Abstract
In this paper, we discuss and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then go over a specific version of Cramer’s rule which is also related to the pseudo-inverse of the system matrix. In these two methods, the constant vector plays an implicit role in solvability of the system. Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process. Each of these methods yields the maximal solution if it exists. Importantly, we show that the maximal solutions obtained from these methods as well as the previously studied LU-method are all identical.
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References
Aceto, L., Esik, Z., Ingólfsdóttir, A.: Equational theories of tropical semirings. Theoret. Comput. Sci. 298(3), 417–469 (2003)
Akian, M., Gaubert, S., Guterman, A.: Tropical Cramer determinants revisited. Trop. Idempotent Math. Appl. 616, 1–45 (2014)
Butkovič, P.: Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl. 367, 313–335 (2003)
Golan, J.S.: Semirings and their Applications. Kluwer Academic, Dordrecht (1999)
Jamshidvand, S., Ghalandarzadeh, S., Amiraslani, A., Olia, F.: Solving Linear Systems over Idempotent Semifields through \(LU\)-factorization, arXiv preprint arXiv:1904.13256, (2019)
Krivulin, N.: Complete algebraic solution of multidimensional optimization problems in tropical semifield. Journal of logical and algebraic methods in programming 99, 26–40 (2018)
Krivulin, N.: Solution of linear equations and inequalities in idempotent vector spaces. Int. J. Appl. Math. Inf. 7(1), 14–23 (2013)
McEneaney, W.M.: Max-Plus Methods for Nonlinear Control and Estimation. Springer, Berlin (2006)
Sararnrakskul, R.I., Sombatboriboon, S., Lertwichitsilp, P.: Invertible matrices over semifields. East-West J. Math. 12(1), 159 (2010)
Tan, Y.J.: Determinants of matrices over semirings. Linear Multilinear Algebra 62(4), 498–517 (2014)
Tan, Y.J.: Inner products on semimodules over a commutative semiring. Linear Algebra Appl. 460, 151–173 (2017)
Tan, Y.J.: On invertible matrices over commutative semirings. Linear Multilinear Algebra 61(6), 710–724 (2013)
Vandiver, H.S.: Note on a simple type of algebra in which the cancellation law of addition does not hold. Bull. Am. Math. Soc. 40(12), 914–920 (1934)
Wilding, D.: Linear algebra over semirings, Doctoral dissertation, The University of Manchester (United Kingdom), (2015)
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Olia, F., Ghalandarzadeh, S., Amiraslani, A. et al. Analysis of linear systems over idempotent semifields. Math Sci 14, 137–146 (2020). https://doi.org/10.1007/s40096-020-00324-x
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DOI: https://doi.org/10.1007/s40096-020-00324-x