NUMERICAL ANALYSIS
MA531
1. Course Description
Topics include iterative methods of solving equations; interpolation and polynomial approximation; numerical differentiation and integration; numerical solution of differential equations; solution of linear systems by direct and iterative methods; matrix inversion and calculation of eigenvalues and eigenvectors of matrices. Selected algorithms may be programmed.
2. Goals of the
Course
a. To strengthen the students’ grasp of basic notions in analysis and algebra, e.g., the idea of a sequence, limit recursion relation, definite integral, matrix techniques in algebra.
b. To help the student see the connection between an algorithm as a computational procedure, and the mathematical foundations.
c. To help the student appreciate the type of algorithmic approach that enables a problem to be handled by a computer.
3. Instructional
Procedure
a. Lecture/discussion
b. Small group study
c. Use of computer software/write computer algorithms by FORTRAN or C
4. Course Content
I. Number Systems and Errors
a. The Representation of Integers
b. The Representation of Fractions
c. Floating-point Arithmetic
d. Computational Methods for Error Estimation
II. Interpolation by Polynomial
a. Polynomial Forms
b. Existence and Uniqueness of the Interpolating Polynomial
c. The Divided-Difference Table
d. Interpolation at an Increasing Number of Interpolation Points
e. The Error of the Interpolating Polynomial
III. The Solution of Nonlinear Equations
a. A Survey of Iterative Methods
b. Fortran Programs for Some Iterative Methods
c. Fixed-point Iteration
d. Convergence Acceleration for Fixed-point Iteration
e. Quadratic
Convergence and
f. Polynomial Equations: Real Roots
IV. Differentiation and Integration
a. Numerical Differentiation
b. Numerical Integration: Some Basic Rules
c. Numerical Integration: Gaussian Rules
d. Numerical Integration: Composite Rules
V. The Solution of Differential Equations
a. Simple Difference Equations
b. Numerical Integration by Taylor Series
c. Error Estimates and Convergence of Euler’s Method
d. Runge-Kuta Methods
5. Evaluation
Measures for Determining students’ Grades
Total points is 100. The component is the following:
Homework 25%
Computer projects 25%
Midterm exam 25%
Final exam 25%
Minimum passing 60%
All programs submitted for grading must be suitably documented showing both input and output along with a complete listing of the program. No late assignments will be accepted without the permission of the instructor. The midterm and the final exam are scheduled to be given.
6. Bibliography
I.
Required
Text
a. ELEMENTARY NUMERICAL ANALYSIS: An Algorithmic Approach,
3rd Edition by
Conte and de Boor, published by McGraw-Hill Book Company, 1980
b. NUMERICAL ANALYSIS, 6th Edition, by Richard L. Burden and J.
Faires, published by Brooks/Cole Publishing Company, 1997
c. NUMERICAL ANALYSIS, by Johnson, L., and R. D. Riess, Publishing
Company
Addison-Wesley, Read,
II.
Supporting
Bibliography
a. The C programming language, 2nd Edition, by Kernighan and Ritchie,
published by Prentice Hall Inc., 1988
b. Learning C, by Neill Graham, published by McGraw-Hill, Inc., 1992
c. Analytical, Numerical, and computational methods for science and
engineering, by Gene H. Hostetter, Mohammed S. Santine, and Paul D’Carpio-Montalvo, published by Prentice Hall, 1991
d. Numerical methods for differential equations: Fundamental Concepts for
Scientific and Engineering Applications, by Michael A. Celia and William G.Gray, published by Prentice Hall, 1992
e. Numerical methods for science and engineering, 2nd Edition, by Hamming,
R. W., published
by McGraw-Hill, Inc.,