Math 497/583
Approximation Theory and Spectral Methods
Fall 2015

Course Info

Overview

We discuss the practical and theoretical aspects of approximating and interpolating functions and data using polynomials, trigonometric series, and radial basis functions. These ideas, which form the basis for spectral methods, will then be applied to solving some differential equations.

Grading

Text Books

L. N. Trefethen, Approximation Theory and Approximation Practice (ATAP), SIAM 2013
Visit the book webpage for L. N. Trefethen, Spectral Methods in MATLAB (SMM), SIAM 2000
Note that both of these books can be purchased directly from SIAM and SIAM members get a significant discount. All Boise State Students can join SIAM for free. If you are interested in applied or computational mathematics, it's a no brainer to join SIAM.

Software

MATLAB

MATLAB is available in most computing labs around the university, including the mathematics computing lab. A student version can be purchased at a big discount from the bookstore. Employees of the university can also get a free copy of MATLAB on their machines.

Chebfun

Chebfun is an open source software package "for computing with functions". The software is a fundamental companion to the ATAP book and will play a fundamental role in this course. It is written in MATLAB and runs from the MATLAB integrated environment. Please download Chebfun and make sure it works on your machine using the instructions in exercise 1.1 from ATAP.

Lectures

Day Topics
25-Aug-2015
No lecture, but do the following:
  1. Read ATAP Ch. 1
  2. Download Chebfun and run chebtest as instructed in exercise 1.1.
  3. Do exercise 1.2. You will use publish on the homework problems.
  4. Look over the Chebfun examples, download the m-file for a couple that interest you and run them in MATLAB.
27-Aug-2015
No lecture, but do the following:
  1. Read ATAP Ch 2
  2. Do exercise 2.7
01-Sep-2015
  1. Review of complex variables
03-Sep-2015 Meet in the Lab (MB 136)
  1. Finish review of complex variables
  2. Chapter 2: Chebyshev points and interpolants
  3. Chapter 3: Chebyshev polynomials and series
08-Sep-2015
  1. Chapter 3: Chebyshev polynomials and series
  2. Chapter 4: Interpolants, projections, and aliasing
10-Sep-2015 Meet in the Lab (MB 136)
  1. Chapter 4: Interpolants, projections, and aliasing
  2. Chapter 5: Barycentric interpolation formula
15-Sep-2015
  1. Chapter 5: Barycentric interpolation formula
  2. Chapter 6: Weierstrass approximation theorem
    Bernstein polynomials
17-Sep-2015 Meet in the Lab (MB 136)
  1. Chapter 7: Convergence for differentiable functions
    Fractional convergence rates
  2. Chapter 8: Convergence for analytic functions
22-Sep-2015
  1. Chapter 8: Convergence for analytic functions
24-Sep-2015 Meet in the Lab (MB 136)
  1. Chapter 9: Gibbs Phenomenon
  2. Chapter 10: Best approximation
29-Sep-2015
  1. Chapter 10: Best approximation
  2. Chapter 11: Hermite integral formula
01-Oct-2015 Meet in the Lab (MB 136)
  1. Chapter 11: Hermite integral formula (HermiteIntegralFormula.m)
  2. Chapter 12: Potential theory and approximation
06-Oct-2015
  1. Chapter 12: Potential theory and approximation
08-Oct-2015 Meet in the Lab (MB 136)
  1. Chapter 12: Potential theory and approximation
  2. Chapter 13: Equispaced points, Runge phenomenon (runge.m)
13-Oct-2015
  1. Chapter 14: High order polynomial interpolation
  2. Chapter 15: Lebesgue constants
15-Oct-2015 Meet in the Lab (MB 136)
  1. Chapter 16: Best and near-best.
20-Oct-2015
  1. Chapter 17: Orthogonal polynomials
22-Oct-2015 Meet in the Lab (MB 136)
  1. Chapter 17: Orthogonal polynomials (OrthogonalPoly.m)
  2. Chapter 18: Polynomial roots and colleague matrices
27-Oct-2015
  1. Chapter 18: Polynomial roots and colleague matrices
  2. Chapter 19: Gaussian quadrature
29-Oct-2015 Meet in the Lab (MB 136)
  1. Chapter 19: Gaussian quadrature
03-Nov-2015
  1. Chapter 19: Gaussian quadrature
  2. Fourier series and trigonometric interpolation (reference)
05-Nov-2015 Meet in the Lab (MB 136)
  1. Fourier series and trigonometric interpolation (reference)
10-Nov-2015
  1. Fourier series and trigonometric interpolation (reference)
12-Nov-2015 No lecture
17-Nov-2015
  1. SMM Chapter 1: Differentiation matrices
19-Nov-2015 Meet in the Lab (MB 136)
  1. SMM Chapter 3: Periodic grids: ApproximatingDerivatives.m, SMM p6.m
01-Dec-2015
  1. SMM Chapter 3: Periodic grids: p6mod.m, p6fd.m
  2. SMM Chapter 4: Smoothness and spectral accuracy: p7mod.m
03-Dec-2015 Meet in the Lab (MB 136)
  1. SMM Chapter 4: The beauty of spectral accuracy: Acoustic wave example (acoustic.m)
  2. SMM Chapter 6: Chebyshev differentiation matrices (p12mod.m)
10-Dec-2015 Meet in the Lab (MB 136)
  1. SMM Chapter 7: Boundary value problems (p13mod.m, p15mod.m, p16mod.m, p17mod.m)

Homework

Homework assignments will involve a mix of analytical and computational work.

Format

All homework will be turned in electronically through a Dropbox folder I set up for you.

All written work should be submitted as a PDF file. Ideally, written work should be typeset using LaTeX (see LaTeX references below). However, although not advised, you can also use other typesetting software (Word, Pages, and LibreOffice have "plugins" for equations), or as a last resort you can scan your hand written pages.

All MATLAB code should be submitted as an m-file that I should be able to run to reproduce your results. Document your code to describe what it does and explain it in your written solutions.

The ideal way to write up your homework assignments is to use "publish" in MATLAB (using LaTeX mode). This allows you to embed equations with code and results in one easy to follow document. The whole ATAP book was written using publish. The best way to see how publish works is to find examples of output from ATAP that you like and then look at the corresponding m-file from ATAP webpage. Additionally, you may look at the many beautiful Chebfun examples, all of which were produced with publish and have m-files that can be downloaded.

Assignments

Final project

In lieu of a final exam, students will complete a final project consisting of a written report (5-10 pages) and oral presentation (15 minutes) on a topic related to the course. The purpose of the project is to explore some topic we cover or discussed in the book in more detail and then to teach the other students about it.

More information about the project will be posted in the fullness of time.

References

Approximation theory

MATLAB

Complex variables

Spectral Methods

Mathematical writing and LaTeX


Please e-mail me regarding any problems with the links on this page.