Verification methods for dense and sparse systems of equations

In this paper we describe verification methods for dense and large sparse systems of linear and nonlinear equations. Most of the methods described have been developed by the author. Other methods are mentioned, but it is not intended to give an overview over existing methods. Many of the results are published in similar form in research papers or books. In this monograph we want to give a concise and compact treatment of some fundamental concepts of the subject. Moreover, many new results are included not being published elsewhere. Among them are the following. A new test for regularity of an interval matrix is given. It is shown that it is significantly better for classes of matrices. Inclusion theorems are formulated for continuous functions not necessarily being differentiable. Some extension of a nonlinear function w.r.t. a point x̃ is used which may be a slope, Jacobian or other. More narrow inclusions and a wider range of applicability (significantly wider input tolerances) are achieved by (i) using slopes rather than Jacobians, (ii) improvement of slopes for transcendental functions, (iii) a two-step approach proving existence in a small and uniqueness in a large interval thus allowing for proving uniqueness in much wider domains and significantly improving the speed, (iv) use of an Einzelschrittverfahren, (v) computing an inclusion of the difference w.r.t. an approximate solution. Methods for problems with parameter dependent input intervals are given yielding inner and outer inclusions. An improvement of the quality of inner inclusions is described. Methods for parametrized sparse nonlinear systems are given for expansion matrix being (i) M-matrix, (ii) symmetric positive definite, (iii) symmetric, (iv) general. A fast interval library having been developed at the author’s institute is presented being significantly faster compared to existing libraries.