Max-Plus Linear Algebra in Maple and Generalized Solutions for First-Order Ordinary BVPs via Max-Plus Interpolation

If we consider the real numbers extended by minus infinity with the operations maximum and addition, we obtain the max-algebra or the max-plus semiring. The analog of linear algebra for these operations extended to matrices and vectors has been widely studied. We outline some facts on semirings and max-plus linear algebra, in particular, the solution of maxplus linear systems. As an application, we discuss how to compute symbolically generalized solutions for nonlinear first-order ordinary boundary value problems (BVPs) by solving a corresponding maxplus interpolation problem. Finally, we present the Maple package MaxLinearAlgebra and illustrate the implementation and our application with some examples. 1 Semirings and Idempotent Mathematics The max-algebra or max-plus semiring (also known as the schedule algebra) Rmax is the set R∪ {−∞} with the operations a⊕ b = max{a, b} and a⊙ b = a+ b. So for example, 2 ⊕ 3 = 3 and 2 ⊙ 3 = 5. Moreover, we have a ⊕ −∞ = a and a ⊙ 0 = a so that −∞ and 0 are respectively the neutral element for the addition and for the multiplication. Hence Rmax is indeed a semiring, a ring “without minus”, or, more precisely, a triple (S,⊕,⊙) such that (S,⊕) is a commutative additive monoid with neutral element 0, (S,⊙) is a multiplicative monoid with neutral element 1, we have distributivity from both sides, and 0⊙ a = a⊙ 0 = 0. Other examples of semirings are the natural numbers N, the dual Rmin of Rmax (the set R ∪ {∞} and min instead of max), the ideals of a commutative ring with sum and intersection of ideals as operations or the square matrices over a semiring; see [Gol99] for the theory of semirings in general and applications. The semirings Rmax and Rmin are semifields with a (−1) = −a. Moreover, they are idempotent semifields, that is, a⊕ a = a. Note that nontrivial rings cannot be idempotent since then we would have 1 + 1 = 1 and so by subtracting one also 1 = 0. Idempotent semirings are actually “as far away as possible” from being a ring because in such semirings a ⊕ b = 0 ⇒ a = b = 0. Hence zero is the only element with an additive inverse. There is a standard partial order on idempotent semirings defined by a b if a⊕ b = b. For Rmax this is the usual order on R. Due to this order, the theory of idempotent semirings and modules is closely related to lattice theory. Moreover, it is a crucial ingredient for the development of idempotent analysis [KM97], which studies functions with values in an idempotent semiring. The idempotent analog of algebraic geometry over Rmin and Rmax respectively is known as tropical algebraic geometry [RGST05]. For a recent survey on idempotent mathematics and an extensive bibliography we refer to [Lit05]. Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences. E-mail: [email protected]. This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322. I would like to thank Martin Burger for his suggestions to study semirings in connection with nonlinear differential equations and Symbolic Computation and for useful discussions. I also extend my thanks to Markus Rosenkranz and our project leaders Bruno Buchberger and Heinz W. Engl for helpful comments.