(Created 2009-08-11.)
 COMPUTER ALGEBRA FMA115

Higher education credits: 6. Grading scale: TH. Level: A (Second level). Language of instruction: The course might be given in English. Optional for: D4, E3, F4, Pi2. Course coordinator: Director of Studies Lars Christer Böiers, Lars_Christer.Boiers@math.lth.se, Matematik. Recommended prerequisits: FMA410 Calculus in one variable, FMA420 Linear algebra. Assessment: Oral examination. Approved home work. Further information: The course is given every second year, next time in spring 2010. Home page: http://www.maths.lth.se/matematiklth/vitahyllan/vitahyllan.html.

Aim
The aim of the course is to introduce some basic concepts and algorithms on which modern computer algebra is based, and to explain how computer algebra programs, e.g. Maple, work and how they can be applied effectively. The course should also give basic knowledge in classical algebra. The aim is also to help the student develop his och her ability in problem solving.

Knowledge and understanding
For a passing grade the student must

be able to explain some basic ways of representing and efficiently handling numbers, polynomials and rational functions in a computer algebra program.

be able to explain how some central, and in applications important, computer algebra algorithms work.

show good knowledge about those basic concepts in abstract algebra which are required in order to understand and use the algorithms treated in the course.

Skills and abilities
For a passing grade the student must

independently, and with the use of computer algebra programs, be able to apply the algorithms treated in the course on relevant problems in algebraic computing.

in the context of problem solving using computer algebra programs be able to explain how the basic algorithm works with the specific problem.

in writing as well as orally show good ability independently to explain mathematical reasoning in a structured and logically clear way.

Contents

• Main algorithms: representation and efficient handling of numbers, polynomials and rational functions. Factorization in Z, Z[x] and Zp[x].

• Gröbner bases and nonlinear systems of equations. Practical applications.

• Isolation of real roots. Sturm sequences, continued fractions.

• Modular methods: Hensel lifting, Berlekamp's algorithm.

• Symbolic summation and integration. Gosper's algorithm.

Literature
Childs, L.N.: A Concrete Introduction to Higher Algebra, Springer 2000. ISBN 0-387-98999-4.
Lecture notes with supplementary material.