zhegvd(3)

NAME

ZHEGVD - compute all the eigenvalues, and optionally, the

eigenvectors of a complex generalized Hermitian-definite eigen

problem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or

B*A*x=(lambda)*x

SYNOPSIS

SUBROUTINE  ZHEGVD(  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( * )
    DOUBLE         PRECISION RWORK( * ), W( * )
    COMPLEX*16     A( LDA, * ), B( LDB, * ), WORK( * )

PURPOSE

ZHEGVD computes all the eigenvalues, and optionally, the

eigenvectors of a complex generalized Hermitian-definite eigen

problem, 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. 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 hexadeci

mal 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*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 con

tains the matrix Z of eigenvectors. The eigenvectors are normal

ized 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 trian

gle (if UPLO='U') or the lower triangle (if UPLO='L') of A, in

cluding 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. 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) DOUBLE PRECISION array, dimen

sion (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 (LI

WORK)

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 i-th argument had an ille

gal value
> 0: ZPOTRF or ZHEEVD returned an error code:
<= N: if INFO = i, ZHEEVD 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.

FURTHER DETAILS

Based on contributions by

Mark Fahey, Department of Mathematics, Univ. of Ken

tucky, USA

LAPACK version 3.0 15 June 2000

ZHEGVD(3)