ZHBGVD - 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 ZHBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB,
LDBB, W, Z, LDZ, WORK, LWORK, RWORK,
LRWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK,
LRWORK, LWORK, N
INTEGER IWORK( * )
DOUBLE PRECISION RWORK( * ), W( * )
COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), WORK( *
), Z( LDZ, * )
PURPOSE
ZHBGVD 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. If eigenvectors are desired, it
uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assump
tions about floating point arithmetic. It will work on
machines with a guard digit in add/subtract, or on those
binary machines without guard digits which subtract like
the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
conceivably fail on hexadecimal or decimal machines with
out guard digits, but we know of none.
ARGUMENTS
JOBZ (input) CHARACTER*1
= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.
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
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*16 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*16 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 ZPBSTF.
LDBB (input) INTEGER
The leading dimension of the array BB. LDBB >=
KB+1.
W (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
Z (output) COMPLEX*16 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.
(LWORK)
On exit, if INFO=0, WORK(1) returns the optimal
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. If N <= 1,
LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= N.
If JOBZ = 'V' and N > 1, LWORK >= 2*N**2.
If LWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LWORK is issued by XERBLA.
RWORK (workspace/output) DOUBLE PRECISION array,
dimension (LRWORK)
On exit, if INFO=0, RWORK(1) returns the optimal
LRWORK.
LRWORK (input) INTEGER
The dimension of array RWORK. If N <= 1,
LRWORK >= 1. If JOBZ = 'N' and N > 1, LRWORK >=
N. If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N +
2*N**2.
If LRWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the RWORK array, returns this value as the first
entry of the RWORK array, and no error message
related to LRWORK is issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension
(LIWORK)
On exit, if INFO=0, IWORK(1) returns the optimal
LIWORK.
LIWORK (input) INTEGER
The dimension of array IWORK. If JOBZ = 'N' or N
<= 1, LIWORK >= 1. If JOBZ = 'V' and N > 1,
LIWORK >= 3 + 5*N.
If LIWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the IWORK array, returns this value as the first
entry of the IWORK array, and no error message
related to LIWORK is issued by XERBLA.
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:
diagonal elements of an intermediate tridiagonal
form did not converge to zero; > N: if INFO = N
+ i, for 1 <= i <= N, then ZPBSTF
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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