MATH/CMPSC 455
Introduction to
Numerical Analysis I
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
COURSE BRIEF DESCRIPTION
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.
TENTATIVE OUTLINE
Fundamentals
1)
2)
3)
4)
5)
(Chapter 0)
Introduction
Evaluating a Polynomial
Binary Numbers (0,1)
Floating Point Representation
Loss of Significance
TENTATIVE OUTLINE
Solving
Equations (Chapter 1)
Bisection Method
2) Fix Point Iteration
3) Newton's Method
4) Secant Method
1)
TENTATIVE OUTLINE
TENTATIVE OUTLINE
System
1)
2)
3)
4)
5)
of Equations (Chapter 2)
Gaussian Elimination
LU Factorization
Linear Iterative Methods
Conjugate Gradient Method
System of Nonlinear Equations
TENTATIVE OUTLINE
Interpolation
(Chapter 3)
Lagrange Interpolation
2) Cubic Splines
3) Bezier Curves
1)
TENTATIVE OUTLINE
Numerical
Integration (Chapter 5)
Newton-Cotes Formulas
2) Romberg Integration
3) Adaptive Quadrature
4) Gaussian Quadrature
1)
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%)
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