Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes

Abstract

In max algebra it is well known that the sequence of max algebraic powers A k , with A an irreducible square matrix, becomes periodic after a finite transient time T ( A ) , and the ultimate period γ is equal to the cyclicity of the critical graph of A . In this connection, we study computational complexity of the following problems: (1) for a given k , compute a periodic power A r with r ≡ k ( mod γ ) and r ⩾ T ( A ) , (2) for a given x , find the ultimate period of { A l ⊗ x } . We show that both problems can be solved by matrix squaring in O ( n 3 log n ) operations. The main idea is to apply an appropriate diagonal similarity scaling A ↦ X - 1 AX , called visualization scaling, and to study the role of cyclic classes of the critical graph.