An approach to solving nonlinear algebraic systems

Abstract

A new method for solving nonlinear algebraic systems with two (one) unknowns is suggested which makes it possible without knowing initial approximations to find all zero-dimensional roots of a system and a regular linear pencil of matrices polynomially dependent on a parameter whose eigenvalues coincide with one-dimensional roots of the original nonlinear system.The method is based on the relationship between solutions to a nonlinear algebraic system and eigenvalues of certain algebraic spectral problems. This relationship is established and algorithms are given to solve auxiliary spectral problems required in solving nonlinear systems.The paper describes methods for solving systems of nonlinear algebraic equations which enable us to find the roots without knowing initial approximations to them.Section 1 establishes a relationship between the solution of a nonlinear algebraic system and eigenvalues of certain spectral algebraic problems and suggests a new method for solving nonlinear algebraic systems. The method makes it possible to find all zero-dimensional solutions to the system and also a linear regular pencil polynomially dependent on a parameter whose eigenvalues coincide with onedimensional roots of the nonlinear algebraic system. Section 2 suggests an algorithm for computing all the roots of the nonlinear algebraic system with one unknown. The algorithm is based on the relationship between the sought roots and eigenvalues of a linear pencil.The description and justification of the algorithm suggested are given for systems of nonlinear algebraic equations with one and two unknowns. However, the approach remains the same for systems with the number of unknowns greater than two. No restrictions are imposed on the number of equations of the system to be solved.Section 3 considers the procedure for refining the approximately computed isolated roots of the system. Section 4 presents some recommendations and describes algorithms for solving certain auxiliary spectral algebraic problems required in solving nonlinear algebraic systems.Sections outlines the known elimination method for solving systems of two nonlinear algebraic equations with two unknowns which is based on the resultant theory.Section 6 suggests a modification of the elimination method reducing its main steps to solution of spectral problems for polynomial pencils.

Solution of systems of nonlinear algebraic equations in three variables. Methods and algorithms. 3

“…(4ii) Find the solutions #~i of SNAE (4.5), for instance, by applying the algorithm from [13]. Then the triples (#~i, #~, #~) provide the zero-dimensional roots of system (1.4) generated by the point (#~, #~) under consideration of the discrete spectrum of 9.1k(#2, #3).…”

“…The second method for solving SNAEs-3 is an extended and modified version of the method for solving SNAEs-2 suggested in [13]. Consider this modification in the context of solving SNAEs-2.…”

“…The first and third methods are extensions of the methods for solving SNAEs-2 suggested in [1]. The second method is a modified version of the method for solving SNAEs-2 proposed in [13]. In the context of SNAEs-2, this modification is considered in See.…”

An approach to solving nonlinear algebraic systems. 2

“…The method proposed by the authors in [1] for solving SNAE of two variables also requires the computation of eigenvalues of some polynomial matrix, but it produces extraneous (parasitic) roots and so demands an increased precision for computation and additional analysis to establish whether the computed roots are roots of the original SNAE.…”

“…The first method decribed below does not produce extraneous roots and so is more economic than the method in [1]. The second method is based on the so-called AW-2 factorization [7] of a certain two-parameter polynomial matrix constructed by means of the equations of the original SNAE with the subsequent solution of several nonlinear algebraic systems in one variable.…”