# NAG Library Routine Document

## 1Purpose

f08ubf (dsbgvx) 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.

## 2Specification

Fortran Interface
 Subroutine f08ubf ( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, q, ldq, vl, vu, il, iu, m, w, z, ldz, work, info)
 Integer, Intent (In) :: n, ka, kb, ldab, ldbb, ldq, il, iu, ldz Integer, Intent (Inout) :: jfail(*) Integer, Intent (Out) :: m, iwork(5*n), info Real (Kind=nag_wp), Intent (In) :: vl, vu, abstol Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), bb(ldbb,*), q(ldq,*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: w(n), work(7*n) Character (1), Intent (In) :: jobz, range, uplo
#include <nagmk26.h>
 void f08ubf_ (const char *jobz, const char *range, const char *uplo, const Integer *n, const Integer *ka, const Integer *kb, double ab[], const Integer *ldab, double bb[], const Integer *ldbb, double q[], const Integer *ldq, const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], double z[], const Integer *ldz, double work[], Integer iwork[], Integer jfail[], Integer *info, const Charlen length_jobz, const Charlen length_range, const Charlen length_uplo)
The routine may be called by its LAPACK name dsbgvx.

## 3Description

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
 $zT A z = λ and zT B z = 1 .$

## 4References

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

## 5Arguments

1:     $\mathbf{jobz}$ – Character(1)Input
On entry: 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:     $\mathbf{range}$ – Character(1)Input
On entry: 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:     $\mathbf{uplo}$ – Character(1)Input
On entry: 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{ka}$ – IntegerInput
On entry: 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: ${\mathbf{ka}}\ge 0$.
6:     $\mathbf{kb}$ – IntegerInput
On entry: 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: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
7:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: 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 ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)\text{.}$
On exit: the contents of ab are overwritten.
8:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08ubf (dsbgvx) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{ka}}+1$.
9:     $\mathbf{bb}\left({\mathbf{ldbb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array bb must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: 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 ${\mathbf{bb}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{b}\right)\text{.}$
On exit: the factor $S$ from the split Cholesky factorization $B={S}^{\mathrm{T}}S$, as returned by f08uff (dpbstf).
10:   $\mathbf{ldbb}$ – IntegerInput
On entry: the first dimension of the array bb as declared in the (sub)program from which f08ubf (dsbgvx) is called.
Constraint: ${\mathbf{ldbb}}\ge {\mathbf{kb}}+1$.
11:   $\mathbf{q}\left({\mathbf{ldq}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: 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.
12:   $\mathbf{ldq}$ – IntegerInput
On entry: the first dimension of the array q as declared in the (sub)program from which f08ubf (dsbgvx) is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldq}}\ge 1$.
13:   $\mathbf{vl}$ – Real (Kind=nag_wp)Input
14:   $\mathbf{vu}$ – Real (Kind=nag_wp)Input
On entry: 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 ${\mathbf{range}}=\text{'V'}$, ${\mathbf{vl}}<{\mathbf{vu}}$.
15:   $\mathbf{il}$ – IntegerInput
16:   $\mathbf{iu}$ – IntegerInput
On entry: 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 ${\mathbf{n}}>0$, $1\le {\mathbf{il}}\le {\mathbf{iu}}\le {\mathbf{n}}$.
17:   $\mathbf{abstol}$ – Real (Kind=nag_wp)Input
On entry: 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 routine 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).
18:   $\mathbf{m}$ – IntegerOutput
On exit: the total number of eigenvalues found. $0\le {\mathbf{m}}\le {\mathbf{n}}$.
If ${\mathbf{range}}=\text{'A'}$, ${\mathbf{m}}={\mathbf{n}}$.
If ${\mathbf{range}}=\text{'I'}$, ${\mathbf{m}}={\mathbf{iu}}-{\mathbf{il}}+1$.
19:   $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
20:   $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobz}}=\text{'V'}$, then
• if ${\mathbf{info}}={\mathbf{0}}$, the first m columns of $Z$ contain the eigenvectors corresponding to the selected eigenvalues, 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 an eigenvector fails to converge (${\mathbf{info}}={\mathbf{1}} \text{to} {\mathbf{n}}$), 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.
Note:  you must ensure that at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$ columns are supplied in the array z; if ${\mathbf{range}}=\text{'V'}$, the exact value of m is not known in advance and an upper bound of at least n must be used.
21:   $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08ubf (dsbgvx) is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
22:   $\mathbf{work}\left(7×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
23:   $\mathbf{iwork}\left(5×{\mathbf{n}}\right)$ – Integer arrayWorkspace
24:   $\mathbf{jfail}\left(*\right)$ – Integer arrayOutput
Note: the dimension of the array jfail must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: 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.
25:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and 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.
${\mathbf{info}}=1 \text{to} {\mathbf{n}}$
The algorithm failed to converge; $〈\mathit{\text{value}}〉$ eigenvectors did not converge. Their indices are stored in array jfail.
${\mathbf{info}}>{\mathbf{n}}$
If ${\mathbf{info}}={\mathbf{n}}+〈\mathit{\text{value}}〉$, for $1\le 〈\mathit{\text{value}}〉\le {\mathbf{n}}$, f08uff (dpbstf) returned ${\mathbf{info}}=〈\mathit{\text{value}}〉$: $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

## 7Accuracy

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.

## 8Parallelism and Performance

f08ubf (dsbgvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ubf (dsbgvx) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 routine is f08upf (zhbgvx).

## 10Example

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
 $A = 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03 and B = 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 .$

### 10.1Program Text

Program Text (f08ubfe.f90)

### 10.2Program Data

Program Data (f08ubfe.d)

### 10.3Program Results

Program Results (f08ubfe.r)