CSCI 5606, Assignment 2

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Problem 1. Consider the bisection method starting with the interval [1.5, 3.5]
(a) What is the width of the interval at the nth step of this method?
(b) What is the maximum distance possible between the root r and the midpoint of this
interval?
Problem 2. Write and test a program to implement the bisection method. Test the program
on the following functions and intervals
(a) x
−1 − tan(x) on [0, π/2].
(b) x
−1 − 2
x on [0, 1].
(c) Find a root of
x
8 − 36x
7 + 546x
6 − 4536x
5 + 22449x
4 − 67284x
3 + 118124x
2 − 109584x + 40320 = 0
in the interval [5.5,6.5].
(d) In (c), change 36 to 36.001 and repeat the test.
Problem 3. Write a program for Newton’s method and test it on the following problems
(a) Find the roots of x − tan x = 0 near 4.5 and 7.7.
(b) Solve the equation x
3 + 3x = 5x
2 + 7, starting at x0 = 5.
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