Suppose that f(3) = f'(c) = 0, so that f has a double root at 5,...

 

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Suppose that f(3) = f'(c) = 0, so that f has a double root at 5, and that f" is defined and continuous in a neighborhood of 3. Let (x) be a sequence obtained by Newton's method. (a) Show that 1 (8 - xx)² ƒ" (1k) — (5-xx) 1 2 3-*x+1 where 1 and X both lie between and xx. f" (nk) f" (xx) (b) Suppose, further, that 0 <m <f"(x) < M for all x in the interval [3-6,5+6] for some 8 > 0, where M < 2m; show that if xo lies in this interval the iteration converges to 5, and that convergence is linear with rate constant S=. (Hint: you cannot directly use the theorem given in the lecture notes to conclude it.) Suppose that f(3) = f'(c) = 0, so that f has a double root at 5, and that f" is defined and continuous in a neighborhood of 3. Let (x) be a sequence obtained by Newton's method. (a) Show that 1 (8 - xx)² ƒ" (1k) — (5-xx) 1 2 3-*x+1 where 1 and X both lie between and xx. f" (nk) f" (xx) (b) Suppose, further, that 0 <m <f"(x) < M for all x in the interval [3-6,5+6] for some 8 > 0, where M < 2m; show that if xo lies in this interval the iteration converges to 5, and that convergence is linear with rate constant S=. (Hint: you cannot directly use the theorem given in the lecture notes to conclude it.)