Simple bounds for zeros of systems of equations

Abstract

Let $F:D( \subseteq \mathbb{R}^n ) \to \mathbb{R}^n$be a continuous function, and suppose that for some X_O ∈ D and some nonsingular matrix A the vector δ_O:=A⁻¹ F(x_O) is "small". Assuming the existence of a nonnegative vector c ∈ ℝⁿ such that

$$\left| {F(x) - F(x_0 ) - A(x - x_0 )} \right| \leqslant \left\| {\delta _0 } \right\|c$$

for all x in a suitable neighbourhood S of x_O, a simple condition is given which guarantees that S contains a zero $\hat x$of F. The resulting bounds are shown to be quite accurate. They contain as a special case the bounds obtainable from a theorem of Kantorovic.

It is discussed how to compute the required vector c, and how to make efficient use of sparsity. The practical use of the bounds is demonstrated by an extensive example, the finite difference equations obtained by discretizing the minimal surface equation.