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Chapter 1 Taylor Series Truncation Error and Significant Figures

# Chapter 1 Taylor Series Truncation Error and Significant Figures

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1 CHAPTER 1: Taylor Series, Truncation Error, and Significant Figures A numerical method is a mathematical procedure that produces an approximate solution to an equation or a system of equations. Through most of undergraduate algebra, calculus, and differential equations, engineers and applied scientists focus on finding exact (analytical) solutions to equations. These solutions usually take the form of algebraic expressions. A numerical method instead produces numbers. When a mathematical problem has a solution, but the analytical solution does not exist, cannot be found, or is too complex to be useful, then we use a numerical method to find the numbers that satisfy the equations. Many of the equations that describe the real world of science and engineering do not have analytical solutions, so numerical methods are required. Because a numerical method produces numbers rather than expressions, solving a problem with a numerical method means determining significant figures. Not knowing the significant figures in a numerical solution means not knowing the solution. The numbers in a numerical solution are seldom exact, but instead contain some amount of error. The error arises from two sources, truncation error and round-off error. In a Taylor series approximation, limiting the number of terms in the series causes the truncation error. Finite word length in computer memory causes the round-off error. We address the issue of truncation error in this chapter but delay a discussion of round-off error for Chapter 2. The approach we develop in this chapter allows us to assess the accuracy of our numerical approximations even when no exact solution exists for comparison. To illustrate all of these concepts, as well as to develop an important tool for deriving and analyzing many of the numerical methods introduced in the rest of the text, this chapter explores the Taylor series expansion. Taylor's theorem is the cornerstone of numerical methods. In this chapter we also use the Taylor series to explain the very important concepts of truncation error and significant figures. Throughout the subsequent chapters of this book we make extensive use of the Taylor Series to derive many of the numerical methods we introduce and explore. We also use the Taylor Series to understand how a numerical method converges to an approximation of the solution with the desired number of significant figures. In the process we explore some important practical applications of the Taylor Series, such as its use in calculators and computer. You should not leave this chapter without understanding Taylor series, truncation error, convergence, and significant figures! Finally, we will use Excel and Visual Basic (VB) in this chapter to approximate a function using its truncated Taylor series. This illustrates how an analysis of truncation error can be used to estimate significant figures, but it also serves as a tutorial for the reader who is unfamiliar with Excel or with basic programming. During

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Chapter 1 Taylor Series Truncation Error and Significant Figures

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