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Unformatted text preview: Homework # 2 Due Thursday, 01/24 1. Find the solution of x 3 = x 2 + x + 1 using: (a) the bisection method (What initial interval did you choose? How did you choose it?) (b) Newtons method (What initial iterate did you choose? How did you choose it?) (c) the secant method (What initial iterates did you choose? How did you choose them?) In each case, use the stopping criterium | p n- p n- 1 | 10- 6 . How many iterates were needed in each case to reach this accuracy. Which method is the fastest to converge? The slowest? (Hint: Before to do any coding, plot the function with matlab so that you can see roughly where is the zero.) 2. Repeat problem 1., but this time with the equation e x = 1 . 1 + x 2 3. Newtons method is the commonly used method for calculating square roots on a computer. (a) What equation would you solve in order to find a ? (b) Show that in this case, Newtons method reduces to the following iteration: p n +1 = 1 2 p n + a p n (1) (c) Explain with a picture why, for any initial iterate...
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- Spring '10