37 research outputs found
The general trapezoidal algorithm for strongly regular maxâmin matrices
AbstractThe problem of the strong regularity for square matrices over a general maxâmin algebra is considered. An O(n2logn) algorithm for recognition of the strong regularity of a given nĂn matrix is proposed. The algorithm works without any restrictions on the underlying maxâmin algebra, concerning the density, or the boundedness
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
(K,L)-eigenvectors in max-min algebra
Using the concept of (K,L)-eigenvector, we investigate the structure of the
max-min eigenspace associated with a given eigenvalue of a matrix in the
max-min algebra (also known as fuzzy algebra). In our approach, the max-min
eigenspace is split into several regions according to the order relations
between the eigenvalue and the components of x. The resulting theory of
(K,L)-eigenvectors, being based on the fundamental results of Gondran and
Minoux, allows to describe the whole max-min eigenspace explicitly and in more
detail.Comment: New title and abstract, several minor correction
Tolerance problems for generalized eigenvectors of interval fuzzy matrices
summary:Fuzzy algebra is a special type of algebraic structure in which classical addition and multiplication are replaced by maximum and minimum (denoted and , respectively). The eigenproblem is the search for a vector (an eigenvector) and a constant (an eigenvalue) such that , where is a given matrix. This paper investigates a generalization of the eigenproblem in fuzzy algebra. We solve the equation with given matrices and unknown constant and vector . Generalized eigenvectors have interesting and useful properties in the various computational tasks with inexact (interval) matrix and vector inputs. This paper studies the properties of generalized interval eigenvectors of interval matrices. Three types of generalized interval eigenvectors: strongly tolerable generalized eigenvectors, tolerable generalized eigenvectors and weakly tolerable generalized eigenvectors are proposed and polynomial procedures for testing the obtained equivalent conditions are presented
Solvability of a Bounded Parametric System in Max-Ćukasiewicz Algebra
The max-Ćukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Ćukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Ćukasiewicz systems with interval coefficients. Furthermore, Ćukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system