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# Convergence to a Fixed Point of a Function and Matlab Program for the Newton-Raphson Method.

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1. Let g: R&#8594;R+ be such a function that g&#8712; C^1(R) and for all x &#8712; R, -1 <g'(x) < 0.
Show that the sequence Xn+1 : = g(Xn) converges to the unique fixed point of the function g, regardless of chioce Xo &#8712; R.
[ Note : Observe that the domain of function g is not a compact interval.]

2. Write a matlab program (Newton - Raphson ) for finding the root of the function f(x) = x^5 - 2x^3 + x + 2. Be as much accurate as you as you can. To proceed, store this function in an m-file, say f.m, and its derivative in df.m. Label the whole procedure newt.m.

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1. let g: R→R+ be such a function that g∈ C^1(R) and for all x ∈ R, -1 <g'(x) < 0.
Show that the sequence Xn+1 : = g(Xn) converges tp the unique fixed point of the ...

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