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hw9 - A has α on the main diagonal α 2 on the diagonal...

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Math 128a, fall 2011, Chorin, theory homework 9, due the week of Nov. 13. 1. Consider iterations of the form x n +1 = g ( x n ). For the following functions g ( x ), find the intervals of x 0 (the starting guess) for which these iterations converge, and for each such interval, what solution α of α = g ( α ) it converges to: (i) g ( x ) = x 3 ; (ii) g ( x ) = x - ( x 3 - 2) / (3 x 2 ), (iii) g ( x ) = x - ( e x - 2) /e x . 2. Suppose you want solve the linear system of equations Ax = f , where A is an N by N matrix, f is an a given vector, and x is your solution. Consider the iteration scheme x n +1 = x n + ( Ax n - f ), in two cases: (i) A = αI , where I is the identity (1 on the main diagonal and 0 elsewhere), and (ii)
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Unformatted text preview: A has α on the main diagonal, α 2 on the diagonal just above the main diagonal, and 0 elsewhere. In each case, find values of α and starting vectors x for which the iteration converges. (Hint: in case (ii), think of matrix norms). 3. Suppose you want to find the minimum of the function f ( x,y ) = (cos( x )-1) 2 + ( x + y + 7) 2 . Write down the Newton scheme for doing this (I am not asking you to run it or to analyze its convergence, just to write down the equations; you do not have to invert the matrices either). 1...
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