$. The functions which must be called to perform the reduction to tridiagonal form and apply $Q$ are:
 full matrix nag_lapack_dsytrd (f08fe) and nag_lapack_dormtr (f08fg) full matrix, packed storage nag_lapack_dsptrd (f08ge) and nag_lapack_dopmtr (f08gg) band matrix nag_lapack_dsbtrd (f08he) with ${\mathbf{vect}}=\text{'V'}$ and .
This function uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. nag_lapack_dstegr (f08jl) can usually compute all the eigenvalues and eigenvectors in $O\left({n}^{2}\right)$ floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal functions in this chapter when all the eigenvectors are required, particularly so compared to those functions that are based on the $QR$ algorithm.

## 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
Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps. SIAM J. Appl. Math. 25 858–899
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)
Indicates which eigenvalues should be returned.
${\mathbf{range}}=\text{'A'}$
All eigenvalues will be found.
${\mathbf{range}}=\text{'V'}$
All eigenvalues in the half-open interval $\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
${\mathbf{range}}=\text{'I'}$
The ilth through iuth eigenvectors 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
${\mathbf{e}}\left(1:{\mathbf{n}}-1\right)$ contains the subdiagonal elements of the tridiagonal matrix $T$. ${\mathbf{e}}\left({\mathbf{n}}\right)$ need not be set.
5:     $\mathrm{vl}$ – double scalar
6:     $\mathrm{vu}$ – double scalar
If ${\mathbf{range}}=\text{'V'}$, vl and vu contain the lower and upper bounds respectively 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 ${\mathbf{range}}=\text{'I'}$, il and iu contains the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If ${\mathbf{range}}=\text{'A'}$ or $\text{'V'}$, il and iu are not referenced.
Constraints:
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$;
• if ${\mathbf{range}}=\text{'I'}$ and ${\mathbf{n}}=0$, ${\mathbf{il}}=1$ and ${\mathbf{iu}}=0$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array d.
$n$, 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
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 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 ${\mathbf{jobz}}=\text{'V'}$, then if ${\mathbf{info}}={\mathbf{0}}$, the columns of z contain the orthonormal eigenvectors of the matrix $T$, 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$
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 ${\mathbf{isuppz}}\left(2×i\right)$.
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.
${\mathbf{info}}>0$
If ${\mathbf{info}}={\mathbf{1}}$, the $\mathrm{dqds}$ algorithm failed to converge, if ${\mathbf{info}}={\mathbf{2}}$, inverse iteration failed to converge.

## Accuracy

See the description for abstol. See also Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

The total number of floating-point operations required to compute all the eigenvalues and eigenvectors is approximately proportional to ${n}^{2}$.
The complex analogue of this function is nag_lapack_zstegr (f08jy).

## Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
 $T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .$
abstol is set to zero so that the default tolerance of $n\epsilon {‖T‖}_{1}$ is used.
```function f08jl_example

fprintf('f08jl example results\n\n');

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

% All eigenvalues and eigenvectors of A
jobz = 'V';
range = 'A';
vl = 0;
vu = 0;
il = int64(0);
iu = int64(0);
[w, ~, m, w, z, isuppz, info] = ...
f08jl( ...
jobz, range, d, e, vl, vu, il, iu);

% 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');
disp(w(1:m)');
disp('Eigenvectors');
disp(z);

```
```f08jl 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

```