Question

Given the following function

f(x)= 5x^4 − 32x^3 +120x^2 −182x + 86,

1) evaluate one of its roots using the fixed point iteration and using an initial guess of x = 0. If you approach this problem by adding x and subtracting x (i.e., f(x) + x – x = 0 that is x =x + f(x)), the method will fail. Explain why analytically.

(b) In part (a), suggest a remedy that makes the simple fixed-point iteration work.

Please show all steps

Answer 1

1) Solution :- a) Giren fonction Pow= 589-32x3+120 x?_182x+86 Now find a root of f(x)=0 there we rewrite f(x)=o as x= $(x)} I where | $(9) <1 fx is initial guess . De we rewrite P(a)= 524_32x3+12022-1823 +86 by using x= x+f(x) . =x+504-32x3+12032-182x +86 het x = $(x) where as $(x) = 5x9_32x364120xY-1812+86 Now derive the above. Sapuation '(a) + 2003-96X9+2402 - 181 het, Xboto, *'(0)= -181 => 1 $(0) > According to above condition fixed point iteration method fails. Here we have to find 0(0) Such that 1$'(x)} </ where as te in mod of Prilial guess x=0 b)