5. Consider the fixed point method g(x) = 2+ (x-2)5 which has fixed points at x...

 

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5. Consider the fixed point method g(x) = 2+ (x-2)5 which has fixed points at x = 1, 2, 3. (a) Perform the fixed point iteration with a tolerance of 10-10, a max number of steps of 20, and starting point x1 = 2.9. Compute the errors En n - n-1 and find a value c such that the error ratio En ES 'n-1 is roughly constant (not going to 0 or infinity). Display the error ratios in a table. What order of convergence are you observing? Is this expected? Why? (b) Now repeat part a using your Newton's method for your fixed point iteration f(x) f'(x) g(x)= =x- for f(x) = (x - 2)5. What order convergence are you observing? Is this expected? Why? (c) Now repeat part a using Newton's method modified for roots with higher multiplicity which is the fixed point iteration g(x) =x- f(x) f'(x) f'(x) -f(x)f"(x)" What order of convergence are you observing now? Extra Credit: In problem 4b you are using fixed point iterations to find a root of f(x) = x + 4x - 10. Find a new fixed point iteration 95(x) (that's different from 4b i, ii, iii, and iv), show how you found it and make the tables from 4b and 4c with the results using this new fixed point iteration. Comment on the convergence, if it converges, and talk about what this says about your new fixed point iteration. 5. Consider the fixed point method g(x) = 2+ (x-2)5 which has fixed points at x = 1, 2, 3. (a) Perform the fixed point iteration with a tolerance of 10-10, a max number of steps of 20, and starting point x1 = 2.9. Compute the errors En n - n-1 and find a value c such that the error ratio En ES 'n-1 is roughly constant (not going to 0 or infinity). Display the error ratios in a table. What order of convergence are you observing? Is this expected? Why? (b) Now repeat part a using your Newton's method for your fixed point iteration f(x) f'(x) g(x)= =x- for f(x) = (x - 2)5. What order convergence are you observing? Is this expected? Why? (c) Now repeat part a using Newton's method modified for roots with higher multiplicity which is the fixed point iteration g(x) =x- f(x) f'(x) f'(x) -f(x)f"(x)" What order of convergence are you observing now? Extra Credit: In problem 4b you are using fixed point iterations to find a root of f(x) = x + 4x - 10. Find a new fixed point iteration 95(x) (that's different from 4b i, ii, iii, and iv), show how you found it and make the tables from 4b and 4c with the results using this new fixed point iteration. Comment on the convergence, if it converges, and talk about what this says about your new fixed point iteration.