
CHEGVD(l) ) CHEGVD(l)
NAME
CHEGVD  compute all the eigenvalues, and optionally, the eigenvectors of a complex gener
alized Hermitiandefinite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or
B*A*x=(lambda)*x
SYNOPSIS
SUBROUTINE CHEGVD( ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK, LWORK, RWORK, LRWORK,
IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, ITYPE, LDA, LDB, LIWORK, LRWORK, LWORK, N
INTEGER IWORK( * )
REAL RWORK( * ), W( * )
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
PURPOSE
CHEGVD computes all the eigenvalues, and optionally, the eigenvectors of a complex gener
alized Hermitiandefinite 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 defi
nite. If eigenvectors are desired, it uses a divide and conquer algorithm.
The divide and conquer algorithm makes very mild assumptions about floating point arith
metic. 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 XMP, Cray YMP, Cray C90, or
Cray2. It could conceivably fail on hexadecimal or decimal machines without guard digits,
but we know of none.
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 array, dimension (LDA, N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading NbyN upper trian
gular part of A contains the upper triangular part of the matrix A. If UPLO =
'L', the leading NbyN 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 array, dimension (LDB, N)
On entry, the Hermitian matrix B. If UPLO = 'U', the leading NbyN upper trian
gular part of B contains the upper triangular part of the matrix B. If UPLO =
'L', the leading NbyN 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) REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
WORK (workspace/output) COMPLEX array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. If N <= 1, LWORK >= 1. If JOBZ =
'N' and N > 1, LWORK >= N + 1. If JOBZ = 'V' and N > 1, LWORK >= 2*N + 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) REAL array, dimension (LRWORK)
On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
LRWORK (input) INTEGER
The dimension of the 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 the array IWORK. If N <= 1, LIWORK >= 1. If JOBZ
= 'N' and N > 1, LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
> 0: CPOTRF or CHEEVD returned an error code:
<= N: if INFO = i, CHEEVD failed to converge; i offdiagonal 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.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
LAPACK version 3.0 15 June 2000 CHEGVD(l) 
