Relationship Theorem between Nonlinear Polynomial Equations and the Corresponding Jacobian Matrix

DOI10.1515/IJNSNS.2000.1.1.7zbMATH Open0966.65055arXivmath/9906054OpenAlexW1970144888MaRDI QIDQ4528725

Publication date: 11 February 2001

Published in: International Journal of Nonlinear Sciences and Numerical Simulation (Search for Journal in Brave)

Abstract: This paper provides a general proof of a relationship theorem between nonlinear analogue polynomial equations and the corresponding Jacobian matrix, presented recently by the present author. This theorem is also verified generally effective for all nonlinear polynomial algebraic system of equations. As two particular applications of this theorem, we gave a Newton formula without requiring the evaluation of nonlinear function vector as well as a simple formula to estimate the relative error of the approximate Jacobian matrix. Finally, some possible applications of this theorem in nonlinear system analysis are discussed.

Full work available at URL: https://arxiv.org/abs/math/9906054

zbMATH Keywords

Newton method nonlinear initial value problems Jacobian matrix nonlinear stability analysis nonlinear polynomial equations

Mathematics Subject Classification ID

Numerical computation of solutions to systems of equations (65H10) Ordinary representations and characters (20C15) Nonlinear ordinary differential equations and systems (34A34) Stability and convergence of numerical methods for ordinary differential equations (65L20) Numerical methods for initial value problems involving ordinary differential equations (65L05) Real polynomials: location of zeros (26C10)

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