MATH 4650-001 & SC 4656-001: Numerical Analysis I
Computer Project 1. Convergence of the secant method movie.
Fall 2002, University of Colorado Denver

INSTRUCTOR:
Prof. Andrew Knyazev
Office: CU (Dravo Bldg) 644. Phone: (303) 556-8102.
Office hours: Tuesday 4-5 pm, or by appointment.
WWW: http://math.ucdenver.edu/~aknyazev

TEXTBOOK: Numerical Mathematics and Computing, 4th Edition by Ward Cheney and David Kincaid.
Published by Brooks/Cole Publishing Company
Fourth Edition, 1999
ISBN 0-534-35184-0. 560 pages

ASSIGNED COMPUTER PROJECTS:
A project consists of software development, testing and analysis. There will be two projects, each is worth 20% of the final grade. Both projects are individual, no group projects will be allowed.

COMPUTING:
Individual projects will be accepted in one of the following languages: Matlab, C, C++, Fortran and Java. My recommendation is to use MATLAB, since it can link executable files, it has graphics already built in and it is an interpretive language originally developed to handle mathematical problems.
The best place to work on the project is the MERC lab. Please register for Math 1999 lab to reserve a computer. Each student will have an account on math (the name of the computer), and other (except for beowulf) Linux-based networked computers of the CCM graduate lab soon after the census date.

Our first project requires high-end graphics tools, which are easily available in Matlab, but might not be simple in other computer languages. Please use Matlab unless you are an expert in programming and know what you are doing. I would only be able to provide help with Matlab for the project.

THE PROJECT DESCRIPTION:

• SOFTWARE DEVELOPMENT:
  1. Write a code of the Secant Method, Section 3.3. Please use the algorithm presented in the book on p. 124. Here is an example of a Matlab secant.m function input and output, which is recommended to use. Here is another example, a Matlab code for bisection. Make sure that your code outputs, in addition to the output of the algorithm on p. 124, the sequence of approximations to the zero and the sequence of the corresponding function values as in the examples above.
  2. Before you run the code, you need to create a file f.m, which describes the function, and put it in the same directory. For a sample file click here. Finally you can run it like in the following example: run_me.m.
     Note: if you are doing it on math, use the command
     cp /home/faculty/aknyazev/public_html/teaching/02/4650/p1/* .
     to copy all files you need into the current directory.
  3. The second part of the main function run_me.m calls a function get_convergence_movie.m which generates the movie movie.avi. When you are done with writing, debugging and testing your new code secant.m, you should use run_me.m and get_convergence_movie.m to generate your own movies for different functions and initial points.
• TESTING
  1. First of all, you should test your code for the example provided in the textbook on p. 125. Make sure your code is getting the same approximations as given in the texbook on p. 125. Explain, why your numbers are close, but not exactly the same as in the example.
  2. Test you code for different functions and initial approximations, suggested in Computer Problems 3.3 section of the text on pp. 131-132.
  3. Compare your numerical results with those obtained by the Secant Method LIVE for several examples.
• ANALYSIS
  1. Find an example when the secant method fails to produce an approximation because of division by zero and illustrate the failure by running your code for this example.
  2. Try to find an example when the secant method does not converge in 100 iterations.
     (Hint: perturb the initial points that have led to division by zero in the previous question).
  3. Set a small secant_tolerance=1e-14; and run the code
     [r,sequence_x,sequence_fx]=secant('f',x_0,x_1,n_max,secant_tolerance);
     for several different function, when it converges. For each run draw the picture of the convergence history using
     close all
     semilogy(abs(sequence_x-r))
     print -dpng slope.png
     (the last command creates a picture file slope.png in your working directory, which you may want to include in your project report) and conclude (from the way the picture looks) if you have a linear or superlinear convergence for these cases. (Hints: How should a history of the errors look like in semilogy scale when the convergence is linear? Draw a similar picture for the bisection, using the code provided.)
• THE PROJECT REPORT
  1. The report should be done electronically on a Web page created specifically for the project. Feel free to use your account on math.ucdenver.edu for the Web page.
  2. The project report should have links to all function written for the project and to all movies generated.
  3. The report needs to include a detailed description of the testing procedures and results, including complete descriptions of the movies generated.
  4. The report must include the results of the analysis.
  5. On the due day, please email me the link to your main Web page or directory with the project report and do not change it after that.

Sample student reports by Dmitriy Tserekhman, Mike Lewis, and Jason Gurgel.

INTERNET RESOURCES:

• MATLAB:
  • MathWorks
  • Getting Started with Matlab, Indiana University
  • Matlab Hypertext Reference, Portland State University
  • Matlab Quick Reference Guide, University of Waterloo
  • A Practical Introduction by Mark S. Gockenbach
  • MATLAB Primer. Third Edition by Kermit Sigmon
• My previous similar projects in NA II:
  • Convergence Movies
  • The Song of Convergence