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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsbgvx (f08ub)

## Purpose

nag_lapack_dsbgvx (f08ub) computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form
 $Az=λBz ,$
where $A$ and $B$ are symmetric and banded, and $B$ is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

## Syntax

[ab, bb, q, m, w, z, jfail, info] = f08ub(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol, 'n', n)
[ab, bb, q, m, w, z, jfail, info] = nag_lapack_dsbgvx(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol, 'n', n)

## Description

The generalized symmetric-definite band problem
 $Az = λ Bz$
is first reduced to a standard band symmetric problem
 $Cx = λx ,$
where $C$ is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
 $zT A z = λ and zT B z = 1 .$

## 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
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
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
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

## 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:     $\mathrm{uplo}$ – string (length ≥ 1)
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangles of $A$ and $B$ are stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangles of $A$ and $B$ are stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
4:     $\mathrm{ka}$int64int32nag_int scalar
If ${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, ${k}_{a}$, of the matrix $A$.
If ${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, ${k}_{a}$, of the matrix $A$.
Constraint: ${\mathbf{ka}}\ge 0$.
5:     $\mathrm{kb}$int64int32nag_int scalar
If ${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, ${k}_{b}$, of the matrix $B$.
If ${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, ${k}_{b}$, of the matrix $B$.
Constraint: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
6:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least ${\mathbf{ka}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of the $n$ by $n$ symmetric band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{a}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left({k}_{a}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{a}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)\text{.}$
7:     $\mathrm{bb}\left(\mathit{ldbb},:\right)$ – double array
The first dimension of the array bb must be at least ${\mathbf{kb}}+1$.
The second dimension of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The upper or lower triangle of the $n$ by $n$ symmetric positive definite band matrix $B$.
The matrix is stored in rows $1$ to ${k}_{b}+1$, more precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of $B$ within the band must be stored with element ${B}_{ij}$ in ${\mathbf{bb}}\left({k}_{b}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{b}\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $B$ within the band must be stored with element ${B}_{ij}$ in ${\mathbf{bb}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)\text{.}$
8:     $\mathrm{vl}$ – double scalar
9:     $\mathrm{vu}$ – 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 ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
10:   $\mathrm{il}$int64int32nag_int scalar
11:   $\mathrm{iu}$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 ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
12:   $\mathrm{abstol}$ – 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, where $T$ is the tridiagonal matrix obtained by reducing $C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold , not zero. If this function returns with ${\mathbf{info}}={\mathbf{1}} \text{to} {\mathbf{n}}$, 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 second dimension of the arrays ab, bb. (An error is raised if these dimensions are not equal.)
$n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.

### Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab will be ${\mathbf{ka}}+1$.
The second dimension of the array ab will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The contents of ab are overwritten.
2:     $\mathrm{bb}\left(\mathit{ldbb},:\right)$ – double array
The first dimension of the array bb will be ${\mathbf{kb}}+1$.
The second dimension of the array bb will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The factor $S$ from the split Cholesky factorization $B={S}^{\mathrm{T}}S$, as returned by nag_lapack_dpbstf (f08uf).
3:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – double array
The first dimension, $\mathit{ldq}$, of the array q will be
• if ${\mathbf{jobz}}=\text{'V'}$, $\mathit{ldq}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise $\mathit{ldq}=1$.
The second dimension of the array q will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobz}}=\text{'V'}$, the $n$ by $n$ matrix, $Q$ used in the reduction of the standard form, i.e., $Cx=\lambda x$, from symmetric banded to tridiagonal form.
If ${\mathbf{jobz}}=\text{'N'}$, q is not referenced.
4:     $\mathrm{m}$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 ${\mathbf{range}}=\text{'I'}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
5:     $\mathrm{w}\left({\mathbf{n}}\right)$ – double array
The eigenvalues in ascending order.
6:     $\mathrm{z}\left(\mathit{ldz},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$ and $1$ otherwise.
If ${\mathbf{jobz}}=\text{'V'}$, z contains the matrix $Z$ of eigenvectors, with the $i$th column of $Z$ holding the eigenvector associated with ${\mathbf{w}}\left(i\right)$. The eigenvectors are normalized so that ${Z}^{\mathrm{T}}BZ=I$.
If ${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
7:     $\mathrm{jfail}\left({\mathbf{info}}\right)$int64int32nag_int array
The dimension of the array jfail will be ${\mathbf{info}}$
If ${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{info}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if ${\mathbf{info}}={\mathbf{1}} \text{to} {\mathbf{n}}$, jfail contains the indices of the eigenvectors that failed to converge.
If ${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
8:     $\mathrm{info}$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: jobz, 2: range, 3: uplo, 4: n, 5: ka, 6: kb, 7: ab, 8: ldab, 9: bb, 10: ldbb, 11: q, 12: ldq, 13: vl, 14: vu, 15: il, 16: iu, 17: abstol, 18: m, 19: w, 20: z, 21: ldz, 22: work, 23: iwork, 24: jfail, 25: 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  ${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
If ${\mathbf{info}}=i$, then $i$ eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.
${\mathbf{info}}>{\mathbf{n}}$
nag_lapack_dpbstf (f08uf) returned an error code; i.e., if ${\mathbf{info}}={\mathbf{n}}+i$, for $1\le i\le {\mathbf{n}}$, then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

## Accuracy

If $B$ is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of $B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of $B$ would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

The total number of floating-point operations is proportional to ${n}^{3}$ if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{range}}=\text{'A'}$, and assuming that $n\gg {k}_{a}$, is approximately proportional to ${n}^{2}{k}_{a}$ if ${\mathbf{jobz}}=\text{'N'}$. Otherwise the number of floating-point operations depends upon the number of eigenvectors computed.
The complex analogue of this function is nag_lapack_zhbgvx (f08up).

## Example

This example finds the eigenvalues in the half-open interval $\left(0.0,1.0\right]$, and corresponding eigenvectors, of the generalized band symmetric eigenproblem $Az=\lambda Bz$, where
 $A = 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03 and B = 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 .$
```function f08ub_example

fprintf('f08ub example results\n\n');

% Symmetric banded matrices A and B stored in symmetric banded format
uplo = 'U';
ka = int64(2);
ab = [0,     0,     0.42,  0.63;
0,     0.39,  0.79,  0.48;
0.24, -0.11, -0.25, -0.03];
kb = int64(1);
bb = [0,     0.95, -0.29, -0.33;
2.07,  1.69,  0.65,  1.17];

% Eigenvalues in range [0,1] and corresponding eigenvectors of Ax = lambda Bx
jobz  = 'Vectors';
range = 'Values in range';
vl = 0;          vu = 1;
il = int64(0); iu = il;
abstol = 0;
[~, ~, ~, m, w, z, jfail, info] = ...
f08ub( ...
jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol);

fprintf('Number of eigenvalues found = %4d\n\n',m);
disp('Selected eigenvalues');
disp(w(1:m)');

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

disp('Corresponding eigenvectors');
disp(z(:,1:m));

```
```f08ub example results

Number of eigenvalues found =    1

Selected eigenvalues
0.0992

Corresponding eigenvectors
0.6729
-0.1009
0.0155
-0.3806

```