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Unformatted text preview: AMATH 341 / CM 271 / CS 371 Assignment 4 Due : Monday February 22, 2010 Instructor: K. D. Papoulia 1. (a) Consider the function f ( x ) = x/ √ x 2 + 1. This function has a unique root at x ∗ = 0. Does Newton’s method converge to the root? Implement it in Matlab, and show that it will indeed converge to the root provided x is suﬃciently close to 0. Also, plot f in Matlab using ezplot . When the method is converging, what is the relationship between | x k − x ∗ | and | x k − 1 − x ∗ | —does the convergence appear to be quadratic? (b) There is a definite cutoff outside of which Newton’s method will not converge. Try to find the cutoff with your Matlab program, and then carry out some hand analysis to confirm it. Hand in: Newton’s method for this problem written (by hand) as a formula, the m-file that implements it, a few sample runs, your computations that try to find the cutoff, and a hand analysis of the cutoff. 2. It seems to be much more diﬃcult to characterize the initial starting points for which the secant method will or will not converge for the function in the previous example. Implement the secant method and see by trial and error if you can find the largest γ such that the secant method will converge provided | x | , | x 1 | ≤ γ and x ̸ = x 1 . Show several examples of convergence when | x | , | x 1 | ≤ γ but an example of divergence when | x | or | x 1 | exceeds γ. 3. Consider the degree-4 polynomial whose roots are 2 . 2, 2 . 3, 2 . 4, 2 . 5. The standard- form coeﬃcients a of this polynomial, a Matlab vector, may be found with Matlab’s poly function. Compute the coeﬃcients using poly and then implement Newton’s method for finding roots of this polynomial; use the termination test on p. 119 of Moler’s book. (a) Show by exhibiting Matlab runs that Newton’s method can find all of these roots provided that it is initialized reasonably close to the root. About how many significant (decimal) digits of accuracy do the roots have? (b) The roots can also be found by Matlab’s roots function. About how many significant digits of accuracy do these roots have? (c) In my computations of (a) and (b), it appears that the roots from Newton’s method are slightly more accurate than the roots from the roots function. Another way to check the accuracy of the roots is to recompute the polynomial’s coeﬃcients from the computed roots (use the poly function again but this time on the computed roots). Call these recomputed coeﬃcients a2 . Then we use norm(a-a2) as a measure of accuracy. According to this test, which seems more accurate, Newton or roots ?...
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This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.
- Spring '10