SSTEVR(l) ) SSTEVR(l)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. Eigenvalues 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 orthogonal eigenvectors are
computed from various "good" L D L^T representations (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 follows. 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 param‐
eter ABSTOL.
For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
agonal 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 requested 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 environments which do not
handle NaNs and infinities in the ieee standard 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 eigenvalues 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 tridiagonal 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 approxi‐
mate eigenvalue is accepted as converged when it is determined
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 reduc‐
ing 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 eigenvalues 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 Diag‐
onally Dominant Matrices", LAPACK Working Note #7, for a dis‐
cussion of which matrices define their eigenvalues to high rel‐
ative 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 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 con‐
tain 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 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).
ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices indi‐
cating the nonzero elements in Z. The i-th eigenvector 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 (LIWORK)
On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
mal) 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 illegal 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(l)
