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Berkeley MATH 128A - Math128A - Numerical Analysis Sample Midterm I

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Prof. Ming Gu, 861 Evans, tel: 2-3145Email: [email protected]://www.math.berkeley.edu/∼mgu/MA128A2008SMath128A: Numerical Analysis SampleMidterm IThis is a closed book, closed notes exam. You need to justify every one of youranswers. Completely correct answers given without justification will receive littlecredit. Do as much as you can. Partial solutions will get partial credit. Look overthe whole exam to find problems that you can do quickly. You need not simplify youranswers unless you are specifically asked to do so.Problem Maximum Score Your Score1 42 243 244 245 24Total 1001. (4 Points)Your Name:Your SID:Your GSI:Math128A: Numerical Analysis Sample Midterm I 22. (24 Points) Show that the cubic equation 2x3− 6x + 1 = 0 has a real root in the interval[01]. Perform one step of Bisection method with this interval. Reformulate this equation asa fixed point equation, and perform one step of fixed point iteration.Math128A: Numerical Analysis Sample Midterm I 33. (24 Points) Let x0< x1< x2. Find a second degree polynomial P (x) such thatP (x0) = f0, P (x1) = f1, and P0(x2) = f02.Hint: Write P (x) asP (x) = α + β(x − x0) + γ(x − x0)(x − x1)and then determine the coefficients from the given conditions.Math128A: Numerical Analysis Sample Midterm I 44. (24 Points) Let a > 0. The cubic root a1/3is the unique positive root of the equationx3− a = 0.(a) Define the Newton iteration for solving this equation.(b) Given a = 2 and x0= 1, compute the first two iterates x1and x2in the Newtoniteration.(c) Define the order of convergence for any convergent iteration. Show this iteration is 2ndorder convergent.Math128A: Numerical Analysis Sample Midterm I 55. (24 Points) Define the absolute error, the relative error, and the number of significant


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