chapter2 - 28 Chapter 2 Numerical Exploration 2.1...

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Chapter 2 Numerical Exploration § 2.1 Discovering Limits The for -loop and the while -loop § 2.2 Floating Point Terrain The floating point representation, exponent, mantissa, overflow, underflow, ma- chine precision, eps , inf , realmax , roundoff error § 2.3 Confirming Conjectures Analyzing errors § 2.4 Interactive Frameworks The type char , testing All work and no play does not a computational scientist make. It is essential to be able to play with a computational idea before moving on to its formal codification and development. This is very much a comment about the role of intuition. A computational experiment can get our mind moving in a creative direction. In that sense, merely watching what a program does is no different then watching a chemistry experiment unfold: it gets us to think about concepts and relationships. It builds intuition. The chapter begins with a small example to illustrate this point. The area of a circle is computed as a limit of regular polygon areas. We “discover” π by writing and running a sequence of programs. Sometimes our understanding of an established result is solidified by experiments that confirm its correctness. In § 2.2 we check out a theorem from number theory that says 3 2 k +1 +2 k is divisible by 7 for all positive integers k . To set the stage for more involved “computational trips” into mathematics and science, we explore the landscape of floating point numbers. The terrain is finite and dangerous . Our aim is simply to build a working intuition for the limits of floating point arithmetic. Formal models are not developed. We’re quite happy just to run a few well chosen computational experiments that show the lay of the land and build an appreciation for the inexactitude of real arithmetic. 29
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30 Chapter 2. Numerical Exploration Figure 2.1 Regular n -gons Figure 2.2 Inscribed n -gon The design of effective problem-solving environments for the computational scientist is a research area of immense importance. The goal is to shorten the path from concept to computer program. We have much to say about this throughout the text, In § 2.4 we develop the notion of an interactive framework that fosters the exploration of elementary computational ideas. 2.1 Limits A polygon with n equal sides is called a regular n -gon . Figure 2.1 illustrates two cases. Given n equally spaced points around a circle C , there are two ways to construct a regular n -gon. One is simply to connect the points in order. Each point is then a vertex of the n -gon which is said to be inscribed in C . See Figure 2.2 . On the other hand, the tangent lines at each point define a regular n -gon that circumscribes C . See Figure 2.3 . If C has radius one, then the areas of these two regular n -gons are given by A n = ( n/ 2) sin(2 π/n ) (Inscribed)
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2.1. Limits 31 Figure 2.3 Circumscribed n -gon B n = n tan( π/n ) (Circumscribed) These formulas can be derived by chopping the n -gon into n equal triangles and summing their areas. Now for any value of n that satisfies n 3, the fragment % Fragment A innerA= (n/2)*sin(2*pi/n);
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