5 - Root-finding techniques for nonlinear equations

Summary

Introduction

As previously discussed in Chapter 2, a system of linear equations can be expressed as Ax = b, where A and b are not functions of x. On the other hand, a system of equations written as Ax = b, in which A and/or bis a function of x, contains one or more nonlinear equations. A polynomial in x of order ≥ 2 has nonlinear dependency on the variable x and is an example of a nonlinear equation. Equations involving trigonometric and/or transcedental functions such as sin(x), cos(x), log(x), e^x, and coth(x) are also nonlinear in x. The solution x of a nonlinear equation f(x) = 0 is termed a root of the equation or a zero of the function f(x). Application of numerical techniques to obtain the roots of nonlinear equations is the focus of this chapter.

Nonlinear equations that are not amenable to an analytical solution may be solved using iterative numerical schemes. A system of consistent linear equations will always yield a solution. In fact, linear equations that are consistent but not underdetermined will always have a unique solution, regardless of the technique used to solve the system of linear equations. On the other hand, a numerical method used to solve a nonlinear equation may not necessarily converge to a solution. Often, certain conditions must be met before a particular numerical technique can successfully produce a solution.