Implement Newton\'s method. Use it to find the solution to 14 digits

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(10 points) Programming: fired-point iteration/Newton's method. Consider the nonlinear equation cosx + 1/(1 + c-22) = 0. Find the smallest positive root, using initial guess xo = 3. 

(i) Implement Newton's method. Use it to find the solution to 14 digits (absolute error). How many steps do you need? 

(ii) Consider the following iterations: 

• rati = arccos(-1/(1 + • xko = 0.5In(-1/(1 + 1/ =6'0)- 1 

(a) Show that they are fixed-point iterations whose fixed-points are a solution to the original equation. 

(b) Do they converge? If so, what is the rate of convergence? Use the theory of fixed-point iteration to analyze them. (You can use the numerical value of the root obtained by Newton's method.) .

(c) Implement them and verify your conclusions numerically. 

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