Mathematics – Rings and Algebras
Scientific paper
2009-03-23
Mathematics
Rings and Algebras
38 pages
Scientific paper
In max algebra it is well-known that the sequence A^k, with A an irreducible square matrix, becomes periodic at sufficiently large k. This raises a number of questions on the periodic regime of A^k and A^k x, for a given vector x. Also, this leads to the concept of attraction cones in max algebra, by which we mean sets of vectors whose ultimate orbit period does not exceed a given number. This paper shows that some of these questions can be solved by matrix squaring (A,A^2,A^4, ...), analogously to recent findings concerning the orbit period in max-min algebra. Hence the computational complexity of such problems is of the order O(n^3 log n). 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. For powers of a visualized matrix in the periodic regime, we observe remarkable symmetry described by circulants and their rectangular generalizations. We exploit this symmetry to derive a concise system of equations for attraction cpne, and we present an algorithm which computes the coefficients of the system.
No associations
LandOfFree
Cyclic classes and attraction cones in max algebra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Cyclic classes and attraction cones in max algebra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cyclic classes and attraction cones in max algebra will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-346823