A New Approach Based on Interval Analysis and B-splines Properties for Solving Bivariate Nonlinear Equations Systems

This paper is addressing the problem of solving non linear systems of equations. It presents a new algorithm based on use of B-spline functions and Interval-Newton’s method. The algorithm generalizes the method designed by Grandine for solving univariate equations. First, recursive bisection is used to separate the roots, then a combination of bisection/Interval Newton’s method is used to refine them. Bisection is ensuring robustness while Newton’s iteration is guaranteeing fast convergence. The algorithm is making great benefit of geometric properties of B-spline functions to avoid unnecessary calculations in both root-separating and root-refining steps. Since B-spline functions can provide an accurate approximation for a wide range of functions (provided they are smooth enough), the algorithm can be made available for those functions by prior conversion/approximation to B-spline basis. It has successfully been used for solving various bivariate nonlinear equations systems.

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