Consider the iteration In+1=4-(1+4k)r,+k where k ER. (a) Show that, for any kR, a 2 is...

 

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Consider the iteration In+1=4-(1+4k)r,+k where k ER. (a) Show that, for any k€R, a 2 is a fixed point of this iteration. (b) Find the values of k for which the iteration will converge to the fixed point a = 2 (provided that the initial guess r, is chosen sufficiently close to a). (c) For which value of k will the convergence be quadratic? Justify your answer. Consider the iteration In+1=4-(1+4k)r,+k where k ER. (a) Show that, for any k€R, a 2 is a fixed point of this iteration. (b) Find the values of k for which the iteration will converge to the fixed point a = 2 (provided that the initial guess r, is chosen sufficiently close to a). (c) For which value of k will the convergence be quadratic? Justify your answer.