NAME

ZHEGV - compute all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A∗x=(lambda)∗B∗x, A∗Bx=(lambda)∗x, or B∗A∗x=(lambda)∗x

SYNOPSIS

SUBROUTINE ZHEGV(
ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, INFO )

CHARACTER JOBZ, UPLO

INTEGER INFO, ITYPE, LDA, LDB, LWORK, N

DOUBLE PRECISION RWORK( ∗ ), W( ∗ )

COMPLEX∗16 A( LDA, ∗ ), B( LDB, ∗ ), WORK( ∗ )

PURPOSE

ZHEGV computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form A∗x=(lambda)∗B∗x,  A∗Bx=(lambda)∗x,  or B∗A∗x=(lambda)∗x.  Here A and B are assumed to be Hermitian and B is also
positive definite.
 

ARGUMENTS

ITYPE   (input) INTEGER
Specifies the problem type to be solved:
= 1:  A∗x = (lambda)∗B∗x
= 2:  A∗B∗x = (lambda)∗x
= 3:  B∗A∗x = (lambda)∗x

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.

A       (input/output) COMPLEX∗16 array, dimension (LDA, N)
On entry, the Hermitian matrix A.  If UPLO = ’U’, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A.  If UPLO = ’L’, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.
 
On exit, if JOBZ = ’V’, then if INFO = 0, A contains the matrix Z of eigenvectors.  The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z∗∗H∗B∗Z = I; if ITYPE = 3, Z∗∗H∗inv(B)∗Z = I. If JOBZ = ’N’, then on exit the upper triangle (if UPLO=’U’) or the lower triangle (if UPLO=’L’) of A, including the diagonal, is destroyed.

LDA     (input) INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

B       (input/output) COMPLEX∗16 array, dimension (LDB, N)
On entry, the Hermitian matrix B.  If UPLO = ’U’, the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B.  If UPLO = ’L’, the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B.
 
On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U∗∗H∗U or B = L∗L∗∗H.

LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).

W       (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

WORK    (workspace/output) COMPLEX∗16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK   (input) INTEGER
The length of the array WORK.  LWORK >= max(1,2∗N-1). For optimal efficiency, LWORK >= (NB+1)∗N, where NB is the blocksize for ZHETRD returned by ILAENV.

RWORK   (workspace) DOUBLE PRECISION array, dimension (max(1, 3∗N-2))

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value
> 0:  ZPOTRF or ZHEEV returned an error code:
<= N:  if INFO = i, ZHEEV 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 the leading minor of order i of 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