chbgvx(3)

NAME

CHBGVX - compute all the eigenvalues, and optionally, the

eigenvectors of a complex generalized Hermitian-definite banded

eigenproblem, of the form A*x=(lambda)*B*x

SYNOPSIS

SUBROUTINE CHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB,
BB,  LDBB,  Q,  LDQ,  VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
RWORK, IWORK, IFAIL, INFO )
    CHARACTER      JOBZ, RANGE, UPLO
    INTEGER        IL, INFO, IU, KA, KB, LDAB, LDBB,  LDQ,
LDZ, M, N
    REAL           ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    REAL           RWORK( * ), W( * )
    COMPLEX        AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, *
), WORK( * ), Z( LDZ, * )

PURPOSE

CHBGVX computes all the eigenvalues, and optionally, the

eigenvectors of a complex generalized Hermitian-definite banded

eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are as

sumed to be Hermitian and banded, and B is also positive defi

nite. Eigenvalues and eigenvectors can be selected by specifying

either all eigenvalues, a range of values or a range of indices

for the desired eigenvalues.

ARGUMENTS

JOBZ (input) CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found;
= 'V': all eigenvalues in the half-open interval

(VL,VU] will be found; = 'I': the IL-th through IU-th eigenvalues

will be found.

UPLO (input) CHARACTER*1

= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.

N (input) INTEGER

The order of the matrices A and B. N >= 0.

KA (input) INTEGER

The number of superdiagonals of the matrix A if

UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.

KB (input) INTEGER

The number of superdiagonals of the matrix B if

UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.

AB (input/output) COMPLEX array, dimension (LDAB, N)

On entry, the upper or lower triangle of the Her

mitian band matrix A, stored in the first ka+1 rows of the array.

The j-th column of A is stored in the j-th column of the array AB

as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j

ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for

j<=i<=min(n,j+ka).

On exit, the contents of AB are destroyed.

LDAB (input) INTEGER

The leading dimension of the array AB. LDAB >=

KA+1.

BB (input/output) COMPLEX array, dimension (LDBB, N)

On entry, the upper or lower triangle of the Her

mitian band matrix B, stored in the first kb+1 rows of the array.

The j-th column of B is stored in the j-th column of the array BB

as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j

kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for

j<=i<=min(n,j+kb).

On exit, the factor S from the split Cholesky fac

torization B = S**H*S, as returned by CPBSTF.

LDBB (input) INTEGER

The leading dimension of the array BB. LDBB >=

KB+1.

Q (output) COMPLEX array, dimension (LDQ, N)

If JOBZ = 'V', the n-by-n matrix used in the re

duction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lamb

da)*x, and consequently C to tridiagonal form. If JOBZ = 'N',

the array Q is not referenced.

LDQ (input) INTEGER

The leading dimension of the array Q. If JOBZ =

'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).

VL (input) REAL

VU (input) REAL If RANGE='V', the lower and

upper bounds of the interval to be searched for eigenvalues. VL <

VU. Not referenced if RANGE = 'A' or 'I'.

IL (input) INTEGER

IU (input) INTEGER If RANGE='I', the indices

(in ascending order) of the smallest and largest eigenvalues to

be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if

N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL (input) REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged when it is de

termined to lie in an interval [a,b] of width less than or equal

to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is

less than or equal to zero, then EPS*|T| will be used in its

place, where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing AP to tridiagonal form.

Eigenvalues will be computed most accurately when

ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not

zero. If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to 2*SLAM

CH('S').

M (output) INTEGER

The total number of eigenvalues found. 0 <= M <=

N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W (output) REAL array, dimension (N)

If INFO = 0, the eigenvalues in ascending order.

Z (output) COMPLEX array, dimension (LDZ, N)

If JOBZ = 'V', then if INFO = 0, Z contains the

matrix Z of eigenvectors, with the i-th column of Z holding the

eigenvector associated with W(i). The eigenvectors are normalized

so that Z**H*B*Z = I. If JOBZ = 'N', then Z is not referenced.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= 1,

and if JOBZ = 'V', LDZ >= N.

WORK (workspace) COMPLEX array, dimension (N)

RWORK (workspace) REAL array, dimension (7*N)

IWORK (workspace) INTEGER array, dimension (5*N)

IFAIL (output) INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M ele

ments of IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge. If JOBZ =

'N', then IFAIL is not referenced.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value
> 0: if INFO = i, and i is:
<= N: then i eigenvectors failed to converge.

Their indices are stored in array IFAIL. > N: if INFO = N + i,

for 1 <= i <= N, then CPBSTF
returned INFO = i: B is not positive definite.

The factorization of B could not be completed and no eigenvalues

or eigenvectors were computed.

FURTHER DETAILS

Based on contributions by

Mark Fahey, Department of Mathematics, Univ. of Ken

tucky, USA

LAPACK version 3.0 15 June 2000

CHBGVX(3)