Exclusion regions for parameter-dependent systems of equations

Abstract

This paper presents a new algorithm based on interval methods for rigorously constructing inner estimates of feasible parameter regions together with enclosures of the solution set for parameter-dependent systems of nonlinear equations in low (parameter) dimensions. The proposed method allows to explicitly construct feasible parameter sets around a regular parameter value, and to rigorously enclose a particular solution curve (resp. manifold) by a union of inclusion regions, simultaneously. The method is based on the calculation of inclusion and exclusion regions for zeros of square nonlinear systems of equations. Starting from an approximate solution at a fixed set $p$ of parameters, the new method provides an algorithmic concept on how to construct a box $\mathbf{s}$ around $p$ such that for each element $s\in \mathbf{s}$ in the box the existence of a solution can be proved within certain error bounds.