Microfluidics: Modeling, Mechanics and Mathematics

Chapter 26

Numerical Solutions to Nonlinear Systems: Newton’s Method

26.1 Introduction

Having derived some of the most common methods for solving linear systems of equations, we will now turn to a significantly more common problem, i.e., finding solutions to nonlinear systems of equations. In these equations the independent variables are given in the form of nonlinear equations, e.g., as sine or cosine functions, polynomials, and exponentials. Again, we will encounter a system of equations of the form

$\begin{array}{l}{f}_0\left(x_0,x_1,\dots ,x_n\right)=0\hfill \\ f_1\left(x_0,x_1,\dots ,x_n\right)=0\hfill \\ f_2\left(x_0,x_1,\dots ,x_n\right)=0\hfill \\ \dots \hfill \\ f_n\left(x_0,x_1,\dots ,x_n\right)=0\hfill \end{array}$

  (Eq. 26.1)

where f_n are nonlinear functions. Again, we are interested in the ...