Numerical Methods for Approximating Roots
Numerical mathematics is the way to solve mathematical problems in real life. With it, we find solutions that are "close enough". That is why numerical mathematics is an important part of any program of study requiring applied mathematics. So, of course, it is included in undergraduate engineering, computer sciences and mathematics programs, but it also included in programs in life sciences, economics and finance.
The basis of numerical mathematics is finding "good enough" approximate solutions to equations or, in other words, approximating the roots or zeros of functions. Typically, four basic numerical methods for approximating roots are taught. These are the bisection method, the secant method, the regula falsi method and Newton-Raphson's (or simply Newton's) method.
Unfortunately, the methods are often taught as four completely distinct methods and rarely are problems solved completely and step-by-step. In this book, we explain the differences between the methods and show how to apply the methods in a consistent way. This makes them easier to remember and understand. Also, students are often confused about when to stop, that is, about when the approximation is "good enough", so we explain the two types of "good enough" and to tell the when and how of each type. Finally and most importantly, through videos we give worked examples and show step-by-step from start to finish how to use the methods both with technology (a spreadsheet) and "by hand" (with just a calculator).
The mathematics involved in applying the methods are basic - substituting, solving and simplifying on a first level algebra course. The exception is Newton-Raphson's method where one must find the derivative of the given function (and any mathematics application will do this if the function has a derivative). All of the methods do require that the function be continuous or "unbroken" since this guarantees that between a positive and negative value, a zero (root) must happen, but nothing of this is required in applying the methods.
We mention that we discuss, but do not analyze or prove stability, non-convergence, convergence rates, problems with multiple roots, etc. Our goal is to ensure that the student understands the two main types of approximations, the four basic numerical methods (bisection, secant, regula falsi and Newton's) for approximating roots of functions and can apply them correctly either by using a spreadsheet or other mathematics application or by-hand using a calculator.
An Introduction to Numerical Methods for Approximating Roots
Solving equations or, in other words, finding the roots (zeros) of functions is important to anyone working in applied mathematics. However, many times we cannot find the exact values for the solutions and so we must approximate those using numerical methods. On the other hand, good approximations are usually sufficient in real life.
There are four basic numerical methods for approximating roots of functions. They are the basis for all root-finding algorithms and although many hybrid methods have been developed over the years, the mathematics behind these four methods and their processes and the precautions required for using them are valid and thus important today.
All numerical methods for approximating roots have (a) an algorithm or recipe for calculating a sequence of (hopefully) better and better approximations and (b) a way of testing this "betterness" or "closeness" of the approximations so as to know when to stop calculating.
Each number in the sequence is calculated with the algorithm using the previous number or numbers in the sequence. This is called iterating and each number in the sequence is called an iteration. When we are given the problem, we are told what type of approximation is required and how "close" our approximation must be in the form of an error size. After we calculate each iteration, we test its "closeness" against the error size and when we reach it, we stop iterating.
The characteristics of a numerical method are (a) what is required of the function or equation in order to use the method, (b) what type(s) of errors can we test for with the method, (c) will the method always work and (d) how fast does it work (how many iterations).
The four methods we look at are: the bisection method, the secant method, the regula falsi method and Newton-Raphson's method (or simply Newton's method).