ECE412 Computer Algebra II

• Methods for the isolation of the real roots of polynomials with integer coefficients: The bisection methods by Sturm (1828) and Vincent-Collins-Akritas (1976), and the Vincent-Akritas-Strzebonski continued fraction method (2005).
• The resultant and gcd computation: Classical algorithms for computing polynomial remainder sequences (prs), resultants, subresultants, the Sylvester-Habicht pseudodivisions subresultant prs method, the matrix-triangularization subresultant prs method.
• Groebner bases: Polynomial ideals, term orderings and reduction, Groebner bases and S-polynomials, Buchberger’s algorithm.
• Symbolic summation.
• Symbolic integration (Rational Functions): Basic concepts of Differential Algebra, rational and logarithmic part of the integral.

The purpose of this course is an in-depth presentation of the algorithms and the mathematics involved in the various Computer Algebra Systems and to teach the students how to use the latter to symbolically solve scientific problems. More specifically, the course offers:

• Knowledge: Problem recognition and choice of the right method to solve it.
• Understanding: Detailed statement of the problem and choice of the appropriate system to solve it.
• Application: Experimentation and discovery of new facts.
• Analysis: Split a complex problem into simpler sub-problems.
• Synthesis: Re-organization and synthesis of the sub-problems.
• Evaluation: Comparison of the various methods and choice of the most appropriate.