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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 , $Az=λBz ,$
where A$A$ and B$B$ are symmetric and banded, and B$B$ 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 $Az = λ Bz$
is first reduced to a standard band symmetric problem
 Cx = λx , $Cx = λx ,$
where C$C$ 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 . $zT A z = λ and zT B z = 1 .$

## 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:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz = 'N'${\mathbf{jobz}}=\text{'N'}$
Only eigenvalues are computed.
jobz = 'V'${\mathbf{jobz}}=\text{'V'}$
Eigenvalues and eigenvectors are computed.
Constraint: jobz = 'N'${\mathbf{jobz}}=\text{'N'}$ or 'V'$\text{'V'}$.
2:     range – string (length ≥ 1)
If range = 'A'${\mathbf{range}}=\text{'A'}$, all eigenvalues will be found.
If range = 'V'${\mathbf{range}}=\text{'V'}$, all eigenvalues in the half-open interval (vl,vu]$\left({\mathbf{vl}},{\mathbf{vu}}\right]$ will be found.
If range = 'I'${\mathbf{range}}=\text{'I'}$, the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'${\mathbf{range}}=\text{'A'}$, 'V'$\text{'V'}$ or 'I'$\text{'I'}$.
3:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangles of A$A$ and B$B$ are stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangles of A$A$ and B$B$ are stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
4:     ka – int64int32nag_int scalar
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, ka${k}_{a}$, of the matrix A$A$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, ka${k}_{a}$, of the matrix A$A$.
Constraint: ka0${\mathbf{ka}}\ge 0$.
5:     kb – int64int32nag_int scalar
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the number of superdiagonals, kb${k}_{b}$, of the matrix B$B$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the number of subdiagonals, kb${k}_{b}$, of the matrix B$B$.
Constraint: kakb0${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
6:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least ka + 1${\mathbf{ka}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The upper or lower triangle of the n$n$ by n$n$ symmetric band matrix A$A$.
The matrix is stored in rows 1$1$ to ka + 1${k}_{a}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(ka + 1 + ij,j)​ for ​max (1,jka)ij${\mathbf{ab}}\left({k}_{a}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{a}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + ka).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)\text{.}$
7:     bb(ldbb, : $:$) – double array
The first dimension of the array bb must be at least kb + 1${\mathbf{kb}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The upper or lower triangle of the n$n$ by n$n$ symmetric positive definite band matrix B$B$.
The matrix is stored in rows 1$1$ to kb + 1${k}_{b}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of B$B$ within the band must be stored with element Bij${B}_{ij}$ in bb(kb + 1 + ij,j)​ for ​max (1,jkb)ij${\mathbf{bb}}\left({k}_{b}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{b}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of B$B$ within the band must be stored with element Bij${B}_{ij}$ in bb(1 + ij,j)​ for ​jimin (n,j + kb).${\mathbf{bb}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)\text{.}$
8:     vl – double scalar
9:     vu – double scalar
If range = 'V'${\mathbf{range}}=\text{'V'}$, the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'I'$\text{'I'}$, vl and vu are not referenced.
Constraint: if range = 'V'${\mathbf{range}}=\text{'V'}$, vl < vu${\mathbf{vl}}<{\mathbf{vu}}$.
10:   il – int64int32nag_int scalar
11:   iu – int64int32nag_int scalar
If range = 'I'${\mathbf{range}}=\text{'I'}$, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'${\mathbf{range}}=\text{'A'}$ or 'V'$\text{'V'}$, il and iu are not referenced.
Constraints:
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n = 0${\mathbf{n}}=0$, il = 1${\mathbf{il}}=1$ and iu = 0${\mathbf{iu}}=0$;
• if range = 'I'${\mathbf{range}}=\text{'I'}$ and n > 0${\mathbf{n}}>0$, 1 il iu n $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
12:   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 [a,b] $\left[a,b\right]$ of width less than or equal to
 abstol + ε max (|a|,|b|) , $abstol+ε max(|a|,|b|) ,$
where ε $\epsilon$ is the machine precision. If abstol is less than or equal to zero, then ε T1 $\epsilon {‖T‖}_{1}$ will be used in its place, where T$T$ is the tridiagonal matrix obtained by reducing C$C$ to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am (   ) , not zero. If this function returns with INFO = 1ton${\mathbf{INFO}}={\mathbf{1}} \text{to} {\mathbf{n}}$, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am (   ) . See Demmel and Kahan (1990).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the arrays ab, bb. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrices A$A$ and B$B$.
Constraint: n0${\mathbf{n}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldab ldbb ldq ldz work iwork

### Output Parameters

1:     ab(ldab, : $:$) – double array
The first dimension of the array ab will be ka + 1${\mathbf{ka}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldabka + 1$\mathit{ldab}\ge {\mathbf{ka}}+1$.
The contents of ab are overwritten.
2:     bb(ldbb, : $:$) – double array
The first dimension of the array bb will be kb + 1${\mathbf{kb}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldbbkb + 1$\mathit{ldbb}\ge {\mathbf{kb}}+1$.
The factor S$S$ from the split Cholesky factorization B = STS$B={S}^{\mathrm{T}}S$, as returned by nag_lapack_dpbstf (f08uf).
3:     q(ldq, : $:$) – double array
The first dimension, ldq, of the array q will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldq max (1,n) $\mathit{ldq}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldq1$\mathit{ldq}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, the n$n$ by n$n$ matrix, Q$Q$ used in the reduction of the standard form, i.e., Cx = λx$Cx=\lambda x$, from symmetric banded to tridiagonal form.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, q is not referenced.
4:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn$0\le {\mathbf{m}}\le {\mathbf{n}}$.
If range = 'A'${\mathbf{range}}=\text{'A'}$, m = n${\mathbf{m}}={\mathbf{n}}$.
If range = 'I'${\mathbf{range}}=\text{'I'}$, m = iuil + 1${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
5:     w(n) – double array
The eigenvalues in ascending order.
6:     z(ldz, : $:$) – double array
The first dimension, ldz, of the array z will be
• if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, ldz max (1,n) $\mathit{ldz}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ldz1$\mathit{ldz}\ge 1$.
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, and at least 1$1$ otherwise
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, z contains the matrix Z$Z$ of eigenvectors, with the i$i$th column of Z$Z$ holding the eigenvector associated with w(i)${\mathbf{w}}\left(i\right)$. The eigenvectors are normalized so that ZTBZ = I${Z}^{\mathrm{T}}BZ=I$.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, z is not referenced.
7:     jfail( : $:$) – int64int32nag_int array
Note: the dimension of the array jfail must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If jobz = 'V'${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{INFO}}={\mathbf{0}}$, the first m elements of jfail are zero;
• if INFO = 1ton${\mathbf{INFO}}={\mathbf{1}} \text{to} {\mathbf{n}}$, jfail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N'${\mathbf{jobz}}=\text{'N'}$, jfail is not referenced.
8:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [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.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ 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 INFO = 1ton${\mathbf{INFO}}=1 \text{to} {\mathbf{n}}$
If info = i${\mathbf{info}}=i$, then i$i$ 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 info = n + i${\mathbf{info}}={\mathbf{n}}+i$, for 1in$1\le i\le {\mathbf{n}}$, then the leading minor of order i$i$ of B$B$ is not positive definite. The factorization of B$B$ could not be completed and no eigenvalues or eigenvectors were computed.

## Accuracy

If B$B$ 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$B$ differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B$B$ 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 n3 ${n}^{3}$ if jobz = 'V'${\mathbf{jobz}}=\text{'V'}$ and range = 'A'${\mathbf{range}}=\text{'A'}$, and assuming that nka $n\gg {k}_{a}$, is approximately proportional to n2 ka ${n}^{2}{k}_{a}$ if jobz = 'N'${\mathbf{jobz}}=\text{'N'}$. 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

```function nag_lapack_dsbgvx_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
ka = int64(2);
kb = int64(1);
ab = [0, 0, 0.42, 0.63;
0, 0.39, 0.79, 0.48;
0.24, -0.11, -0.25, -0.03];
bb = [0, 0.95, -0.29, -0.33;
2.07, 1.69, 0.65, 1.17];
vl = 0;
vu = 1;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[abOut, bbOut, q, m, w, z, jfail, info] = ...
nag_lapack_dsbgvx(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol)
```
```

0         0    0.3912    0.9434
0    0.5117    0.9991   -0.0630
0.1159    1.1546    0.0265   -0.8160

bbOut =

0    0.6603   -0.3886   -0.3051
1.4387    1.0502    0.7463    1.0817

q =

0.6950   -0.2817   -0.1099   -0.3154
0    0.6139    0.2395    0.6873
0    1.3440   -0.1596   -0.4580
0    0.3791    0.8280   -0.4334

m =

1

w =

0.0992
0
0
0

z =

0.6729         0         0         0
-0.1009         0         0         0
0.0155         0         0         0
-0.3806         0         0         0

jfail =

0

info =

0

```
```function f08ub_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
ka = int64(2);
kb = int64(1);
ab = [0, 0, 0.42, 0.63;
0, 0.39, 0.79, 0.48;
0.24, -0.11, -0.25, -0.03];
bb = [0, 0.95, -0.29, -0.33;
2.07, 1.69, 0.65, 1.17];
vl = 0;
vu = 1;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[abOut, bbOut, q, m, w, z, jfail, info] = ...
f08ub(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol)
```
```

0         0    0.3912    0.9434
0    0.5117    0.9991   -0.0630
0.1159    1.1546    0.0265   -0.8160

bbOut =

0    0.6603   -0.3886   -0.3051
1.4387    1.0502    0.7463    1.0817

q =

0.6950   -0.2817   -0.1099   -0.3154
0    0.6139    0.2395    0.6873
0    1.3440   -0.1596   -0.4580
0    0.3791    0.8280   -0.4334

m =

1

w =

0.0992
0
0
0

z =

0.6729         0         0         0
-0.1009         0         0         0
0.0155         0         0         0
-0.3806         0         0         0

jfail =

0

info =

0

```