Math 640: Numerical Analysis

Math 640 - Numerical Analysis
Spring - 2007

Course Goals:

To familiarize students with additional types of problems (beyond those studied in MATH 541) where numerical methods are used
to approximate solutions; to analyze, compare, and contrast the basic numerical method algorithms in these areas; to investigate
real life applications of these numerical methods; to further develop students’ ability to implement and utilize numerical methods
in MATLAB or other mathematical software; and for students to hone their presentation skills by becoming proficient with LATEX,
a professional typesetting system for scientific documents.

Course Description & Topics:

This course will focus on numerical techniques in:

solving linear systems of equations (direct and iterative methods)
solving differential equations

solving boundary-value problems for ordinary differential equations (shooting methods, finite difference methods)
solving partial differential equations (explicit and implicit methods)

solving systems of nonlinear equations (quasi-Newton, steepest descent)
approximation theory (least squares approximation, Fast Fourier Transforms)

Convergence and stability issues associated with each technique will also be discussed. There will be a significant component
of the class that comes from implementing or using these methods to complete homework projects.

Daily homework listings can be found at the following link: Homework Assignments

Materials for the course have been sorted by class meetings and topics:

Syllabus: in PDF and in LaTeX

Miscellaneous Materials for Class
LaTeX Samples:

Sample LaTeX file (LaTeXExample.tex) and image to go with it
Guidelines and Example for Concept Map
Final Project Description and Deadlines

Test #1 Material: Solving Systems of Linear Equations
Part 1: Direct Methods

Handout on Gaussian Elimination Algorithm (PDF) and (LaTeX)
Maple Worksheet on Gaussian Elimination and Pivoting Strategies: PartialPivoting.mw
Homework #1:

Assignment in PDF and LaTeX along with needed image (ElasticBeamPix.jpg)
MATLAB code: NaiveGauss.m, NaiveGaussLE.m, NaiveGaussLE_ErrChecking.m, BeamSetup.m, BigSystem.m, BigSysProblem.m, bdry.m

Maple Worksheet on LU Decomposition Example
Handout on Special Types of Matrices and Their Decomposition (PDF) and (LaTeX)

Part 2: Iterative Methods

Handout on Norms and Eigenvalues (PDF) and (LaTeX)
Handout on Jacobi, Gauss-Seidel and SOR Iterative Methods (PDF) and (LaTeX)
Handout on the Condition Number and Error Analysis (PDF) and (LaTeX)
Handout on the Conjugate Gradient Method (PDF) and (LaTeX) and Maple Worksheet (mw)
Homework #2:

Assignment in PDF and LaTeX
MATLAB code needed for Room Temp Problem: Asm_Rm_Tmp.m, bdry1.m, bdry2.m, bdry3.m, bdry4.m, k.m, vx.m, vy.m

Test # 1 Review Topics: in PDF and LaTeX

Test #2 Material: Differential Equations (Boundary Value Problems and PDEs)
Part 1: Boundary Value Problems (Chapter 11)

Homework #3:

Assignment in PDF and LaTeX with illustration and a second illustration
MATLAB code needed for Linear Shooting Method (RK4.m)
A skeleton of some MATLAB code for Linear Shooting Method using Subfunctions (LinearShootingStarter.m)

Part 2: Partial Differential Equations (Chapter 12)

Handout on types of PDE's and Elliptic PDE's (Section 12.1) (PDF) and (LaTeX)
Material on Room Temperature problem

Test #2 Review Topics: in PDF and LaTeX

Test #3 Material: Approximation Theory and Nonlinear Systems of Equations
Part 1: Approximation Theory (Chapter 8)

Handout on Discrete Least Squares Approximation and Orthogonal Polynomials (PDF) and (LaTeX)
Maple Worksheet on Least Squares Approximation with Orthogonal Polynomials and Weighting Functions (mw)
Handout on Trigonometric Polynomials (PDF)
Maple Worksheet on Discrete and one on Continuous Trigonometric Polynomials
(Optional) Homework # 5:

Discrete Least Squares Polynomial and Trigonometric Polynomials (PDF) and (LaTeX)

Review topics for Test 3 (PDF)

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