From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci) Subject: Re: rate of converge -- help Date: 10 Feb 2000 10:17:48 GMT Newsgroups: sci.math.num-analysis Summary: [missing] In article <38A2229E.F0714871@cae.ca>, dans writes: |> Hi, |> |> For the following equation : x^n (x^2 - 9) = 0, where n=1,2,3 ....., |> I want to calculate the rate of convergence using the Newton Method. |> Now, I see that for n > 1 I have multiple root at x = 0. |> The Newton method is : x(k+1) = x(k) - g(x(k))/g'(x(k)) where g'(x(k)) |> != 0. |> Then I get the following equation: |> (n+1) x(k)^3 - 9 (n -1) x(k) |> x(k+1) = ----------------------------------- |> (n+2) x(k)^2 - 9n |> |> But from here, I am unable to see what rate of convergence ..... looks very much like students homework. hence i give you a hint only (you should find this in your lecture notes) theorem: let x* be a fixed point of the iteration map phi(x) and phi be sufficiently often differentiable at x*. then if 0< |phi' (x*)| < 1 then convergence is linear (Q-linear, i.e. |x_k-x*|<=L |x_{k-1}-x*| , 00 , then the Q-order is p. hint: if f is analytic and if x* is a k-th order zero of f then g(x)=f(x)/f'(x) has a single zero at x* and is analytic in some neighborhood of x* hope that helps peter