I.  Number and Title of Course

 MAT 461  Numerical Analysis
 

II. Reasons for Revision

We have updated the bibliography and topical outline to better address and reflect the current work in this area.  The incorporation of technology as a tool for learning is one aspect of the revision.  We have also redefined the major objectives of the course in terms of student outcomes.  The course continues to serve the following purposes in our program:

A. Many problems in mathematics and it applications have solutions which are either infeasible or impossible to find exactly.  This course provides a rigorous foundation in numerical methods which the student may use to approximate these solutions.

B. Our mathematics majors have extensive experience early in their coursework with a programmable graphing calculator and a computer algebra system.  The students are therefore adequately prepared to pursue a challenging, realistic upper division course in approximation techniques.
 

III. Major Objectives of the Course

 The student will:

A. Use iteration techniques to estimate solutions to linear and nonlinear equations.

B. Numerically differentiate, integrate, interpolate, and solve differential equations.

C. Analyze the errors resulting from methods of approximation.

D. Utilize current technology to accomplish objectives A, B and C.

IV. Topical Outline

A. Solving Nonlinear Equations

1. Computer arithmetic and errors
2. Bisection method
3. Newton’s method
4. Fixed-point iteration
5. Bairstow’s method

1. LU decomposition
2. Condition numbers and errors
3. Iterative methods

a.  Jacobi method
b.  Gauss-Seidel iteration
c.  Relaxation method

4. Systems of nonlinear equations

 1.  Lagrangian polynomials
 2.  Newton divided differences
 3.  Cubic spline interpolation
 4.  Least-squares approximations

 1.  Derivatives from difference tables
 2.  Trapezoidal rule
 3.  Simpson’s rule
 4.  Gaussian quadrature

 1.  Chebyshev polynomials
 2.  Economized power series
 3.  Approximation with rational functions

 1.  Taylor series method
 2.  Euler methods
 3.  Multistep methods
 4.  Runge-Kutta methods
 5.  Errors and error propagation

V.   Bibliography

N. S. Asaithambi, Numerical Analysis, Theory and Practise, Saunders College Publishing, New York, 1995.
K. Atkinson, Elementary Numerical Analysis, John Wiley & Sons, New York, 1993.
R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, New York, 1997.
W. Cheney and D. Kincaid , Numerical Mathematics and Computing, Brooks/Cole, New York, 1985.
C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison-Wesley, New York, 1994.
G. H. Golub ( editor), Studies in Numerical Analysis, Vol. 24, MAA Studies in Mathematics, The Mathematical Association of America, 1984.
G. H. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996.
M. J. Maron,  Numerical Analysis: A Practical Approach, Macmillan, New York, 1982.
J. H. Mathews, Numerical Methods for Computer Science, Engineering and Mathematics, Prentice Hall, New Jersey, 1987.
J. L. Morris, Computational Methods in Elementary Numerical Analysis, John Wiley & Sons, New York, 1983.
D. G. Moursund and C. S. Duris, Elementary Theory and Application of Numerical Analysis, McGraw-Hill, New York, 1967.
S. M. Pizer with V. L. Wallace, To Compute Numerically: Concepts and Strategies, Little/Brown, Boston, 1983.
B. F. Plybon, An Introduction to Applied Numerical Analysis, PWS Publishing, Boston, 1992.
Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996.
T. E. Shoup, Applied Numerical Methods for the Microcomputer, Prentice-Hall, New Jersey, 1984.
W. A. Smith, Elementary Numerical Analysis, Prentice-Hall, New Jersey, 1986.
D. I. Steinberg, Computational Matrix Algebra, McGraw-Hill, New York, 1974.
S. Yakowitz and F. Szidarovszky, An Introduction to Numerical Computations, Macmillan, New York, 1989.
 

VI.  Presentation and Evaluation

A. Presentation by means of lectures, discussions, and individual projects.

B. Evaluation by means of written exams, homework, class participation, problem sets, and computer projects.
 

VII. Prerequisites

  MAT 202, MAT 263 and MAT 264
 

VIII. Credit

  3 credits:  (3:0)
 

IX.  Departmental Approval

This course proposal was examined in accordance with recommended procedures and
approved by the Department of Mathematics Curriculum Committee on
_____________________.    _____________________________________
             Signature of Department Chair   Date
 

X.   Catalog Description

Numerical solutions (and error analyses) to linear and nonlinear equations, interpolation, curve fitting, function approximation, numerical differentiation and integration, differential equations.

A. An M.A. or M.S. in Mathematics and an interest and expertise in Numerical Analysis.

B. Some of the current faculty who may teach this course are:
 James Guyker, Ph.D. in Mathematics
 Peter Mercer, Ph.D. in Mathematics

XII. Support Services Required

The Mathematics Department Computer Laboratory in Bishop 340 and present classroom facilities are adequate for this course.