MAP 6436 -- Numerical Analysis -- Fall 2009

MAP 6436 -- Numerical Analysis -- Fall 2009

Link to this page: http://www.math.fau.edu/kalies/map6436.htm

Homework 3
Final Project Information
Download M-files for Trefethen's book HERE.
Link to some perhaps interesting M-files HERE. (Thanks, Richard.)
Matlab Crash-course 2 From October 2, 2009. (Thanks, Richard.)
Matlab Crash-course 1 From September 18, 2009. (Thanks, Richard.)
Homework 2 Due October 19, 2009.
Miscellaneous Exercises
Homework 1 Due September 23, 2009.
euler.m
vf.m

Syllabus is still under construction and subject to change.

Instructor: Dr. Bill Kalies
Office: Science & Engineering Bldg., Room 242
Phone: (561) 297-1107
Fax: (561) 297-2436
Email: wkalies@fau.edu

Time and location: M W 4:00 - 5:20 PM in BU 102. Starting Wed 9/14 the room will be GS 120.

Office hours: M W 1:30 - 2:50 OR by appointment.

References:

  Undergraduate standard textbooks:
    Numerical Methods and Analysis by Buchanan and Turner
    Numerical Analysis, 8th ed. by Burden and Faires (library has 2nd, 4th, and 7th eds.)
    Numerical Analysis, 3rd ed. by Kincaid and Cheney (library has 2nd ed.)

  Other references:
    Spectral Methods in Matlab by Trefethen
    Numerical Linear Algebra by Trefethen and Bau
    An Introduction to Numerical Methods for Differential Equations by Ortega and Poole
    Computational Differential Equations by Eriksson, Estep, Hansbo, and Johnson

Course Description: Generally speaking, "numerical analysis" is the design and analysis of algorithms to approximate or represent continuous functions in a finite, discrete manner so that they can be studied on a computer. Sometimes it is the function itself that is the goal, such as in solving differential equations or curve-fitting data, and sometimes the goal is to find features of a function such as in root-finding, integration, or optimization.

There are standard methods and algorithms for many common problems, such as Newton's method for root-finding, least-squares for curve-fitting, Newton-Cotes formulas for integration, conjugate gradients for optimization etc. to name a few. This course will NOT be a survey of these methods. Instead we will focus on the problem of numerical differentiation and solving differential equations. Along the way we will need to study some of the standard methods for problems such as solving linear equations (numerical linear algebra), numerical integration, polynomial interpolation, etc.

In designing good numerical aglorithms generally three factors must be taken into account: (1) convergence -- theoretically does the algorithm provide a better approximation if you compute more and how fast does it converge -- (2) stability -- can the algorithm be implemented in a way that avoids numerical artifacts and loss of significance -- and (3) complexity -- does the number of operations or computation time required by the algorithm or implementation to get a better approximation grow too quickly. This process can involve some deep and interesting theoretical mathematics as well as challenging problems in implementation. Much of current applied mathematics research is in numerical analysis.

This course will involve both theoretical and practical aspects of numerical algorithms. The minimum prerequisites are undergraduate analysis and linear algebra. An undergraduate course in numerical methods and graduate courses in analysis and/or linear algebra might be helpful, but not required. I expect the course to be self-contained. There will be no required textbook; I will pull materials from a variety of sources. Some computer programming will be required, but everything should be able to be implemented in MATLAB fairly easily. If you do not know any computer languages, then MATLAB is easy to learn and is installed on the computers in SE 271.

Topics:

Intro to Numerical Analysis
Error and precision
Floating-point arithmetic and loss of significance
Convergence and order of accuracy
Stability of numerical algorithms
Computational complexity

Numerical Differentiation
Finite differences
Polynomial interpolation
Spectral differentiation
Fourier transform
Chebyshev points
Discrete and fast Fourier transforms

Boundary-Value Problems
Finite difference methods
Spectral methods
Intro to finite elements
Elliptic PDE's

Numerical Integration
Newton-Cotes formulas
Gauss quadrature

Initial Value Problems
Single step (Runge-Kutta) methods
Multistep (Adams) methods

Initial Boundary-Value Problems
Parabolic PDE's
Hyperbolic PDE's

Grades: grades will be determined by three homework assignments, a final group project, and class participation.

Last updated: 11/6/09