MATH 417: Numerical Analysis
Instructors: Prof. Wolfgang Bangerth, Prof. Guido Kanschat
Teaching Assistants: Seungil Kim, Yan Li
Homework assignment 3 – due 9/21/06 and 9/22/06
Problem 1 (Newton’s method). For certain functions, Newton’s method
will always converge in a single step, no matter where we start. What functions
are these, and why is a single step enough? (Hint: think about the graphical
interpretation of Newton’s method, and when it will produce a new iteration
that falls exactly onto the true root of the function.) (2 points)
Problem 2 (Newton’s method). For functions f(x) of one variable x, New-
ton’s method almost always converges very quickly (in a matter of a few itera-
tions). However, almost always is not always, and we can find examples where
Newton’s method converges rather slowly.
Write a program to find the zero x= 1 of the function
f(x) = x25 −1
that uses Newton’s method and starts at x0= 20.
(a) How many iterations do you need to achieve an accuracy of 10−8?
(b) You will observe very slow convergence. Can you explain from the formulas
that express the error enas a function of en−1why convergence is so slow?
(c) Does the method still converge of second order? (6 points)
Problem 3 (Newton’s method for convex functions). Prove that New-
ton’s method converges from any point p0with f0(p0)6= 0 to a root of the
function f, provided that fhas at least one zero, and is globally convex, i.e.
f00(x)>0 for all x.
(4 points)
Problem 4 (Convergence order). Determine the order of convergence and
the asymptotic error constant for the following sequences:
(a) an= 5.0625,2.25,1,4
9,16
81
(b) bn= 2.718,2.175,1.740,1.392,1.113,0.8907
(c) cn= 0.318,0.180,0.0761,0.021,3.04 ·10−3,1.68 ·10−4,2.17 ·10−6.
(3 points)
1