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 unre
duced 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
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 float
ing 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 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 multi
plied 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
ABSTOL (input) REAL
The absolute error tolerance for the eigenvalues.
An approximate 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 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 guar
antee that eigenvalues are computed to high rela
tive 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 Domi
nant Matrices", LAPACK Working Note #7, for a dis
cussion of which matrices define their eigenvalues
to high relative 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 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,
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 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 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
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