Apply Newton's method to $f(x)=(x-2)^4+(x-2)^5$ with initial guess $x_0=3$. We can observe that the sequence converges linearly with rate constant $3/4$. Now apply the iterative mathod $x_{k+1}=x_k-4f(x_k)/f'(x_k)$. This method should converge more rapidly for this problem. But how to prove that the new method converges quadratically and determine the rate constant?
Asked
Active
Viewed 121 times
2
-
Do you understand how to do the first part, that is, how to see linear convergence with rate constant $3/4$? Have you tried plugging the formula for $f$ into the given iterative method to see what happens? – Gerry Myerson Feb 17 '13 at 05:43