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# hw4 - Homework#4 1(a Starting with the initial guess of 1...

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Homework #4 1. (a) Starting with the initial guess of 1, find three additional approximations to the root of f ( x ) = x 2 - 2 using Newton’s method. (b) What is the absolute error of the final approximation? 2. (a) Consider f ( x ) = ( x, x 0 - - x, x < 0 . Starting with the initial guess of x 0 = a > 0, find three additional approximations to the root of f ( x ) using Newton’s method. (b) Give a graphical description of how Newton’s method is arriving at the approxi- mations. (c) Does Newton’s method converge to the exact root at 0 for x 0 sufficiently close to 0? Why does this not violate the theorem on the convergence of Newton’s method? 3. (a) Starting with the initial guess of x 0 = 1, find three additional approximations to the root of f ( x ) = ( x - 2) 2 using Newton’s method. (b) What is the absolute error of the initial guess and each of the three approxima- tions? Note the exact root is x * = 2. (c) What is | x * - x k +1 | / | x * - x k | for k = 0 , 1 , 2? From this, what do you guess to be the order of convergence and asymptotic error constant in this case?

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