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

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 v...

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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
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Abstract 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.
AbstractList 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.
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.
Author Chueca-Díez, Fernando
Gañán-Calvo, Alfonso M.
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  fullname: Gañán-Calvo, Alfonso M.
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10.1016/j.ins.2020.06.042
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10.1016/0022-247X(90)90103-M
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StartPage 1111
SubjectTerms Algebra
Algorithms
Applications of mathematics
Computer Science
Edge detection
Hyperspaces
Methods
Nonlinear equations
Numeric Computing
Numerical Analysis
Partial differential equations
Theory of Computation
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