ZHBGVD(l) ) ZHBGVD(l)NAME
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 assumptions 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 without guard dig‐
its, 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 = '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*16 array, dimension (LDAB, N)
On entry, the upper or lower triangle of the Hermitian 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 Hermitian 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 factorization 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.
WORK (workspace/output) COMPLEX*16 array, dimension (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 illegal value
> 0: if INFO = i, and i is:
<= N: the algorithm failed to converge: i off-diagonal ele‐
ments 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 factoriza‐
tion of B could not be completed and no eigenvalues or eigen‐
vectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
LAPACK version 3.0 15 June 2000 ZHBGVD(l)