Answer the following questions to the best of your ability. Solutions should be

Question

Answer the following questions to the best of your ability. Solutions should be typed. Any plots or graphs should be included with the question (please include the questions in your solutions). For several of the problems listed below you will need to choose your own test functions (Please Choose Your Own Test Functions). Answer the following question in python using sagemathcloud

1. If the root of f (z) 0 is a double root (has multiplicity 2), then Newton's Method can be accelerated by using: n--1 f (zn) Numerically compare the convergence of this scheme with Newton's Method on a func- tion with a known double root.

Explanation / Answer

Consider the function f(x) = x2 - 2*x = 0. This function has a root at x=2. Then, f'(x) = 2x-2.

The term f(x) / f'(x) reduces twice as fast as the original Newton's method. Hence the accelerated Newton's method will converge in 1/2 the steps as the original method.

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