Componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces

Abstract

In this paper we use interval arithmetic tools for the computation of componentwise inclusion and exclusion sets for solutions of quadratic equations in finite dimensional spaces. We define a mapping for which under certain assumptions we can construct an interval vector which is mapped into itself. Using Brouwer's fixed point theorem we conclude the existence of a solution of the original equation in this interval vector. Under different assumptions we can construct an interval vector such that the range of the mapping has no point in common with this interval vector. This implies that there is no solution in this interval vector. Furthermore we consider an iteration method which improves componentwise errorbounds for a solution of a quadratic. The theoretical results of this paper are demonstrated by some numerical examples using the algebraic eigenvalue problem which is probably the best known example of a quadratic equation.