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arXiv · 2210.00384

Using matrix sparsification to solve tropical linear vector equations

Nikolai Krivulin
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Abstract

A linear vector equation in two unknown vectors is examined in the framework of tropical algebra dealing with the theory and applications of semirings and semifields with idempotent addition. We consider a two-sided equation where each side is a tropical product of a given matrix by one of the unknown vectors. We use a matrix sparsification technique to reduce the equation to a set of vector inequalities that involve row-monomial matrices obtained from the given matrices. An existence condition of solutions for the inequalities is established, and a direct representation of the solutions is derived in a compact vector form. To illustrate the proposed approach and to compare the obtained result with that of an existing solution procedure, we apply our solution technique to handle two-sided equations known in the literature. Finally, a computational scheme based on the approach to derive all solutions of the two-sided equation is discussed.

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Nikolai Krivulin. 2024-09-05. Using matrix sparsification to solve tropical linear vector equations. https://doi.org/10.1007/978-3-031-68279-7_12

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