Introduction to Numerical Analysis I MATH/CMPSC 455 Fall 2011 Instructor: Xiaozhe Hu (Shawn)

W HAT IS N UMERICAL A NALYSIS ? “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 E XAMPLE : Algorithm: This is known as the “Babylonian method”, which is about 3600~3800 years old ( BC). Question: Why this algorithm works?

F IELD OF N UMERICAL 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

Practical Problem Mathematical Model Numerical Algorithm Code Code

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. C OURSE B RIEF D ESCRIPTION

Fundamentals (Chapter 0) 1) Introduction 2) Evaluating a Polynomial 3) Binary Numbers (0,1) 4) Floating Point Representation 5) Loss of Significance T ENTATIVE O UTLINE

Solving Equations (Chapter 1) 1) Bisection Method 2) Fix Point Iteration 3) Newton's Method 4) Secant Method

T ENTATIVE O UTLINE

System of Equations (Chapter 2) 1) Gaussian Elimination 2) LU Factorization 3) Linear Iterative Methods 4) Conjugate Gradient Method 5) System of Nonlinear Equations

T ENTATIVE O UTLINE Interpolation (Chapter 3) 1) Lagrange Interpolation 2) Cubic Splines 3) Bezier Curves

T ENTATIVE O UTLINE Numerical Integration (Chapter 5) 1) Newton-Cotes Formulas 2) Romberg Integration 3) Adaptive Quadrature 4) Gaussian Quadrature

TENTATIVE OUTLINE Numerical Solution of Ordinary Differential Equations (Chapter 6) 1) Euler's Method, Taylor Method 2) Runge Kutta Methods

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%)