Question

Suppose you want to find a fixed point of a smooth function g(x) on the interval [a,b]

a. Give conditions which would be sufficient to show that fixed point iteration on g(x), starting with some $p_{0}$$\epsilon$ [a,b], will converge to the fixed point p.

b. When is this convergence only linear?

c. When is this convergence only quadratic?

d. Suppose a smooth function f(x) has a root p with f '(p) != 0. Assuming you choose the initial guess close enough to p, will Newton's method converge linearly or quadratically? Why?

Answer 1

Solution : le Let, g(x) on the interval [a,b] Let g: [a,b] → [a,b] be a differentiable function Such that, 18 = 2 + 1 [q,k] →if p be the fixed point of iteration 2004 = 962m). then the sequence & xn} with a starting point Po € [a, b] will converge to the fixed pornst Po. This is a sufficient condities for the Convergence. Linear Convergence : . If p be the fixed point of the iteration, Konto = 9(2n) and Suppose that g'(P) to then the iteration will have linear rate of convergence. b)

c) Quadratic convergence ;- Let o be the fixed point of the iterations Into = g(xn) & suppose that g'(P)=0 but g1 CP) +0 Quadratic rate of Then the iteration will have a Convergence fa) has a root p with f'CP) +0. reo, the multiplicity of the root pis I. and we know that for simple root the Newton's method convergence quadratically. So ' If you choose the initial guess close top. It will converge quadratically