Final_Fall2010_TakeHome - Introduction to Numerical...

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Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc´ ıa-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presentation of results . Name: Due on Wednesday December 8th, before 12pm. Problem 1: out of 10. Problem 2: out of 10. Problem 3: out of 25. Problem 4: out of 20. Problem 5 (BONUS): out of 20. Problem 6: out of 35. Problem 7: out of 40. Total: out of 140. THESE SHEETS ARE TO BE HANDED IN WITH YOUR EXAM. Page 1 of 5 Please go on to the next page. . .
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Math 104A Final Exam Fall 2010 1. (10 points) Consider the following iterative method to solve f ( x ) = 0: x n +1 = g ( x n ) , where g ( x ) = x - f ( x ) f ( x ) - 1 2 f ( x ) f ( x ) f ( x ) f ( x ) 2 . (a) (5 points) Does the method converge? What is the rate of convergence of the method? Justify your answer. (b) (5 points) Find, using this method, and to within 10 - 10 , the solution to e - x = sin( x ) located in the interval [0 , 1]. 2. (10 points) Find the coefficients a 0 , a 1 , a 2 , and a 3 , such that a 0 f ( x - 2 h ) + a 1 f ( x - h ) + a 2 f ( x + h ) + a 3 f ( x + 2 h ) = f ( x ) + O ( h 4 ) for any smooth function f . Check your answer with f ( x ) = e x cos( x ) at x = 0. Use several values of h and show numerically that the degree of accuracy of the formula that you have obtained is correct.
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