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ME 2016 Sections
Fall Semester 2007
Computing Techniques 3-0-3
Homework Assignment 3
Root Finding Algorithms
Due: Friday, September 14, 2007 on the hour before class
Description and Outcomes
In this assignment, you will explore the properties of different root-finding algorithms. Since
we studied three different root finding algorithms, it is important that you learn about their
properties so that you know when to use which algorithm. In this assignment, you will end up
solving the algorithm for a very large number of problems through the use of an interactive Java
applet that was developed specifically for this course. By applying the algorithms to functions
with different shapes, you will learn for which types of functions you should use which type of
algorithm. Unfortunately, there is no “one-size-fits-all” in this case!
The learning objectives of this assignment are:
• to learn about the properties of different root-finding algorithms
• to learn to recognize features of interest in the equations you are solving
• to learn how to select the best algorithm for solving a particular equation
• to develop as an independent learner by exploring root finding algorithms interactively
Background
The background for this assignment can be found in Chapter 5 of Chapra and Canale. Make sure
to review this chapter and your lecture notes BEFORE you start solving the problems!
Tasks
Task 1: Solve a Root Finding problem with pencil and paper
Consider the following equation:
2sin( ) 5
x
ex=+
With pencil and paper, solve for x in this equation using two different methods:
1) The bisection algorithm starting from an initial bracket of [0,1]. For each iteration,
compute the absolute approximate error. Stop after iteration 3 (i.e., include iterations
0 through 3).
2) The Newton Raphson algorithm starting from an initial guess of x=1. For each
iteration, compute the absolute approximate error. Stop after iteration 3 (i.e., include
iterations 0 through 3).
As previously, organize your computations in table form (a separate table for each
algorithm). For this task, no electronic submission is required.
Task 2: Error Analysis
Based on the errors computed in Task 1, verify the equations that we derived in class for
the errors of the bisection and Newton Raphson algorithms:
For Bisection: 0
2
n
an
x
E
Δ
=
For Newton Raphson: 2
,1 ,
()
2()
r
ti ti
r
fx
E
E
fx
+
′
′
−
=
′
Compare the values predicted by these equations with your calculation in Task 1.
