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7Differentiation & SearchingIn this chapter we add two more tools to our computational toolbox: numerical differentiationand trial-and-error searching. In Unit I we derive the forward-difference,central-difference, and extrapolated-difference methods for differentiation. They willbe used throughout the book, especially for partial differential equations. In Unit IIwe devise ways to search for solutions to nonlinear equations by trial and error andapply our new-found numerical differentiation tools there. Although trial-and-errorsearching may not sound very precise, it is in fact widely used to solve problems whereanalytic solutions do not exist or are not practical. In Chapter 8, “Solving Systemsof Equations with Matrices; Data Fitting,” we extend these search and differentiationtechniques to the solution of simultaneous equations using matrix techniques.In Chapter 9, “Differential Equation Applications,” we combine trial-and-errorsearching with the solution of ordinary differential equations to solve the quantumeigenvalue problem.7.1 Unit I. Numerical DifferentiationProblem: Figure 7.1 shows the trajectory of a projectile with air resistance. Thedots indicate the times t at which measurements were made and tabulated. Yourproblem is to determine the velocity dy/dt ≡ y ′ as a function of time. Note that sincethere is realistic air resistance present, there is no analytic function to differentiate,only this table of numbers.You probably did rather well in your first calculus course and feel competent attaking derivatives. However, you may not ever have taken derivatives of a table ofnumbers using the elementary definitiondy(t)dxdef y(t + h) − y(t)= lim. (7.1)h→0 hIn fact, even a computer runs into errors with this kind of limit because it iswrought with subtractive cancellation; the computer’s finite word length causes thenumerator to fluctuate between 0 and the machine precision ɛ m as the denominatorapproaches zero.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 146