# The multivariate bisection algorithm

## TL;DR

This paper introduces a multivariate bisection algorithm based on the Poincaré-Miranda theorem for solving nonlinear systems in ^n, demonstrating local convergence and providing a numerical implementation in two dimensions.

## Contribution

It proposes a novel multivariate bisection method supported by the Poincaré-Miranda theorem, with proven local convergence and practical numerical implementation.

## Key findings

- Algorithm converges locally for systems satisfying Poincare9-Miranda conditions.
- Numerical implementation successfully applied in two-dimensional cases.
- Provides a new root-finding approach for nonlinear systems in higher dimensions.

## Abstract

The aim of this paper is the study of the bisection method in $\mathbb{R}^n$. In this work we propose a multivariate bisection method supported by the Poincar\'e-Miranda theorem in order to solve non-linear system of equations. Given an initial cube verifying the hypothesis of Poincar\'e-Miranda theorem the algorithm performs congruent refinements throughout its center by generating a root approximation. Throughout 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.