sstevr(3)

NAME

SSTEVR - compute selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix T

SYNOPSIS

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

PURPOSE

SSTEVR computes selected eigenvalues and, optionally,

eigenvectors of a real symmetric tridiagonal matrix T. Eigenval

ues and eigenvectors can be selected by specifying either a range

of values or a range of indices for the desired eigenvalues.

Whenever possible, SSTEVR calls SSTEGR to compute the
eigenspectrum using Relatively Robust Representations.

SSTEGR computes eigenvalues by the dqds algorithm, while orthogo

nal eigenvectors are computed from various "good" L D L^T repre

sentations (also known as Relatively Robust Representations).

Gram-Schmidt orthogonalization is avoided as far as possible.

More specifically, the various steps of the algorithm are as fol

lows. For the i-th unreduced block of T,

(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i

D_i L_i^T

is a relatively robust representation,

(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T

to high

relative accuracy by the dqds algorithm,

(c) If there is a cluster of close eigenvalues,

"choose" sigma_i

close to the cluster, and go to step (a),

(d) Given the approximate eigenvalue lambda_j of L_i

D_i L_i^T,

compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.

The desired accuracy of the output can be specified by the

input parameter ABSTOL.

For more details, see "A new O(n^2) algorithm for the sym

metric tridiagonal eigenvalue/eigenvector problem", by Inderjit

Dhillon, Computer Science Division Technical Report No.

UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : SSTEVR calls SSTEGR when the full spectrum is re

quested on machines which conform to the ieee-754 floating point

standard. SSTEVR calls SSTEBZ and SSTEIN on non-ieee machines

and
when partial spectrum requests are made.

Normal execution of SSTEGR may create NaNs and infinities

and hence may abort due to a floating point exception in environ

ments which do not handle NaNs and infinities in the ieee stan

dard default manner.

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 obtained

by reducing A to tridiagonal form.

See "Computing Small Singular Values of Bidiagonal

Matrices with Guaranteed High Relative Accuracy," by Demmel and

Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL

to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigen

values are computed to high relative accuracy when possible in

future releases. The current code does not make any guarantees

about high relative accuracy, but future releases will. See J.

Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled

Diagonally Dominant Matrices", LAPACK Working Note #7, for a dis

cussion of which matrices define their eigenvalues to high rela

tive accuracy.

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). Note: the us

er 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 ad

vance 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).

ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )

The support of the eigenvectors in Z, i.e., the

indices indicating the nonzero elements in Z. The i-th eigenvec

tor is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ(

2*i ).

WORK (workspace/output) REAL array, dimension (LWORK)

On exit, if INFO = 0, WORK(1) returns the optimal

(and minimal) LWORK.

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= 20*N.

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/output) INTEGER array, dimension (LI

WORK)

On exit, if INFO = 0, IWORK(1) returns the optimal

(and minimal) LIWORK.

LIWORK (input) INTEGER

The dimension of the array IWORK. LIWORK >= 10*N.

If LIWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the IWORK array,

returns this value as the first entry of the IWORK array, and no

error message related to LIWORK is issued by XERBLA.

INFO (output) INTEGER

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

gal value
> 0: Internal error

FURTHER DETAILS

Based on contributions by

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of

California at Berkeley, USA

LAPACK version 3.0 15 June 2000

SSTEVR(3)