Abstract
The aim of this paper is to study the bisection method in Rn. We propose a multivariate bisection method supported by the Poincare–Miranda theorem in order to solve non-linear systems of equations. Given an initial cube satisfying the hypothesis of the Poincare–Miranda theorem, the algorithm performs congruent refinements through its center by generating a root approximation. Through preconditioning we will prove the local convergence of this new root finder methodology and moreover we will perform a numerical implementation for the two dimensional case.
Highlights
The problem of finding numerical approximations to the roots of a non-linear system of equations was the subject of various studies, and different methodologies have been proposed between optimization and Newton’s procedures
The classical Bolzano’s theorem or Intermediate Value theorem ensures that a continuous function that changes sign in an interval has a root, that is, if f : [a, b] → R is continuous and f (a)f (b) < 0 there exists c ∈ (a, b) such that f (c) = 0
Due to the preconditioning at each step we could prove a local convergence theorem and we found an error estimation
Summary
The problem of finding numerical approximations to the roots of a non-linear system of equations was the subject of various studies, and different methodologies have been proposed between optimization and Newton’s procedures. H. Lehmer proposed a method for solving polynomial equations in the complex plane testing increasingly smaller disks for the presence or absence of roots. In the multidimensional case the generalization of this result is the known Poincare– Miranda theorem that ensures that if we have f1, . Xn and the variables are subjected to vary between ai and −ai, if fi(x1, . For different proofs of the Poincare–Miranda theorem in the n-dimensional case, see [3, 10]
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