solving multiple nonlinear algebraic equations newton' method the idea of newton' method is that we have some approximation to the root and seek new (and hopefully betterapproximation by approximating by linear function and solve the corresponding linear system of algebraic equations we approximate the nonlinear problem by the linear problem ( where is just another notation for rf the equation ( is linear system with coefficient matrix and right-hand side vector we therefore write this system in the more familiar form / /where we have introduce symbol for the unknown vector that multiplies the jacobian the -th iteration of newton' method for systems of algebraic equations consists of two steps solve the linear system / with respect to set solving systems of linear equations must make use of appropriate software gaussian elimination is the most commonand in general the most robustmethod for this purpose python' numpy package has module linalg that interfaces the well-known lapack package with high-quality and very well tested subroutines for linear algebra the statement numpy linalg solve(absolves system ax with lapack method based on gaussian elimination when nonlinear systems of algebraic equations arise from discretization of partial differential equationsthe jacobian is very often sparsei most of its elements are zero in such cases it is important to use algorithms that can take advantage of the many zeros gaussian elimination is then slow methodand (muchfaster methods are based on iterative techniques implementation here is very simple implementation of newton' method for systems of nonlinear algebraic equationsimport numpy as np def newton_system(fjxeps)""solve nonlinear system = by newton' method is the jacobian of both and must be functions of at inputx holds the start value the iteration continues until || |eps ""