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Simon Fraser MACM 316 Midterm Questions




Question;Simon Fraser University MACM 316 - Numerical AnalysisSpring term 2013Problem: Consider the following equationf (x) = x? 2?xRecall that:di) (a?x) = a?x ln a ii) ln 2 0.693147dx1. 2 ptsHow many (real) solutions does the original equation f (x) = 0 have? Explain.2. 2 ptGiven a root-finding problem f (p) = 0, define an initial interval [a, b] in order to apply theBisection method.3. 3 ptsDetermine the number of iterations necessary to find a solution accurate to within 10?7 using the Bisection method on the interval defined in (2), i.e. | p? pn |? 10?7.Apply three consecutive iterations of the Bisection method on the interval defined in (2) using three-digit rounding arithmetic.5. 3 ptsApply Newton?s method to recover | f (pn) | < 0.005. Use four-digit chopping arithmetic.Change the original equation to a fixed-point form x = g(x) and define the sequence pn+1 = g(pn).Solution:A fixed-point form is7. 3 ptsTwo fixed-point forms g1(x) and g2(x) have been defined. Which of these forms exhibits more rapid convergence, if g0 (x)? 0.15 and g0 (x)? 0.87 near the fixed point?1 28. 2 ptsWhat should one expect when using the Secant method on a 32-bit architecture, and starting with the two initial approximations p0 = (9A27)16 and p1 = (FECA)16?bonus. 2 ptsDetermine the second Taylor polynomial for f (x) about x0 = 0. ApproximateZ 0.4using P2(x).f (x) dx0


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