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# Undergraduate Course: Computer Algebra (INFR10009)

 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
 Pre-requisites It is RECOMMENDED that students have passed Mathematics for Informatics 3a (MATH08042) AND Mathematics for Informatics 3b (MATH08043) AND Mathematics for Informatics 4a (MATH08044) AND Mathematics for Informatics 4b (MATH08045) Co-requisites Prohibited Combinations Other requirements This course is open to all Informatics students including those on joint degrees. For external students where this course is not listed in your DPT, please seek special permission from the course organiser. Familiarity with computer programming and data structures will be assumed. The course will contain an overview of less familiar algebra, as well as some new concepts.
 Pre-requisites None
 Not being delivered
 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).
 * 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, Addison-Wesley 1981.
 Course URL http://course.inf.ed.ac.uk/ca Graduate Attributes and Skills Not entered Keywords Not entered
 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
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