hw3 - Homework 3 Math 471 Fall 2007 Assigned Monday Due...

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Homework 3 Math 471, Fall 2007 Assigned: Monday, September 24, 2007 Due: Monday, October 1, 2007 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Newton versus Secant, p. 113, # 12) Newton’s method approximates the zero of f ( x ) = x 3 + 2 x 2 - 3 x - 1 on the interval ( - 3 , - 2) to within 9 . 436 × 10 - 11 in 3 iterations and 6 function evaluations. How many iterations and how many function evaluations are needed by the secant method to approximate this zero to a similar accuracy ? Take p 0 = - 2 aand p 1 = - 3. (2) (Finding Multiple Roots, p. 105 # 11; p. 125 # 14, # 15) (a) The function f ( x ) = 27 x 4 + 162 x 3 - 180 x 2 + 62 - 7 has a zero at x = 1 / 3. Perform ten iterations of Newton’s method on this function, starting with p 0 = 0. What is the apparent order of convergence of the sequence of approximations? What is the multiplicity of the zero at x = 1 / 3? Would the sequence generated by the bisection method converge faster?
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