DSTEVR(1) LAPACK driver routine (version 3.2) DSTEVR(1)NAME
DSTEVR - computes selected eigenvalues and, optionally, eigenvectors of
a real symmetric tridiagonal matrix T
SYNOPSIS
SUBROUTINE DSTEVR( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER ISUPPZ( * ), IWORK( * )
DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ,
* )
PURPOSE
DSTEVR 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, DSTEVR calls DSTEMR to compute the
eigenspectrum using Relatively Robust Representations. DSTEMR 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 : DSTEVR calls DSTEMR when the full spectrum is requested on
machines which conform to the ieee-754 floating point standard. DSTEVR
calls DSTEBZ and DSTEIN on non-ieee machines and
when partial spectrum requests are made.
Normal execution of DSTEMR 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (max(1,N-1))
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
plied by a constant factor chosen to avoid over/underflow in
computing the eigenvalues.
VL (input) DOUBLE PRECISION
VU (input) DOUBLE PRECISION 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) DOUBLE PRECISION
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 pre‐
cision. 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 impor‐
tant, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will
guarantee that eigenvalues are computed to high relative accu‐
racy 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 discussion 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) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in
ascending order.
Z (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension
(MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,20*N). If
LWORK = -1, then a workspace query is assumed; the routine only
calculates the optimal sizes of the WORK and IWORK arrays,
returns these values as the first entries of the WORK and IWORK
arrays, and no error message related to LWORK or LIWORK is
issued by XERBLA.
IWORK (workspace/output) INTEGER array, dimension (MAX(1,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 >= max(1,10*N). If
LIWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal sizes of the WORK and IWORK arrays,
returns these values as the first entries of the WORK and IWORK
arrays, and no error message related to LWORK or 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 driver routine (version 3November 2008 DSTEVR(1)
