Undergraduate Course: Computer Algebra (INFR10009)
Course Outline
School  School of Informatics 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 10 (Year 4 Undergraduate) 
Availability  Available to all students 
SCQF Credits  10 
ECTS Credits  5 
Summary  Computer graphics uses various shapes such as ellipsoids for modelling. Consider the following problem: we are given an ellipsoid, a point from which to view it, and a plane on which the viewed image is to appear. The problem is to find the contour of the image as an equation (a numerical solution is not good enough for many applications). The problem does not involve particularly difficult mathematics, but a solution by hand is very difficult in general. This is an example of a problem which can be solved fairly easily with a computer algebra system. These systems have a very wide range of applications and are useful both for routine work and research. From a computer science point of view they also give rise to interesting problems in implementation and the design of algorithms. The considerations here are not only theoretical but also pragmatic: for example there is an algorithm for polynomial factorization which runs in polynomial time; however systems do not use this since other (potentially exponential time) methods work faster in practice. The design of efficient algorithms in this area involves various novel techniques. The material of the course will be related whenever possible to the computer algebra system Maple, leading to a working knowledge of the system. 
Course description 
* Maple: general design principles, user facilities, data structures, use of hashing, etc.
* Brief comparison of systems.
* Algebraic structures: overview, basic concepts and algorithms.
* Arbitrary precision operations on integers, rationals, reals, polynomials and rational expressions.
* Importance of greatest common divisors and their efficient computation for integers and univariate polynomials (using modular methods).
* Multivariate polynomial systems: solution of sets of equations over the complex numbers; construction and use of Groebner bases; relevant algebraic structures and results.
* Reliable solution of systems of polynomial equations in one variable; Sturm sequences, continued fractions method.
Relevant QAA Computing Curriculum Sections: Data Structures and Algorithms, Simulation and Modelling, Theoretical Computing

Information for Visiting Students
Prerequisites  None 
Course Delivery Information
Not being delivered 
Learning Outcomes
1  Use the computer algebra system Maple as an aid to solving mathematical problems.
2  Design and implement in Maple appropriate algorithms from constructive mathematical solutions to problems.
3  Discuss the overall design of the computer algebra system Maple.
4  Evaluate the results obtained from a computer algebra system and discuss possible problems.
5  Explain the gap between ideal solutions and actual systems (the need to compromise for efficiency reasons).
6  Describe and evaluate data structures used in the computer representation of mathematical objects.
7  Discuss the mathematical techinques used in the course and relate them to computational concerns.
8  Discuss and apply various advanced algorithms and the mathematical techniques used in their design.
9  Use the techniques of the course to design an efficient algorithm for a given mathematical problem (of a fairly similar nature to those discussed in the course).

Reading List
* J. von zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, second edition, 2003.
* K. O. Geddes, S. R. Czapor and G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers (1992).
* J.H. Davenport, Y. Siret and E. Tournier, Computer Algebra; systems and algorithms for algebraic computation, Academic Press 1988.
* D.E. Knuth, Seminumerical Algorithms, second dedition, AddisonWesley 1981. 
Contacts
Course organiser  Dr Kyriakos Kalorkoti
Tel: (0131 6)50 5149
Email: kk@inf.ed.ac.uk 
Course secretary  Miss Claire Edminson
Tel: (0131 6)51 4164
Email: C.Edminson@ed.ac.uk 

