A Symbolic-Numerical Method for Finding a Real Solution of an Arbitrary System of Nonlinear Algebraic Equations
Copyright policy )A Symbolic-Numerical Method for Finding a Real Solution of an Arbitrary System of Nonlinear Algebraic Equations
A hybrid symbolic-numerical method is proposed for finding solutions of systems of polynomial equations which may be overdetermined or underdetermined. A reduction algorithm due to \textit{J. F. Ritt} [Differential equations from the algebraic standpoint (1932; Zbl 0005.39404)] is used to derive sets of regular systems from the original system. The regular systems have Jacobians which are of full rank. For such systems a Gauss-Newton method can be applied. A convergence result analogous to the Newton-Kantorovich theory is given for an appropriate starting value for a regular system. Two numerical examples are given involving an underdetermined system and an overdetermined system.
numerical examples, Gauss-Newton method, convergence, Algebra and Number Theory, overdetermined system, Numerical computation of solutions to systems of equations, symbolic-numerical methods, Symbolic computation and algebraic computation, reduction algorithm, Real polynomials: location of zeros, Computational Mathematics, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), underdetermined system, Computational aspects of field theory and polynomials, system of nonlinear algebraic equations
numerical examples, Gauss-Newton method, convergence, Algebra and Number Theory, overdetermined system, Numerical computation of solutions to systems of equations, symbolic-numerical methods, Symbolic computation and algebraic computation, reduction algorithm, Real polynomials: location of zeros, Computational Mathematics, Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral), underdetermined system, Computational aspects of field theory and polynomials, system of nonlinear algebraic equations
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