chegvx(3)

NAME

CHEGVX - compute selected eigenvalues, and optionally,

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 CHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B,
LDB, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ,  WORK,  LWORK,  RWORK,
IWORK, IFAIL, INFO )
    CHARACTER      JOBZ, RANGE, UPLO
    INTEGER         IL,  INFO,  ITYPE,  IU, LDA, LDB, LDZ,
LWORK, M, N
    REAL           ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    REAL           RWORK( * ), W( * )
    COMPLEX        A( LDA, * ), B( LDB, * ), WORK( * ), Z(
LDZ, * )

PURPOSE

CHEGVX computes selected eigenvalues, and optionally,

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. Eigenvalues and eigenvectors can be

selected by specifying either a range of values or a range of in

dices for the desired eigenvalues.

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.

RANGE (input) CHARACTER*1

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval

(VL,VU] will be found. = 'I': the IL-th through IU-th eigenval

ues will be found.

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 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, the lower triangle (if UPLO='L') or the

upper triangle (if UPLO='U') of A, including the diagonal, is de

stroyed.

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 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).

VL (input) REAL

VU (input) REAL If RANGE='V', the lower and

upper bounds of the interval to be searched for eigenvalues. VL <

VU. Not referenced if RANGE = 'A' or 'I'.

IL (input) INTEGER

IU (input) INTEGER If RANGE='I', the indices

(in ascending order) of the smallest and largest eigenvalues to

be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if

N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL (input) REAL

The absolute error tolerance for the eigenvalues.

An approximate eigenvalue is accepted as converged when it is de

termined to lie in an interval [a,b] of width less than or equal

to

ABSTOL + EPS * max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is

less than or equal to zero, then EPS*|T| will be used in its

place, where |T| is the 1-norm of the tridiagonal matrix obtained

by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when

ABSTOL is set to twice the underflow threshold 2*SLAMCH('S'), not

zero. If this routine returns with INFO>0, indicating that some

eigenvectors did not converge, try setting ABSTOL to 2*SLAM

CH('S').

M (output) INTEGER

The total number of eigenvalues found. 0 <= M <=

N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W (output) REAL array, dimension (N)

The first M elements contain the selected eigen

values in ascending order.

Z (output) COMPLEX array, dimension (LDZ, max(1,M))

If JOBZ = 'N', then Z is not referenced. If JOBZ

= 'V', then if INFO = 0, the first M columns of Z contain the or

thonormal eigenvectors of the matrix A corresponding to the se

lected eigenvalues, with the i-th column of Z holding the eigen

vector associated with W(i). The eigenvectors are normalized as

follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3,

Z**T*inv(B)*Z = I.

If an eigenvector fails to converge, then that

column of Z contains the latest approximation to the eigenvector,

and the index of the eigenvector is returned in IFAIL. Note: the

user must ensure that at least max(1,M) columns are supplied in

the array Z; if RANGE = 'V', the exact value of M is not known in

advance and an upper bound must be used.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= 1,

and if JOBZ = 'V', LDZ >= max(1,N).

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. LWORK >=

max(1,2*N-1). For optimal efficiency, LWORK >= (NB+1)*N, where

NB is the blocksize for CHETRD returned by ILAENV.

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) REAL array, dimension (7*N)

IWORK (workspace) INTEGER array, dimension (5*N)

IFAIL (output) INTEGER array, dimension (N)

If JOBZ = 'V', then if INFO = 0, the first M ele

ments of IFAIL are zero. If INFO > 0, then IFAIL contains the

indices of the eigenvectors that failed to converge. If JOBZ =

'N', then IFAIL is not referenced.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value
> 0: CPOTRF or CHEEVX returned an error code:
<= N: if INFO = i, CHEEVX failed to converge; i

eigenvectors failed to converge. Their indices are stored in ar

ray IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then the

leading minor of order i of B is not positive definite. The fac

torization 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

CHEGVX(3)