Some closer bounds of Perron root basing on generalized Perron complement

Abstract

This paper is concerned with the bounds of the Perron root ρ ( A ) of a nonnegative irreducible matrix A . Two new methods utilizing the relationship between the Perron root of a nonnegative irreducible matrix and its generalized Perron complements are presented. The former method is efficient because it gives the bounds for ρ ( A ) only by calculating the row sums of the generalized Perron complement P t ( A / A [ α ] ) or even the row sums of submatrices A [ α ] , A [ β ] , A [ α , β ] and A [ β , α ] . And the latter gives the closest bounds (just in this paper) of ρ ( A ) . The results obtained by these methods largely improve the classical bounds. Numerical examples are given to illustrate the procedure and compare it with others, which shows that these methods are effective.