Math 351 - Introduction to Numerical Analysis - Sect 001 - Summer 2009

• Instructor: Enrique Thomann, Kidder 368E, 541.737.5160 (phone), thomann@science.oregonstate.edu (e-mail), http://www.math.oregonstate.edu/people/view/thomann (URL).
• Office Hours: MW 10:00 - 11:00, F: 9:30 -10:30, or by appointment.
• Textbook: An Introduction to Numerical Analysis, Third Ed., Kendall Atkinson and Weimin Han, John Wiley & Sons, Inc. 2004.
• Time and Place Lectures, MWF 11:00 - 12:20, Rogers 332.
• Course Objectives:  Solving concrete applied problems usually requires a numerically approximation of the true solution. In this process, two basic questions naturally arise. One is to find an appropriate numerical tool or method to approximate and solve the problem at hand (e.g., evaluating an integral from data consisting of a table of values. ) The other is to understand the sources of error, numerical or not. The main objective of Numerical Analysis is to help you develop skill that help in choosing an appropriate method and to be aware of its limitations. The main topics in this course are covered in chapters 1 through 6 and parts of chapter 7 of the text. Although we will follow the text closedly, you should attend classes regularly or get class notes since some of the material that we will cover is not completely developed in the text. A partial list of these topics is given below.
  • Errors due to finite precision. Loss of significant digits.
  • Roots of nonlinear equations. Bisection, secant, Newton's and fixed point methods. Error analysis.
  • Polynomial Interpolation. Newton's divided differences. Near-MinMax approximation. Least square approximations. Error estimates.
  • Numerical Integration. Trapezoidal and midpoint methods. Error analysis. Richardson extrapolation and Romberg integration.
  • Numerical Linear Algebra. Basic matrix factorization and error analysis.
  One of the objectives of the course is to develop a good understanding of what causes errors in numerical calculations and how to mitigate its effect. Consequently, the a priori and a posteriori error analysis in numerical calculations is one of the main focus of this class. The homework and examples will help illustrate and recognize some of the common sources of error.
• Grading:  Homework: 50 %. Midterm; Monday, July 27th in Kidder 108 20%. Note New Date and Place Final, Friday, August 14th: 30%.
• MATLAB:  Through the course most of the calculations will be done using MATLAB, one of the finest numerically oriented products available. The text provides examples of MATLAB codes, as well as a brief introduction in Appendix D. You can access MATLAB in any of the following ways.
  • Using the Department of Mathematics Computer Lab located in the Mathematics Learning Center (MLC) in Kidder 108.
  • Using the computer Lab in the basement of Milne Hall.
  • Check your department network or computer Lab.
  While not a requirement for this course, and depending on your future plans, you might consider obtaining a Student Edition version of this program.
• General Information:  Every week I will post the sections of the book that we have covered in class as well as homework or possible handouts. Make a habit of checking regularly.
• Week of June 22: Review of Taylor Series (Chapter 1). Floating Point representation and Loss of Significant Digits (Chapter 2). Numerical Differentiation and loss of significant digits (Section 5.4).
• Week of June 29: Root Finding, Bisection, Newton's and Secant method. Effect of Multiple roots. (Sections 3.1, 3.2, 3.3, 3.5)
• Week of July 6: Fixed point methods, stability of roots. Polynomial interpolation. Lagrange interpolation formula, Newton divided difference. Error estimates. Runge example.
• Week of July 13: Polynomial interpolation. Newton divided difference. Error estimates. Runge example. Chebyshev Polynomials, Near Min Max problem. Least square problem. Interpolation using splines.
• Week of July 20: Least square problem. Numerical integration. Trapezoidal, Simpsons, Midpoint and Romberg method.
• Week of July 27: Midterm on Monday July 27 in Kidder 108 There are no regular classes this week.
• Week of August 3: Finish up with numerical integration and start with numerical linear algebra.
• Homework Assignments: 
  • The first homework, due July 1st, is posted here.
  • The second homework, due July 10, is posted here. Until we cover fixed point methods (Monday July 6th,) you can start working on Parts 1 and 2 of each of the three cases considered in the homework. Have a great July 4th holiday.
  • The third homework, due July 17, is posted here. It covers the accelerated fixed point method using the Aitken extrapolation idea described in the text.
  • The fourth homework, due July 22, consists of the following problems from the text.
    • From section 4.1; problems 17 and 20.
    • From section 4.2; problems 11 and 13.
    • Problem2 from section 4.7.
  • The fifth homework, due August 7th, is posted here.
  • The last homework, due August 12, is posted here.
• Information regarding the Midterm:  The midterm will take place on Monday July 27 at the Mathematics Learning Center (MLC) located in Kidder 108. You can take the midterm at a time of your convenience, keeping in mind that the MLC is open between 10:00 AM and 3:45 PM. The time limit to complete the test is two hours, so you should start sometime before 1:45 PM. You are allowed to your use your class notes, textbook and a calculator. The midterm will cover the material from Chapters 1, 2, 3 and 4 that have been discussed in class. To help you review for the midterm, here is a collection of suggested problems and a copy of the cover page page.
• Information regarding the Final:  The final exam will take place during the regular class time on Friday Aug 14. The final will concentrate on the topics of numerical integration and numerical linear algebra that have been covered in class. Please use the follwing list of suggested problems to review for the final exam problems.
• Class Material: 
  • Single precision format Table 2.2
  • Loss of Significant digits, function evaluation Table 2.7
  • Loss of Significant digits, numerical differentiation Table 5.13
  • Newton's method - quadratic convergence. Example
  • Newton's and Secant method. Example log Plot
  • Fixed Point method, regular Example
  • Fixed Point method, accelerated code and graph
  • Illustration of error in Polynomial interpolation Code and graph
  • Runge Example Code and graphs
  • Runge Example using Chebyshev Nodes Code and graphs
  • Splines example Code and graph
  • Numerical Integration.
    Trapezoidal method
    • regular and
    • fast convergence.
    Simpsons method
    • regular and
    • slow convergence.
    Richardson Extrapolation used on the case of slow convergence.
    Romberg integration code.
    Midpoint method and Romberg_like method based on it. code.
  • Numerical Linear Algebra,
    • Hilbert matrix example
    • Normal equations from homework example