APPLICATIONS OF MATH

MA 220

Catalogue Description

The course includes topics in the theory of optimization. The topics include the maximization and/or minimization of univariate functions (using methods such as exhaustive search, interval search, random search, and Fibonacci search), and multivariate functions (using techniques such as the method of steepest descent). The linear programming problem is introduced and the simplex method for solving it. Topics are covered in the context of decision making. 3 credits. Prerequisites: MATH 191 Calculus II or equivalent

Goals

The purpose of this course is to give the student an idea of the steps an applied mathematician follows in solving a problem in a realistic situation.

A. The recognition and analysis of the non-mathematical origin of the problem.

B. Formulation of a mathematical model of the problem situation.

C. Solution of the mathematical problem and relevant computations.

D. The interpretation of the results of the solution in terms of the original problem.

A major objective is to introduce the student to problem solving in the context of the decision making process, namely the construction of a model of whatever it is a decision must be made about, determination of objectives and criteria with respect to the model, the determination of constraints on the possible choices, and finally, optimization within the bounds set by the criteria and constraints. The course can be offered on either an elementary level or an advanced level. On an elementary level, the course would be useful to majors in the social sciences.

Procedures

A. Lectures covering the theory & areas of application.

B. Verification of algorithms using facilities of the Math Resource Center

C. Assignment of problems

D. Interactive use of computer systems

Course Content

A. Maximization and Minimization of Univariate Functions

1. Exhaustive search

2. Interval search

3. Random search

4. Fibonacci search

B. Maximization and Minimization of Multivariate Functions

1. Exhaustive search

2. Interval search

3. Grid search

4. Steepest descent

C. Matrix Algebra

1. Transpose of a matrix

2. Symmetric matrix

3. Triangular and echelon matrices

4. Determinants

5. Inverse matrix

6. Matrix transformations

D. Systems of Linear Equations

E. Convex Sets

1. Linear inequalities

2. Convex sets

3. Maximum and minimum of a linear function over a convex polygon

F. Linear Programming

1. A general linear programming problem

2. The dual problem

3. Simplex method

4. Degeneracy

Evaluation methods

1. Satisfactory run of programs on the computer

2. Class participation

3. Final examination, periodic quizzes

4. Above requirements made known during first lecture period

Bibliography

Required Text: Cooper & Stein, Methods of Optimization, John Wiley & Sons, 1970.

Contributors to E.C.C.P, Man Made World, McGraw Hill, 1968 (Placed on reserve).

Branch, James R. & Rose, Donald J., (ed.), Sparse Matrix Computations, Academic Publishing, 1976.

Gastinel, Noel, Linear Numerical Analysis, Academic Publishing, 1970.

Haggerty, Gerald B., Elementary Numerical Analysis with Programming, Allyn Publishing, 1972.

Ortega, James M., Numerical Analysis - A Second Course, Academic Publishing, 1972.

Schwartz, H., et.al, Numerical Analysis of Symmetric Matrices, Prentice Hall, 1973.

Stewart, G.W, Introduction to Matrix Computations, Academic Publishing, 1973.