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# hwk3 - -10(start with p = 1(d How many iterations are...

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Homework #3 Due Thursday, 01/31 1. We have seen in class that, if f ( x ) has a root of multiplicity m 2, then Newton’s method converges only with order 1. Here is a modified version of Newton’s method: p n +1 = p n - m f ( p n ) f ( p n ) Show that, if it converges, then it has to converge with order 2. 2. Let f ( x ) = e x - x - 1 (a) Plot f ( x ) with matlab. Where is the zero (just guess with the picture)? (b) Show that the zero is actually where you thought it was, and show that it has multiplicity 2. (c) How many iterations are needed with Newton’s method to find this zero with accuracy 10
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Unformatted text preview: -10 ? (start with p = 1) (d) How many iterations are needed with the modiﬁed version of Newton’s method that we have seen in problem 1.? 3. Textbook: Problem 15 page 27 (a), (b) and (c) 4. Complete the following sentences: The distance between the number from 15.a) and the next largest machine number is 2-m where m = The distance between the number from 15.c) and the next largest machine number is 2-m where m = Show your reasoning (no hand-waving argument, be clear). 1...
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