Download - Introduction to Numerical Analysis I
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%)