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MACM 316 – Computing Assignment 3

 MACM 316 – Computing Assignment 3

Submission Instructions: You must upload one .pdf file in Crowdmark that consists of two pages:
page 1 is your report which should fit all discussions, data and figures into a single page; and page
2 is a listing of your code. The deadline is 11:00pm on the due date. The actual due time is set
to 11:05pm and if Crowdmark indicates that you submitted late, you will be assigned a grade of 0
on this assignment. Your TA has emailed you a Crowdmark link that you should save since it will
allow you to upload your completed assignments.
❼ Please review the Guidelines for Assignments carefully.
❼ Acknowledge any collaborations or assistance from colleagues/TAs/instructor.
❼ If you have any questions about Matlab or aspects of this assignment, then you are strongly
encouraged to attend tutorials and drop-in workshops.
Computing Assignment – Modified Newton’s Method
In this assignment, you will investigate a variation of Newton’s Method called the Modified Newton’s
Method. You will compare the performance of both methods using the following nonlinear function
which has a double root at x = 1 (that is, a root of multiplicity m = 2).
(a) Apply the Newton iteration formula to f(x) using the Matlab code newton.m (posted on Canvas
under Lecture 2a). Use the default convergence tolerance of 10䛈 6 and record the iterations
starting from an initial guess of x0 = 2.
(b) We know from class that a convergent iteration generates a sequence xk for k = 0, 1, 2, . . . that
satisfies
Ek+1 6 αEp
convergence p and rate of convergence α. Use your Newton iterations xk from part (a) and the
exact root x
∗ = 1 to plot the absolute errors Ek+1 versus Ek on a log-log scale. Estimate the
convergence order p and rate α from your plot, and clearly explain how you calculated their
values.
(c) Modify the newton.m code to implement the following iteration
which is known as the Modified Newton’s Method. Repeat your calculation from part (a) start￾ing with the same initial point x0 = 2. Again plot your errors on a log-log scale and estimate
the order of convergence p. Explain any differences between the two iterations. List several
advantages and disadvantages of Modified Newton’s Method relative to Newton’s Method.
(d) How easy would it be to apply the Bisection Method to the function f(x)? Explain.
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