Mathematics 4311 - Numerical Analysis

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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
Math 4311 – Numerical Analysis
Office: MCS 219F
email: [email protected]
Math 4311.010
webpage: www.angelo.edu/faculty/jmontema/ M-F: 9:45-11:15
MCS 214
Other classes:
Important Dates:
Math 4311 M-F @ 9:45- 11:15 MCS 214
Math 1312 M-F @ 8:00 – 9:30 MCS 210
Exam I: June 5
Exam II: June 17
Final Exam: July 2 @ 10:15 – 12:15
Exam : June 27
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:
Grading Distribution
A: 90 and above, B: 80 – 89.9,
C: 70 – 79.9,
D: 60 – 69.9, F: Below 60
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
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