Introduction to Numerical Analysis I

MATH/CMPSC 455

Fall 2011

Instructor: Xiaozhe Hu (Shawn)

WHAT IS NUMERICAL ANALYSIS?“It is the study of algorithms that use numerical approximation for the problem of mathematical analysis”

---- Wikipedia study of algorithms: Create, analyze, and implement

algorithms numerical approximation:

Solving problems numerically and approximately mathematical analysis:

Problems of continuous mathematics. Such problems originate generally from real-world applications of algebra, geometry and calculus,

and they involve variables which vary continuously.

A EXAMPLE:

Algorithm:

This is known as the “Babylonian method”, which is about 3600~3800 years old (1800-1600 BC).

Question: Why this algorithm works?

FIELD OF NUMERICAL ANALYSIS Numerical linear and nonlinear algebra:

solution of systems of linear and nonlinear equations, possible with a very large number of variables

Approximation Theory:approximation of functions and methods based

on using such approximations

Solving differential and integral equations:

Effects of computer hardware

This course is an introduction to basic and classical numerical algorithms. We will describe numerical algorithms for floating point computation, rootfinding, solving linear systems, interpolation and quadrature. We will also discuss the underlying mathematical principles and theories of these numerical methods and their implementations.

COURSE BRIEF DESCRIPTION

Fundamentals (Chapter 0)1) Introduction2) Evaluating a Polynomial3) Binary Numbers (0,1)4) Floating Point Representation5) Loss of Significance

TENTATIVE OUTLINE

TENTATIVE OUTLINE

Solving Equations (Chapter 1)1) Bisection Method2) Fix Point Iteration3) Newton's Method4) Secant Method

TENTATIVE OUTLINE

TENTATIVE OUTLINE

System of Equations (Chapter 2)1) Gaussian Elimination2) LU Factorization3) Linear Iterative Methods4) Conjugate Gradient Method5) System of Nonlinear Equations

TENTATIVE OUTLINE

Interpolation (Chapter 3)1) Lagrange Interpolation2) Cubic Splines3) Bezier Curves

TENTATIVE OUTLINE

Numerical Integration (Chapter 5)1) Newton-Cotes Formulas2) Romberg Integration3) Adaptive Quadrature4) Gaussian Quadrature

Course Text: Numerical Analysis, Timothy Sauer, Addison

Wesley

Office Hours: Tuesday & Thursday 1:30 – 2:30 pm, or by

appointment

Grading Policy: 1. Homework & Computer assignments (50%)2. Midterm exam (20%)3. Final Exam (30%)