# NAG Library Routine Document

## 1Purpose

f08ucf (dsbgvd) computes all the eigenvalues and, optionally, the 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. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

## 2Specification

Fortran Interface
 Subroutine f08ucf ( jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)
 Integer, Intent (In) :: n, ka, kb, ldab, ldbb, ldz, lwork, liwork Integer, Intent (Out) :: iwork(max(1,liwork)), info Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), bb(ldbb,*), z(ldz,*) Real (Kind=nag_wp), Intent (Out) :: w(n), work(max(1,lwork)) Character (1), Intent (In) :: jobz, uplo
#include <nagmk26.h>
 void f08ucf_ (const char *jobz, const char *uplo, const Integer *n, const Integer *ka, const Integer *kb, double ab[], const Integer *ldab, double bb[], const Integer *ldbb, double w[], double z[], const Integer *ldz, double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_uplo)
The routine may be called by its LAPACK name dsbgvd.

## 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 eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, $Z$, satisfies
 $ZT A Z = Λ and ZT B Z = I ,$
where $\Lambda$ is the diagonal matrix whose diagonal elements are the eigenvalues.

## 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
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:     $\mathbf{uplo}$ – 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 $\text{'L'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{ka}$ – 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: ${\mathbf{ka}}\ge 0$.
5:     $\mathbf{kb}$ – 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: ${\mathbf{ka}}\ge {\mathbf{kb}}\ge 0$.
6:     $\mathbf{ab}\left({\mathbf{ldab}},*\right)$ – 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.
7:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f08ucf (dsbgvd) is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{ka}}+1$.
8:     $\mathbf{bb}\left({\mathbf{ldbb}},*\right)$ – 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 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).
9:     $\mathbf{ldbb}$ – IntegerInput
On entry: the first dimension of the array bb as declared in the (sub)program from which f08ucf (dsbgvd) is called.
Constraint: ${\mathbf{ldbb}}\ge {\mathbf{kb}}+1$.
10:   $\mathbf{w}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
11:   $\mathbf{z}\left({\mathbf{ldz}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array z must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobz}}=\text{'V'}$, and at least $1$ otherwise.
On exit: 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.
12:   $\mathbf{ldz}$ – IntegerInput
On entry: the first dimension of the array z as declared in the (sub)program from which f08ucf (dsbgvd) is called.
Constraints:
• if ${\mathbf{jobz}}=\text{'V'}$, ${\mathbf{ldz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldz}}\ge 1$.
13:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
14:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08ucf (dsbgvd) is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
• if ${\mathbf{n}}\le 1$, ${\mathbf{lwork}}\ge 1$;
• if ${\mathbf{jobz}}=\text{'N'}$ and ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×{\mathbf{n}}\right)$;
• if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{lwork}}\ge 1+5×{\mathbf{n}}+2×{{\mathbf{n}}}^{2}$.
15:   $\mathbf{iwork}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{liwork}}\right)\right)$ – Integer arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{iwork}}\left(1\right)$ returns the minimum liwork.
16:   $\mathbf{liwork}$ – IntegerInput
On entry: the dimension of the array iwork as declared in the (sub)program from which f08ucf (dsbgvd) is called.
If ${\mathbf{liwork}}=-1$, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.
Constraints:
• if ${\mathbf{jobz}}=\text{'N'}$ or ${\mathbf{n}}\le 1$, ${\mathbf{liwork}}\ge 1$;
• if ${\mathbf{jobz}}=\text{'V'}$ and ${\mathbf{n}}>1$, ${\mathbf{liwork}}\ge 3+5×{\mathbf{n}}$.
17:   $\mathbf{info}$ – 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; $〈\mathit{\text{value}}〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
${\mathbf{info}}>{\mathbf{n}}$
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

f08ucf (dsbgvd) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ucf (dsbgvd) 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, assuming that $n\gg {k}_{a}$, is approximately proportional to ${n}^{2}{k}_{a}$ otherwise.
The complex analogue of this routine is f08uqf (zhbgvd).

## 10Example

This example finds all the eigenvalues 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 (f08ucfe.f90)

### 10.2Program Data

Program Data (f08ucfe.d)

### 10.3Program Results

Program Results (f08ucfe.r)