The Computing Procedure for the Multiparametric Eigenproblem in Max Algebra

Denote a⊕ b = max(a,b), and a⊗ b = a+ b, for a,b ∈ R and extend this pair of operations to matrices and vectors in the same way as in conventional linear algebra, that is if A = (ai j),B = (bi j),C = (ci j) are real matrices or vectors of compatible sizes then C = A⊗ B if ci j = ⊕ k aik ⊗ bk j for all i, j. For any n× n matrix A = (ai j) and for arbitrary sequence of real parameters α = (α1, . . . ,αp), p ≤ n, the problem of finding all x(α) and λ (A(α)) satisfying A(α)⊗ x(α) = λ (A(α))⊗ x(α) is studied. The problem is calledMultiparametric Eigenproblem (in short: MPE). We introduce some properties of the general MPE and we suggest a pseudopolynomial O(pn5) algorithm for computing all eigenvectors of MPE.