sstevx(3)

NAME

SSTEVX - compute selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix A

SYNOPSIS

SUBROUTINE  SSTEVX(  JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
    CHARACTER      JOBZ, RANGE
    INTEGER        IL, INFO, IU, LDZ, M, N
    REAL           ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    REAL           D( * ), E( * ), W( * ), WORK( *  ),  Z(
LDZ, * )

PURPOSE

SSTEVX computes selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix A. Eigenval

ues and eigenvectors can be selected by specifying either a range

of values 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.

N (input) INTEGER

The order of the matrix. N >= 0.

D (input/output) REAL array, dimension (N)

On entry, the n diagonal elements of the tridiago

nal matrix A. On exit, D may be multiplied by a constant factor

chosen to avoid over/underflow in computing the eigenvalues.

E (input/output) REAL array, dimension (N)

On entry, the (n-1) subdiagonal elements of the

tridiagonal matrix A in elements 1 to N-1 of E; E(N) need not be

set. On exit, E may be multiplied by a constant factor chosen to

avoid over/underflow in computing the eigenvalues.

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.

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').

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) REAL array, dimension (N)

The first M elements contain the selected eigen

values in ascending order.

Z (output) REAL 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 (INFO > 0), then that column of Z con

tains the latest approximation to the eigenvector, and the index

of the eigenvector is returned in IFAIL. If JOBZ = 'N', then Z

is not referenced. 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) REAL array, dimension (5*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

SSTEVX(3)