Description
AMATH 740, CS 770, CM 750
1. (a) Implement Newton’s method. Submit print-out of your code.
(b) Give your own examples when iterations converge well, don’t converge, and converge
slowly. Choose your own way to give the answer: by hand, as a print-out of a Matlab
session, as a table, etc.
2. (a) Derive Newton’s iteration for function f(x)=(x 1)2ex. Show that they are well
defined for x 6= 1. What convergence rate would you expect?
(b) Solve the problem using your code with x0 = 2. Report and discuss the results.
3. (a) Argue that (in general) if f(x) has a multiple root at x⇤, then w(x) = f(x)/f0
(x) has
a simple root at x⇤.
(b) Derive the Newton’s iteration formula for w(x) in terms of f(x) and its derivatives.
(c) Implement the method in (b) and apply it to the function in Q2. Compare the results
and discuss advantages and disadvantages of the two methods.
4. Question 5 in Quarteroni. Also compare the method’s eciency to the eciency of the
secant method in terms of the required number of function evaluations.
5. Question 6 in Quarteroni. j (x) refers to various ways to rewrite f(x) = 0 as g(x) = x, i.e.
j (x) is g(x).
6. Read about the Muller method in Quarteroni, Ch 6.4.3. (you may skip the refinement part.)
1 AMATH 740, CS 770, CM 750
