Math 6407

Numerical Analysis

NAME:	Tim Olson
ADDRESS:	Department of Mathematics University of Florida 454 Little Hall Gainesville, FL 32611-8105
PHONE:	(352) 392-0281 ext.295
FAX:	(352) 392-8357
OFFICE HOURS:	MWF 6'th or by appointment
E-MAIL:	olson@math.ufl.edu

Book 1: An intoduction of numerical methods and analysis by Epperson

or Book 2: An intoduction of numerical analysis, by Atkinson

Prerequisites: Basic linear algebra and calculus

Topics:

Test 1:

Topics: Root Finding Methods

· Bisection Method

· Newton's Method, and the Secant Method

· Fixed Point Methods

· Error Analysis for Fixed Point Methods

· Error Analysis for Newton's Method

· Convergence rate analysis, Order of Convergence

· Multivariate Fixed Point Methods

· Multivariate Newton's Method

Topics: Polynomial Interpolation

· The Lagrange Problem, Solution, and Formula

· The Lagrange Interpolation Error Formula, and Proof

· Problems with Polynomial Interpolation

· Alternative methods, from basic principles

Topics: Fourier analysis

· Basic Fourier Series

· The l^2 to L2 Isometry, and the L2 to L2 isometry

· The relationship between smoothness and decay

· The Shannon Sampling theorem and proof

· Problems with Shannon Sampling

· Aliasing, etc.

· Advanced Shannon Sampling

· The uncertainty principle and consequences

· Convolution and correlation theorems and proofs

Topics: General Orthogonal Systems

· Orthogonality, Cauchy-Schartz, and Gramm-Schmidt

· Orthogonal Polynomials

· Nested Subspaces and Gramm-Schmidt

· The zeros of orthogonal polynomials

Test 2:

Topic: Numerical Integration

· Trapezoid Rule and Error Formula

· Simpson's Rule and Error Formula

· General Newton-Cotes Formulas

· Gaussian Quadrature

Topic: Solution of ODE's

· Euler's Method and Error Formula

· Midpoint Method and Error Formula

· Parasitics associated with the Midpoint Method

· Predictor Corrector Methods

· Implicit Methods

· Trapezoid Method as a Predictor-Corrector

· General Multistep Methods, and their potential parasitics

Suggested Problems and Exercises: Root Finding Methods

· Program Newton's Method using matlab or something else

· Program the Multivariate Newton's Method, and test as above.

· Find and do selected problems from the suggested book, or any numerical analysis book, which correspond to the above topics.