ssyevx(3)

NAME

SSYEVX - compute selected eigenvalues and, optionally,

eigenvectors of a real symmetric matrix A

SYNOPSIS

SUBROUTINE  SSYEVX(  JOBZ, RANGE, UPLO, N, A, LDA, VL, VU,
IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO )
    CHARACTER      JOBZ, RANGE, UPLO
    INTEGER        IL, INFO, IU, LDA, LDZ, LWORK, M, N
    REAL           ABSTOL, VL, VU
    INTEGER        IFAIL( * ), IWORK( * )
    REAL           A( LDA, * ), W( * ), WORK( * ), Z( LDZ,
* )

PURPOSE

SSYEVX computes selected eigenvalues and, optionally,

eigenvectors of a real symmetric matrix A. Eigenvalues and eigen

vectors 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.

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) REAL array, dimension (LDA, N)

On entry, the symmetric 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) 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').

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)

On normal exit, the first M elements contain the

selected eigenvalues 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, 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) REAL 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(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB

is the max of the blocksize for SSYTRD and SORMTR 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.

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

SSYEVX(3)