The distance of potentially stable sign patterns to the unstable matrices
Date
2001
Authors
Lin, Qing
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
The relative distance of a potentially stable sign pattern to the unstable matrices is introduced. The computation of the precise value of this distance is a nonlinear optimization problem. For order 2, an optimal stable matrix in the unique minimally potentially stable sign pattern is determined analytically. This analysis of the order 2 minimally potentially stable sign pattern shows how difficult the computation of this distance can be. Furthermore. it is shown that this distance can be arbitrarily small for potentially stable sign patterns of large order. Therefore, an adequate estimation of this distance becomes increasingly important.
A solution is introduced for such an estimation. Firstly, guided by the heuristic hierarchy of the minimally potentially stable tree sign patterns, graph theory is used to obtain a good minimal subpattern. Secondly, a sequence of algorithms is introduced to estimate the distance to the unstable matrices of the minimal components of this subpattern that. have a properly signed nest. The algorithms are applied to those order 3 and 4 minimally potentially stable tree sign patterns that have a properly signed nest, as well as to order 5 minimally potentially stable rooted tree sign pat.terns. An ad hoc procedure is used to find a good st.able matrix in each of the two minimally potentially stable tree sign patterns of order 4 that. do not have a properly signed nest.