CHAPTER 1 Mathematical Preliminaries and Error Analysis Clearly, the ideal gas law is suspect, but before concluding that the law is invalid in this situation we should examine the data to see whether the error could be attributed to the experimental results. If so, we might be able to determine how much more accurate our experimental results would need to be to ensure that an error of this magnitude did not Analysis of the error involved in calculations is an important topic in numerical analysi and is introduced in Section 1. 2. This particular application is considered in Exercise 28 of that section This chapter contains a short review of those topics from single-variable calculus that will be needed in later chapters. A solid knowledge of calculus is essential for an understand ing of the analysis of numerical techniques, and more thorough review might be needed if you have been away from this subject for a while. In addition there is an introduction to onvergence, error analysis, the machine representation of numbers, and some techniques for categorizing and minimizing computational error. 1.1 Review of calculus Limits and Continuity The concepts of limit and continuity of a function are fundamental to the study of calculus, and form the basis for the analysis of numerical technique Definition 1.1 A function f defined on a set X of real numbers has the limit L at xo, written lim f(x)=L, x→ if, given any real number e>0, there exists a real number 8>0 such that lf(x)-L|<E, whenever x∈Xand0<x-xol<δ. (See Figure 1. 1.) 1.1 y=f(r) Copyright 2010 Cengage Learning. All Rights t materially affect the overall leaming eaperience Cengage Learning reserves the right to remo rty commen may be suppressed from the eBook andor eChaptert'sh. May no be copied, scanned, or duplicated, in whole or in part Due to2 CHAPTER 1 Mathematical Preliminaries and Error Analysis Clearly, the ideal gas law is suspect, but before concluding that the law is invalid in this situation, we should examine the data to see whether the error could be attributed to the experimental results. If so, we might be able to determine how much more accurate our experimental results would need to be to ensure that an error of this magnitude did not occur. Analysis of the error involved in calculations is an important topic in numerical analysis and is introduced in Section 1.2. This particular application is considered in Exercise 28 of that section. This chapter contains a short review of those topics from single-variable calculus that will be needed in later chapters. A solid knowledge of calculus is essential for an understand￾ing of the analysis of numerical techniques, and more thorough review might be needed if you have been away from this subject for a while. In addition there is an introduction to convergence, error analysis, the machine representation of numbers, and some techniques for categorizing and minimizing computational error. 1.1 Review of Calculus Limits and Continuity The concepts of limit and continuity of a function are fundamental to the study of calculus, and form the basis for the analysis of numerical techniques. Definition 1.1 A function f defined on a set X of real numbers has the limit L at x0, written lim x→x0 f (x) = L, if, given any real number ε > 0, there exists a real number δ > 0 such that |f (x) − L| < ε, whenever x ∈ X and 0 < |x − x0| < δ. (See Figure 1.1.) Figure 1.1 x L ε L ε L x0 δ x0 x0 δ y y  f(x) Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.