solve q5 and q7 Suppose that f (x) has a root at x = p. Moreover, assume that |f

Question

solve q5 and q7

Suppose that f (x) has a root at x = p. Moreover, assume that |f' (x) | greaterthanorequalto 1 and |f" (x) | lessthanorequalto 3 Forall x (a) Show that p is a simple root of f (x). (b) If the initial error in Newton's iteration is less that 0.5, find an upper bound of the error at each of the first two steps. Find the next two iterative values of the solution of x^2 - 4 = 0 using the secant method, if the initial guesses are 3 and 4. Consider the fixed point iteration P_n + 1 = g (P_n) with g (P) = P. (a) Show that if g' (P) = 0, then the above iteration will converge (at least) quadratically to P. (b) Use part (a) to show that if P is a multiple root (M > 1) for some function f (x), then the accelerated Newton's iteration will converge quadratically to P. (c) Use part (a) to show that if P is a simple root for some function f (x), then the asymptotic error constant for Newton's iteration is |f" (P) |/2 |f' (P) |

Explanation / Answer

x = p that implies p = x which implies p is a simple root of f(x).

Upper bound of the error will be 0.6 as the Newton's irretattion is less than 0.5

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