zheevx(3)

NAME

ZHEEVX - compute selected eigenvalues and, optionally,

eigenvectors of a complex Hermitian matrix A

SYNOPSIS

SUBROUTINE  ZHEEVX(  JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,  IWORK,  IFAIL,
INFO )
    CHARACTER      JOBZ, RANGE, UPLO
    INTEGER        IL, INFO, IU, LDA, LDZ, LWORK, M, N
    DOUBLE         PRECISION ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    DOUBLE         PRECISION RWORK( * ), W( * )
    COMPLEX*16     A( LDA, * ), WORK( * ), Z( LDZ, * )

PURPOSE

ZHEEVX computes selected eigenvalues and, optionally,

eigenvectors of a complex Hermitian matrix A. Eigenvalues and

eigenvectors can be selected by specifying either a range of val

ues or a range of indices for the desired eigenvalues.

ARGUMENTS

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 triangle of A is stored;
= 'L': Lower triangle of A is stored.

N (input) INTEGER

The order of the matrix A. 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, the lower triangle (if UPLO='L')

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

is destroyed.

LDA (input) INTEGER

The leading dimension of the array A. LDA >=

max(1,N).

VL (input) DOUBLE PRECISION

VU (input) DOUBLE PRECISION If RANGE='V', the

lower and upper bounds of the interval to be searched for eigen

values. 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) DOUBLE PRECISION

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*DLAMCH('S'), not

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

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

CH('S').

See "Computing Small Singular Values of Bidiagonal

Matrices with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

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) DOUBLE PRECISION array, dimension (N)

On normal exit, the first M elements contain the

selected eigenvalues in ascending order.

Z (output) COMPLEX*16 array, dimension (LDZ,

max(1,M))

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

columns of Z contain the orthonormal eigenvectors of the matrix A

corresponding to the selected eigenvalues, with the i-th column

of Z holding the eigenvector associated with W(i). If an eigen

vector fails to converge, then that column of Z contains the lat

est approximation to the eigenvector, and the index of the eigen

vector is returned in IFAIL. If JOBZ = 'N', then Z is not refer

enced. 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*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 max of the blocksize for ZHETRD and for ZUNMTR as re

turned 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) DOUBLE PRECISION 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: if INFO = i, then i eigenvectors failed to

converge. Their indices are stored in array IFAIL.

LAPACK version 3.0 15 June 2000

ZHEEVX(3)