Final_Fall2010_TakeHome - Introduction to Numerical Analysis Math 104A Winter 2009 Instructor Carlos J Garc a-Cervera December 8th 2010 Answer the

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 solvef(x) = 0:xn+1=g(xn),whereg(x) =x-f(x)f(x)-12f(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 toe-x= sin(x)located in the interval [0,1].2. (10 points) Find the coefficientsa0,a1,a2, anda3, such thata0f(x-2h) +a1f(x-h) +a2f(x+h) +a3f(x+ 2h) =f(x) +O(h4)for any smooth functionf. Check your answer withf(x) =excos(x) atx= 0. Use severalvalues ofhand show numerically that the degree of accuracy of the formula that you haveobtained is correct.3. (25 points) The semi perimeters of regular polygons with
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