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Solving linear systems over idempotent semifields through LU-factorization

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Abstract

In this paper, we introduce and analyze a new LU-factorization technique for square matrices over idempotent semifields. In particular, more emphasis is put on “max-plus” algebra here but the work is extended to other idempotent semifields as well. We first determine the conditions under which a square matrix has LU factors. Next, using this technique, we propose a method for solving square linear systems of equations whose system matrices are LU-factorizable. We also give conditions for an LU-factorizable system to have solutions. This work is an extension of similar techniques over fields. Maple® procedures for this LU-factorization are also included.

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Authors and Affiliations

  1. Department of Mathematics, School of STEM, Capilano University, North Vancouver, BC, Canada

    Amirhossein Amiraslani

  2. Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

    Sedighe Jamshidvand, Shaban Ghalandarzadeh, Amirhossein Amiraslani & Fateme Olia

Authors
  1. Sedighe Jamshidvand
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  2. Shaban Ghalandarzadeh
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  3. Amirhossein Amiraslani
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  4. Fateme Olia
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Correspondence to Amirhossein Amiraslani.

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Appendix

Appendix

See Tables 1, 2, 3, 4, 5 and 6.

Table 1 Finding the determinant of a square matrix in max-plus
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Table 2 Finding the permutation matrix of a square matrix in max-plus
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Table 3 Calculation of matrix multiplication in max-plus
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Table 4 Calculating the LU factors of a square matrix in max-plus
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Table 5 Solving the system \(LX= b\) in max-plus
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Table 6 Solving the system \(UX= b\) in max-plus
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Jamshidvand, S., Ghalandarzadeh, S., Amiraslani, A. et al. Solving linear systems over idempotent semifields through LU-factorization. Rend. Circ. Mat. Palermo, II. Ser 70, 769–791 (2021). https://doi.org/10.1007/s12215-020-00529-y

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  • DOI: https://doi.org/10.1007/s12215-020-00529-y

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