Interval Analysis

6Nonlinear systems of equations

A nonlinear system of n equations with n variables x₁, …, x_n is considered here either as a zero problem f(x) = 0 of a function f : D ⊆ ℝⁿ → ℝⁿ or as a fixed point problem x = g(x) with g : D ⊆ ℝⁿ → ℝⁿ. Newton’s method and the general iteration method are two traditional methods to approximate a zero of f, and a fixed point of g, respectively. We will extend them to interval iterations verifying thus a zero or a fixed point within some iterate. We will get to know additional methods in this chapter. To this end we assume that f and g always have the required smoothness – at least continuity. Mostly it is sufficient that f, g ∈ C²(D) holds.

We will start with the interval Newton method with which we will be able ...