An Efficient Algorithm for Nontrivial Eigenvectors in Max-Plus Algebra

Umer, Mubasher; Hayat, Umar; Abbas, Fazal

doi:10.3390/sym11060738

Cited by 7 publications

(7 citation statements)

References 16 publications

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“…Similarly to [7,[9][10][11][12][13][14][15][16][17][20][21][22], by an interval of R(m, n), we mean a subset of R(m, n) of the form Y = (y ij ) for i ∈ M, j ∈ N, where each y ij is an arbitrary interval belonging to R. For each i, j we denote y ij := inf y ij and y ij = sup y ij . Then, we also have…”

Section: Interval Solutionsmentioning

confidence: 99%

“…Moreover, some of the equivalent conditions of strong versions of solvability can be checked in a polynomial number of arithmetic operations. Systems of two-sided (max, plus)-linear equations A ⊗ x = B ⊗ x have been studied by many authors [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The motivation for this research are papers [8,10].…”

Section: Introductionmentioning

confidence: 99%

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Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Myšková

Plávka

2021

Mathematics

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Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

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Section: Interval Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Myšková

Plávka

2021

Mathematics

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“…Let ρ(A) denote the spectral radius of A, i.e., the largest eigenvalue. In Tropical Algebra, the eigenvalue defines the cycle time in a TEG and is still active research topic, with recent results, such as involving nontrivial eigenvectors [19,20]. The symbol is because a b is the analog of a division in Max-Plus algebra.…”

Section: Tropical Algebramentioning

confidence: 99%

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

2020

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Tropical Algebra is used to model the dynamics of Timed Event Graphs (TEG), a particular class of Timed Discrete-Event System (TDES) in which we are interested only in synchronization and delay phenomena. Whenever this TEG has control inputs, we can use them to control the synchronization of the system to achieve some objective. Thus, this paper formulates a framework based on tropical algebra and lexicographic optimization to synchronize a TEG when dealing with many synchronization objectives that are ranked in previous priority order. We call this kind of problem the Tropical Lexicographic Synchronization Optimization (TLSO). This work develops a solution to this problem, based on Tropical Fractional Linear Programming (TFLP) and lexicographic optimization concepts. In this way, the basics of tropical algebra are determined, including essential terms to this paper, such as left and right residuations, and the following stages of the solution to the TLSO problem are explained. Therefore, this work presents a general framework based on structured algebraic models with application to TEG synchronization. By synchronization, we mean balancing and organizing events chronologically in order to achieve the desired goal. So, we are dealing with concepts closely related to symmetry ones. An illustrative numerical example is presented, which demonstrates the implementation of the proposed algorithms. The acquired results confirm the efficiency of the proposed methodology. Codes used for implementing the algorithms are listed in the appendix section of the article.

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“…These nonlinear systems can be described by a max-plus linear time-invariant model, which is called the max-plus linear system. The matrix theory in max-plus algebra has been developed, including the computation for eigenvalues and eigenvectors [12][13][14][15][16][17], The Cayley-Hamilton theorem [18,19], QR decomposition [20] and solvability of linear equations [21][22][23]. Meanwhile, the polynomial theory in max-plus algebra has also been studied.…”

Section: Introductionmentioning

confidence: 99%

Ordered Structures of Polynomials over Max-Plus Algebra

2021

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The ordered structures of polynomial idempotent algebras over max-plus algebra are investigated in this paper. Based on the antisymmetry, the partial orders on the sets of formal polynomials and polynomial functions are introduced to generate two partially ordered idempotent algebras (POIAs). Based on the symmetry, the quotient POIA of formal polynomials is then obtained. The order structure relationships among these three POIAs are described: the POIA of polynomial functions and the POIA of formal polynomials are orderly homomorphic; the POIA of polynomial functions and the quotient POIA of formal polynomials are orderly isomorphic. By using the partial order on formal polynomials, an algebraic method is provided to determine the upper and lower bounds of an equivalence class in the quotient POIA of formal polynomials. The criterion for a formal polynomial to be the minimal element of an equivalence class is derived. Furthermore, it is proven that any equivalence class is either an uncountable set with cardinality of the continuum or a finite set with a single element.

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An Efficient Algorithm for Nontrivial Eigenvectors in Max-Plus Algebra

Cited by 7 publications

References 16 publications

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Tropical Lexicographic Optimization: Synchronizing Timed Event Graphs

Ordered Structures of Polynomials over Max-Plus Algebra

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