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MATH622: Introduction to Functional Analysis
Revised: November 2006 (Erin McNelis)
Course Description & Topics
The course will focus on numerical techniques in:
- solving linear systems of equations (direct and iterative methods)
- approximation theory (least squares approximation, Fast Fourier Transforms)
- solving systems of nonlinear equations (quasi-Newton, steepest descent)
- solving differential equations:
- solving partial differential equations (explicit, implicit, and Finite-Element methods)
- solving boundary-value problems for ordinary differential equations (shooting methods, finite difference methods, Rayleigh-Ritz method)
Course Objectives
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; and to further develop students. ability to implement and utilize numerical methods in MATLAB or other mathematical software.
Required Text
Burden & Faires, Numerical Analysis (8th Ed.), Thompson Brooks/Cole Publishing.
Prerequisite
MATH 541.
Grading Procedure
Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy. Commonly, the final grade will be based upon homework/labwork, in-class tests, a class journal, and a final project.
Attendance Policy
Attendance policy is left to the discretion of individual instructors, subject to general university policy.
Course Outline
Direct Methods for Solving Linear Systems (Chapter 6) [8 days]
- Naïve Gaussian Elimination
- Pivoting Strategies
- Matrix Factorization
Iterative Methods for Solving Linear Systems (Chapter 7) [8 days]
- Norms, Eigenvalues and Errors of Iterative Methods
- Jacobi, Gauss-Seidel, and SOR Methods
- Conjugate Gradient Method
Approximation Theory (Chapter 8) [8 days]
- Discrete Least Squares Approximation
- Orthogonal Polynomials and Least Squares Approximation
- Trigonometric Polynomial Approximation
- Fast Fourier Transforms
Numerical Solutions of Nonlinear Systems (Chapter 10) [7 days]
- Fixed-Point Methods
- Newton and Quasi-Newton's Methods
- Steepest Descent
Solving Partial Differential Equations (Chapter 12) [8 days]
- Elliptic PDEs and Finite Difference Methods
- Parabolic PDEs and Forward and Backward Difference Methods, and Crank-Nicolson
- Hyperbolic PDEs
Boundary Value Problems for Ordinary Differential Equations (Chapter 11) [6 days]
- Shooting Methods
- Finite Difference Methods
- Rayleigh-Ritz Method