Math 351 - Introduction to Numerical Analysis - Sect 001 - Summer 2008

• Instructor: Enrique Thomann, Kidder 358E, 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, Pharmacy 107. Starting Monday July 14, we will meet in 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. 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; Friday, July 18th 20%. Final, Friday, August 13th: 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.
• Homework: 
  • The first homework, due July 2nd, is posted here.
  • The second homework, due July 9th, is posted here.
  • The third homework, due July 16th, is posted here.
  • The fourth homework, due July 30th, consists of Problem 4 from Section 4.6 (on Chebyshev polynomials) and problems 1 and 2 from Section 4.7 (on least square approximation).
  • The fifth homework, due August 6, is posted here.
  • The last homework, due August 13, is posted here. It corrects a minor typo from the version given in class. You should be able to do Problems 1, 2 and 3. Problem 4 will remain as an extra credit problem due the date of the final exam (August 15.)
• 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 23: Review of Taylor Series. Floating Point representation and Loss of Significant Digits.
• Week of June 30: Roofinding. Bisection, Newton's and Secant Method. Fixed Point iterations.
• Week of July 7: Roofinding. Newton's and Secant Method. Fixed Point iterations. Aitken extrapolation. Beginning of Polynomial Interpolation.
• Week of July 14: Polynomial Interpolation. Splines and Runge Example. Midterm
• Week of July 21: Polynomial Interpolation. Minmax problem. Least Squares Approximation. Beginning of Numerical Integration.
• Week of July 28: Numerical Integration, Midpoint, Trapezoidal and Simpson's rule. Richardson extrapolation and Romberg integration. Numerical differentiation. Review of Matrix Algebra.
• Week of August 4: Numerical Linear Algebra; Gaussian Elimination, LU factorization. Error analysis.
• Week of August 11: Numerical Linear Algebra; Error analysis, Matrix Norms and Condition Number. Least square problem.
• Information about the Midterm:  Recall that we will have a midterm on Friday July 18 during the regular class time. You can use your class notes and textbook during the test. You might also find a calculator useful (specially when dealing with factorial numbers...) To help you review for the test, here is a list of sample and suggested problems.
• Information about the Final:  Recall that the final exam will take place on Friday August 15 during the regular class time. You can use your class notes and textbook during the test. You might also find a calculator useful. To help you review for the test, here is a list of sample and suggested problems.
• Class Material: 
  • Newton's method Table 3.2
  • Secant method Table 3.3
  • Example of Fixed Point method accelerated using Aitken extrapolation.
  • Runge Example and polynomial interpolation. Graphs of the error using eleven or twenty one interpolation points.
  • Spline interpolation graph and code.
  • Polynomial interpolation errors using equispaced nodes graph and code.
  • Examples of numerical integration showing regular and fast convergence for the Trapezoidal method.
  • Examples of numerical integration showing slow convergence for the Trapezoidal and Simpson method, and regular convergence for the Simpson's method.
  • Notes on Romberg method of integration
  • One more examples of numerical integration showing slow convergence for the Trapezoidal method and acceleration using Richardson extrapolation.
  • Example of error propagation in numerical differentiation.
  • Example of QR and normal equation.