2023/12/06 by Fernando Chueca-Díez, Alfonso M. Gañán‐Calvo, Chueca-Diez, Fernando +1
Mathematics · Physics and Astronomy · #FOS: Electrical engineering #Iterative Methods for Nonlinear Equations #Model Reduction and Neural Networks #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2312.03927
openalex publication_date 2023/12/06 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28
A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica and Matlab) do not offer a solution, is to find all the real solutions of a system of N nonlinear equations in a certain finite domain of the N-dimensional space of variables. We present an algorithm of minimum length and computational weight to solve this problem, resembling a graphical tool of edge detection in an image extended to N dimensions. Once the hypersurfaces (edges) defined by each nonlinear equation have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in the vicinity of all the hyperpoints that constitute the solutions makes the final Newton-Raphson step rapidly convergent to all the solutions with the desired degree of accuracy. As long as N remains smaller than about five, which is often the case for physical systems that depend on fewer than five parameters, this approach demonstrates excellent effectiveness.