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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dstein (f08jk)

## Purpose

nag_lapack_dstein (f08jk) computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration.

## Syntax

[z, ifailv, info] = f08jk(d, e, m, w, iblock, isplit, 'n', n)
[z, ifailv, info] = nag_lapack_dstein(d, e, m, w, iblock, isplit, 'n', n)

## Description

nag_lapack_dstein (f08jk) computes the eigenvectors of a real symmetric tridiagonal matrix $T$ corresponding to specified eigenvalues, by inverse iteration (see Jessup and Ipsen (1992)). It is designed to be used in particular after the specified eigenvalues have been computed by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$, but may also be used when the eigenvalues have been computed by other functions in Chapters F02 or F08.
If $T$ has been formed by reduction of a full real symmetric matrix $A$ to tridiagonal form, then eigenvectors of $T$ may be transformed to eigenvectors of $A$ by a call to nag_lapack_dormtr (f08fg) or nag_lapack_dopmtr (f08gg).
nag_lapack_dstebz (f08jj) determines whether the matrix $T$ splits into block diagonal form:
 $T = T1 T2 . . . Tp$
and passes details of the block structure to this function in the arrays iblock and isplit. This function can then take advantage of the block structure by performing inverse iteration on each block ${T}_{i}$ separately, which is more efficient than using the whole matrix.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Jessup E and Ipsen I C F (1992) Improving the accuracy of inverse iteration SIAM J. Sci. Statist. Comput. 13 550–572

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{d}\left(:\right)$ – 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 diagonal elements of the tridiagonal matrix $T$.
2:     $\mathrm{e}\left(:\right)$ – 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 off-diagonal elements of the tridiagonal matrix $T$.
3:     $\mathrm{m}$int64int32nag_int scalar
$m$, the number of eigenvectors to be returned.
Constraint: $0\le {\mathbf{m}}\le {\mathbf{n}}$.
4:     $\mathrm{w}\left(:\right)$ – double array
The dimension of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The eigenvalues of the tridiagonal matrix $T$ stored in ${\mathbf{w}}\left(1\right)$ to ${\mathbf{w}}\left(m\right)$, as returned by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$. Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.
Constraint: if ${\mathbf{iblock}}\left(\mathit{i}\right)={\mathbf{iblock}}\left(\mathit{i}+1\right)$, ${\mathbf{w}}\left(\mathit{i}\right)\le {\mathbf{w}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}-1$.
5:     $\mathrm{iblock}\left(:\right)$int64int32nag_int array
The dimension of the array iblock must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The first $m$ elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$. If the eigenvalues were not computed by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$, set ${\mathbf{iblock}}\left(\mathit{i}\right)$ to $1$, for $\mathit{i}=1,2,\dots ,m$.
Constraint: ${\mathbf{iblock}}\left(\mathit{i}\right)\le {\mathbf{iblock}}\left(\mathit{i}+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}-1$.
6:     $\mathrm{isplit}\left(:\right)$int64int32nag_int array
The dimension of the array isplit must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The points at which $T$ breaks up into sub-matrices, as returned by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$. If the eigenvalues were not computed by nag_lapack_dstebz (f08jj) with ${\mathbf{order}}=\text{'B'}$, set ${\mathbf{isplit}}\left(1\right)$ to n.

### 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:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – double array
The first dimension of the array z will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array z will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
The $m$ eigenvectors, stored as columns of $Z$; the $i$th column corresponds to the $i$th specified eigenvalue, unless ${\mathbf{info}}>{\mathbf{0}}$ (in which case see Error Indicators and Warnings).
2:     $\mathrm{ifailv}\left({\mathbf{m}}\right)$int64int32nag_int array
If ${\mathbf{info}}=i>0$, the first $i$ elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to $0$.
3:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{info}}=-i$
If ${\mathbf{info}}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: d, 3: e, 4: m, 5: w, 6: iblock, 7: isplit, 8: z, 9: ldz, 10: work, 11: iwork, 12: ifailv, 13: 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.
W  ${\mathbf{info}}>0$
If ${\mathbf{info}}=i$, then $i$ eigenvectors (as indicated by the argument ifailv above) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.

## Accuracy

Each computed eigenvector ${z}_{i}$ is the exact eigenvector of a nearby matrix $A+{E}_{i}$, such that
 $Ei = Oε A ,$
where $\epsilon$ is the machine precision. Hence the residual is small:
 $A zi - λi zi = Oε A .$
However, a set of eigenvectors computed by this function may not be orthogonal to so high a degree of accuracy as those computed by nag_lapack_dsteqr (f08je).

The complex analogue of this function is nag_lapack_zstein (f08jx).

## Example

See Example in nag_lapack_dormtr (f08fg).
```function f08jk_example

fprintf('f08jk example results\n\n');

% Find eigenvalues 1:2 of symmetric A (Lower triangular part stored)
uplo = 'L';
a = [ 2.07,  0,    0,     0;
3.87, -0.21, 0,     0;
4.20,  1.87, 1.15,  0;
-1.15,  0.63, 2.06, -1.81];

% Reduce to tridiagonal form
[apt, d, e, tau, info] = f08fe( ...
uplo, a);

% Get eigenvalues 1:2 of T (= eigenvalues of A)
vl = 0;
vu = 0;
il = int64(1);
iu = int64(2);
abstol = 0;
[m, ~, w, iblock, isplit, info] = ...
f08jj( ...
'I', 'B', vl, vu, il, iu, abstol, d, e);

% Get corresponding eigenvectors of T
[v, ifailv, info] = f08jk( ...
d, e, m, w, iblock, isplit);

% Transform Q*V to get eigenvectors of A
side = 'Left';
trans = 'No transpose';
[z, info] = f08fg( ...
side, uplo, trans, apt, tau, v);

% 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

fprintf(' Eigenvalues numbered 1 to 2 are:\n   ');
fprintf(' %7.4f',w(1:m));
fprintf('\n\n');

[ifail] = x04ca( ...
'General', ' ', z, 'Corresponding eigenvectors of A');

```
```f08jk example results

Eigenvalues numbered 1 to 2 are:
-5.0034 -1.9987

Corresponding eigenvectors of A
1       2
1   0.5658 -0.2328
2  -0.3478  0.7994
3  -0.4740 -0.4087
4   0.5781  0.3737
```