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 assumed to be Hermitian 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 eigenval
ues.
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.
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
reduction of A*x = (lambda)*B*x to standard form,
i.e. C*x = (lambda)*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
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 refer
enced 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 determined 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*SLAMCH('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)
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
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