Mathematics 4311 - Numerical Analysis

Course Objectives

1.  Gaining factual knowledge including the mathematical notation and terminology used in this course. Learn the vocabulary, symbolism, basic definitions used in numerical analysis including those related to topics learned in calculus and algebra and revisited in this course; limits, continuity, numerical integration, numerical differentiation, ordinary differential equations, systems of linear equations, and polynomial interpolation.

2.  Learning Fundamental Principals. including the laws and theorems arising from the concepts covered in this course. Become familiar with the properties and theorems that result directly from the definitions as well as statements discovered in calculus and extended in this course; for example, Rolle’s Thm., Mean Value Thm., Intermediate Value Thm, Taylor series, theorems on convergence and existence and their error terms.

3.  Learning how to apply course material along with techniques and procedures covered in this course to solve problems. Use the facts, formulas, techniques to develop and use algorithms and theorems to find numerical solutions and bounds on its error to various types of problems including root finding, polynomial approximation, numerical differentiation, numerical integration.

4.  Developing specific skills , competencies, and thought processes sufficient to support further study or work in this field or related fields. Gain the ability to use a software package such as Maple to solve numerical type problems and acquire a level of proficiency in the fundamental concepts and applications necessary for further study in academic areas requiring numerical analysis as a prerequisite for graduate work or for work in occupational fields. These fields might include further study in mathematics, engineering, computer science, or the physical sciences.

Course Content

Textbook: Numerical Analysis, Sixth Edition, by R. Burden – J. Faires. The following chapters including the particular sections listed are covered.

1. Mathematical Preliminaries. Review of Calculus including limits, derivatives, continuity, Rolle’s

Thm., Mean Value Thm. , Extreme Value , Intermediate Value Thm, and extensions of these

theorems. Taylor and McLaurin Series , machine arithmetic, round-off error, representation of

numbers in different bases, floating point form, loss of significance due to errors

2. Solutions of Equations in One Variable. Bisection method, Newton’s method, the secant

method, fixed –point iteration, Newton –Raphson method, error analysis for iterative methods

3. Interpolation and Polynomial Approximation. Weierstrass Approximation Thm, Lagrange and

Taylor Polynomials, Neville’s iterated interpolation, Divided Differences, Cubic spline

interpolation, errors in polynomial interpolation,

4. Numerical Differentiation and Integration. Three and five point formulas of differentiation ,

Richardson’s Extrapolation, numerical quadrature, Trapezoidal and Simpson’s rule Midpoint,

Composite numerical integration, Rhomberg Integration, Adaptive quadrature methods,

bound on errors

5. Initial – Value Problems for Ordinary Differential Equations. Elementary theory, Taylor series

methods, Runge-Kutta methods

Additional Content

(Additional topics as time permits and as to the interest of the instructor)

A1. Methods for Solving Linear Systems. Gaussian Elimination, elimination with pivoting, Matrix

Theory, Matrix factorization,

A2. Iterative Techniques in Matrix Algebra. Jacobi and Gauss Seidel Techniques

A3. Approximation Theory. Splines, discrete least squares , orthogonal polynomials

A4. Monte Carlo Methods and Simulation. Random numbers, estimation of areas and volumes,

simulations.

Juan P. Montemayor
Phone: 942-2317 ext. 232
Office: MCS 219F
email:
webpage:www.angelo.edu/faculty/jmontema/ / Math 4311 – Numerical Analysis
Math 4311.010
M-F: 9:45-11:15
MCS 214
Other classes:
Math 4311 M-F @ 9:45- 11:15 MCS 214
Math 1312 M-F @ 8:00 – 9:30 MCS 210 / Important Dates:
Exam I: June 5 Exam II: June 17 Exam : June 27
Final Exam: July 2 @ 10:15 – 12:15
June 19th: Last Day to Withdraw from University
Office Hours for Summer I - 2003
M-Tu-W- Th – Fri
12:45 - 3:00 / Math Lab
MCS 215
M-F
2:00 – 4:00
Grading:
A: 90 and above, B: 80 – 89.9,
C: 70 – 79.9,
D: 60 – 69.9, F: Below 60 / Grading Distribution
Exam: (3) 20 % each
Final Exam: 20 %
Programs: 15 % all
HW : 5% each
Long Quizzes / Short Quizzes
Homework: / Bonus
none
There will be no make-ups of any kind. No make-ups on quizzes or HW
Enough quizzes will be dropped to account for most absences. There are
only two exams(plus the final) make plans not to miss those days