Math256 Exam Homework
Throughout this question, f(a) is a function with a root at 2 = c (so f(c) = 0). You may assume
that f'(c) ‡ 0 except where the question states otherwise.
(a) Let K(a) be an arbitrary function and consider the Newton-Raphson method applied to the
product f(x)K(x).
(i) Show that the resulting iteration converges quadratically toward c provided the initial
estimate is good, and that K (c) exists.
Find the terms in o'”(a) that do not automatically vanish when 2
= c and then determine
K(2) such that o”(c) = 0.
Take advantage of the fact that f(c) = 0 to avoid messy algebra.
Substitute your formula for K() back into the Newton-Raphson iteration and show
that this leads to Halley’s method.
(b) Now consider the iteration formula
D(x) = 2 – A(x) f(a) – B(a)[f(x)|?.
where A(.) and B(a) may be chosen arbitrarily, except that A(c) and B(c) must exist.
(i) What function A(a) vields at least quadratic convergence in this iteration? Why is this
choice not affected by B(x)?
(i) Find B(x) such that the iteration has cubic convergence (at least), and show that the
resulting formula can be written in the form
f(x)f”‘(2x))
(c) Suppose now that f'(c) = 0, but f”(c) ‡ 0. Find the values of the limits
and hence determine the convergence properties of the iteration formula from part (b) in