NAME

ZHBGV - 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 ZHBGV(
JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, RWORK, INFO )

CHARACTER JOBZ, UPLO

INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, N

DOUBLE PRECISION RWORK( ∗ ), W( ∗ )

COMPLEX∗16 AB( LDAB, ∗ ), BB( LDBB, ∗ ), WORK( ∗ ), Z( LDZ, ∗ )

PURPOSE

ZHBGV 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. 
 

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) COMPLEX∗16 array, dimension (N)

RWORK   (workspace) DOUBLE PRECISION array, dimension (3∗N)

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 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.

  —  LAPACK version 2.0  —  08 October 1994