A fast numerical algorithm for finding all real solutions to a system of N nonlinear equations in a finite domain

• Published in: Numerical algorithms Vol. 99; no. 3; pp. 1111 - 1125
• Main Authors: Chueca-Díez, Fernando, Gañán-Calvo, Alfonso M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025; Springer Nature B.V
• Subjects: Algebra; Algorithms; Applications of mathematics; Computer Science; Edge detection; Hyperspaces; Methods; Nonlinear equations; Numeric Computing; Numerical Analysis; Partial differential equations; Theory of Computation; Domain discretization; All-solutions; Nonlinear systems; Algebraic equations
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-024-01908-7

A highly recurrent traditional bottleneck in applied mathematics, for which the most popular codes (Mathematica, Matlab, and Python as examples) 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 two similar algorithms of minimum length and computational weight to solve this problem, in which one resembles a graphical tool of edge detection in an image extended to n dimensions. To do this, we discretize the n -dimensional space sector in which the solutions are sought. Once the discretized hypersurfaces (edges) defined by each nonlinear equation of the n -dimensional system have been identified in a single, simultaneous step, the coincidence of the hypersurfaces in each n -dimensional tile or cell containing at least one solution marks the approximate locations of all the hyperpoints that constitute the solutions. This makes the final Newton-Raphson step rapidly convergent to all the existent solutions in the predefined space sector with the desired degree of accuracy.