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Numerical Analysis: Fixed Point Iteration 
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Mar2306, 08:59 AM

P: 11

Consider the fixed point iteration formula:
*x_(n+1) = (2/3)[(x_n)^3  1]  3(x_n)^2 + 4x_n = g(x) *Note: "_" precedes a subscript and "^" precedes a superscript (a) Find an interval in which every starting point x_0 will definitely converge to alpha = 1. (b) Show that the order of the above fixed point iteration formula is 2 (quadratic convergence). ======================================= For (a), I took the derivative of g(x) and set it equal to zero. I found that when g'(x) = 2x^2  6x +4 = (2x  2)(x  2)= 0, x = 1, 2. But g'(alpha) = g'(1) = 2  6 + 4 = 0...? I want to say that the interval is (1,2]... For (b), I tried alpha  x_(n + 1) <= c1  x_n^p, where p is the order and c is some constant >= 0. And Newton's method usually converges quadratically... I ended up with: (2/3)(x_n)^3 + 3(x_n)^2  4x_n + (5/3) <= c1  x_n^p I don't know how to conclude that p must be 2... or if this is even right... 


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