Chebyshev Subdivision and Reduction Methods for Solving Multivariable Systems of Equations

We present a new algorithm for finding isolated zeros of a system ofreal-valued functions in a bounded interval in $\mathbb{R}^n$. It uses theChebyshev proxy method combined with a mixture of subdivision, reductionmethods, and elimination checks that leverage special properties of Chebyshevpolynomials. We prove the method has R-quadratic convergence locally nearsimple zeros of the system. We also analyze the temporal complexity and thenumerical stability of the algorithm and provide numerical evidence indimensions up to three that the method is both fast and accurate on a widerange of problems. The algorithm should also work well in higher dimensions.Our tests show that the algorithm outperforms other standard methods on thisproblem of finding all real zeros in a bounded domain. Our Pythonimplementation of the algorithm is publicly available.