关键词: 牛顿-拉弗森法, 泰勒展开式, 数值分析, 非线性方程组

Abstract: Newton-Raphson method， also known as Newton iteration method， is an approximate method for solving equations， applicable to both complex and real number domains. As one of the most important methods in numerical analysis， it is not only suitable for solving differential equations and integral equations， but also for solving nonlinear equations. This article analyzed the basic principles of the Newton-Raphson method and proved its practical application in solving nonlinear equations through case analysis. 

Key words: Newton-Raphson method； Taylor expansion； numerical analysis； nonlinear equations