MATH 417: Numerical Analysis
Instructor: Prof. Wolfgang Bangerth
Teaching Assistants: Dukjin Nam
Homework assignment 2 – due 2/8/2007
Problem 1 (Bisection method). A ball is dropped from a tower. During
its fall it is subject to the forces of gravity and air resistance. Thus, the height
hin ft as a function of time tin sis given by the formula
h(t) = h0−
mg
kt+m2g
k21−e−
kt
m.
Here, h0= 300ft is the height of the tower, m= 1/2lb the mass of the ball,
g= 32.17ft/s2is the gravitation of earth and k= 1/4lb s/ft is the air resistance
coefficient.
Use the bisection method (on paper, with pen and calculator) to find out
when the ball hits the ground with an accuracy of 0.1 seconds. (6 points)
Problem 2 (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 3 (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, i.e. is the relationship
between enand en+1 derived in class true also for this problem?
(d) What answers do you get to the questions in (a)–(c) if you apply the
same program to the function f(x) = x3
−1 instead, again starting from
x0= 20? (8 points)
1
