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Unformatted text preview: Assigned: Friday Oct. 7, 2011 Due Date: Friday Oct. 14, 2011 MATH 174: Homework III Linear Algebra (Part I) Fall 2011 1. The function f ( x ) = cos( x ) has a root at x = / 2. Using the theory we developed for fixed point iterations , find the largest interval around x = / 2 in which we can choose an initial guess for Newtons method and still be guaranteed to converge to / 2. 2. Show that the iteration scheme n +1 = g ( n ) = 2 n- a n + a 2 + 5 a n + 5 , n converges to the fixed point a quadratically (i.e., order of convergence is 2) for all a 6 =- 5. ( HINT: subtract a from both sides, manipulate the expression so that it has the same form as in the definition of order of convergence, then take the limit as n . Note that n a as n . This problem is very similar to Problem #4 of Project #1.) 3. Consider the following 2 matrices, A = a b c d , B = 1 2 2 1 ....
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