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# NAG Toolbox: nag_lapack_dstevx (f08jb)

## Purpose

nag_lapack_dstevx (f08jb) computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix $A$. 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, jfail, info] = f08jb(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)
[d, e, m, w, z, jfail, info] = nag_lapack_dstevx(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)

## Description

nag_lapack_dstevx (f08jb) computes the required eigenvalues and eigenvectors of $A$ by reducing the tridiagonal matrix to diagonal form using the $QR$ algorithm. Bisection is used to determine selected eigenvalues.

## 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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 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 $A$.
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 $A$.
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 $\epsilon {‖A‖}_{1}$ will be used in its place. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this function returns with ${\mathbf{info}}>{\mathbf{0}}$, indicating that some eigenvectors did not converge, try setting abstol to . See Demmel and Kahan (1990).

### 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:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
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 ${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{info}}={\mathbf{0}}$, 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 an eigenvector fails to converge (${\mathbf{info}}>{\mathbf{0}}$), then that column of $Z$ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If ${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
6:     $\mathrm{jfail}\left(:\right)$int64int32nag_int array
The dimension of the array jfail will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{info}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if ${\mathbf{info}}>{\mathbf{0}}$, jfail contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
7:     $\mathrm{info}$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.

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  ${\mathbf{info}}>0$
The algorithm failed to converge; $_$ eigenvectors did not converge. Their indices are stored in array jfail.

## 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 in the half-open interval $\left(0,5\right]$, and the corresponding eigenvectors, of the symmetric tridiagonal matrix
 $A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 .$
```function f08jb_example

fprintf('f08jb example results\n\n');

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

% Eigenvalues in range [0,5] and corresponding eigenvectors
jobz  = 'Vectors';
range = 'Values in range';
vl = 0;
vu = 5;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, ~, m, w, z, jfail, info] = ...
f08jb( ...
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);

```
```f08jb example results

Eigenvalues of A in range:
0.6476
3.5470

Corresponding eigenvectors:
0.9396    0.3388
-0.3311    0.8628
0.0853   -0.3648
-0.0167    0.0879

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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