678 research outputs found
The level set method for the two-sided eigenproblem
We consider the max-plus analogue of the eigenproblem for matrix pencils
Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible
values of lambda), which is a finite union of intervals, can be computed in
pseudo-polynomial number of operations, by a (pseudo-polynomial) number of
calls to an oracle that computes the value of a mean payoff game. The proof
relies on the introduction of a spectral function, which we interpret in terms
of the least Chebyshev distance between Ax and lambda Bx. The spectrum is
obtained as the zero level set of this function.Comment: 34 pages, 4 figures. Changes with respect to the previous version: we
explain relation to mean-payoff games and discrete event systems, and show
that the reconstruction of spectrum is pseudopolynomia
Tropical linear algebra with the Lukasiewicz T-norm
The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped
with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over
this semiring can be developed in the usual way. We observe that any problem of
the max-Lukasiewicz linear algebra can be equivalently formulated as a problem
of the tropical (max-plus) linear algebra. Based on this equivalence, we
develop a theory of the matrix powers and the eigenproblem over the
max-Lukasiewicz semiring.Comment: 27 page
On Integer Images of Max-plus Linear Mappings
Let us extend the pair of operations (max,+) over real numbers to matrices in
the same way as in conventional linear algebra. We study integer images of
max-plus linear mappings. The question whether Ax (in the max-plus algebra) is
an integer vector for at least one x has been studied for some time but
polynomial solution methods seem to exist only in special cases. In the
terminology of combinatorial matrix theory this question reads: is it possible
to add constants to the columns of a given matrix so that all row maxima are
integer? This problem has been motivated by attempts to solve a class of
job-scheduling problems. We present two polynomially solvable special cases
aiming to move closer to a polynomial solution method in the general case
Block Iterative Eigensolvers for Sequences of Correlated Eigenvalue Problems
In Density Functional Theory simulations based on the LAPW method, each
self-consistent field cycle comprises dozens of large dense generalized
eigenproblems. In contrast to real-space methods, eigenpairs solving for
problems at distinct cycles have either been believed to be independent or at
most very loosely connected. In a recent study [7], it was demonstrated that,
contrary to belief, successive eigenproblems in a sequence are strongly
correlated with one another. In particular, by monitoring the subspace angles
between eigenvectors of successive eigenproblems, it was shown that these
angles decrease noticeably after the first few iterations and become close to
collinear. This last result suggests that we can manipulate the eigenvectors,
solving for a specific eigenproblem in a sequence, as an approximate solution
for the following eigenproblem. In this work we present results that are in
line with this intuition. We provide numerical examples where opportunely
selected block iterative eigensolvers benefit from the reuse of eigenvectors by
achieving a substantial speed-up. The results presented will eventually open
the way to a widespread use of block iterative eigensolvers in ab initio
electronic structure codes based on the LAPW approach.Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics
Communication
Controllable and tolerable generalized eigenvectors of interval max-plus matrices
summary:By max-plus algebra we mean the set of reals equipped with the operations and for A vector is said to be a generalized eigenvector of max-plus matrices if for some . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval) entries are studied and complete solutions of the controllable, the tolerable and the strong generalized eigenproblem in max-plus algebra are presented. As a consequence of the obtained results, efficient algorithms for checking equivalent conditions are introduced
On the problem Ax=\lambda Bx in max algebra: every system of intervals is a spectrum
We consider the two-sided eigenproblem Ax=\lambda Bx over max algebra. It is
shown that any finite system of real intervals and points can be represented as
spectrum of this eigenproblem.Comment: 7 pages, minor corrections, change of titl
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