Symbolic Computations (ΠΛ5)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΠΛ5
Semester	2
Course Title	Symbolic Computations
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Specialization
Prerequisite Courses	Undergraduate courses in Data structures, Design and Analysis of Algorithms, Algebraic Structures, (optionally a course in Discrete Mathematics).
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes

The purpose of the course is an in-depth study of computer algebra and the algorithms used for the symbolic processing of mathematical expressions. The goal is the understanding of the algorithms and the applications of computer algebra and the training of the students in critical thinking for problem solving as well as the research process. Many basic computer algebra algorithms as well as advanced ones are examined and analyzed. Application of these algorithms is also discussed. With the completion of the course the student: Knows how mathematical objects are represented Knows the basic algorithms for symbolic algebraic computations as well as some more advanced algorithms Can use specialize software packages for the symbolic processing of mathematical expressions Can apply the necessary symbolic algebra algorithms for the solution of mathematical problems

General Competences

Working Independently Competence in Bibliographic search Application of symbolic algebra procedures and algorithms for the solution of a mathematical problem Use specific software in the area of computer algebra

Syllabus

• Introduction to computer algebra
• Symbolic computations compared to numerical computations.
• Basic algebraic structures.
• Representation of numbers, polynomials (one or many variables), rational expressions, functions, series.
• Simplifications of symbolic mathematical expressions.
• Basic algorithms: Greatest common devisor, Chinese remainder algorithm.
• Basic operations and algorithms on integers and polynomials.
• Integer and polynomial factorization.
• Modular algorithms.
• Linear algebra algorithms, solution of equations and systems.
• Gröbner bases and applications.
• Algorithms for symbolic integration and summation.
• Symbolic solution of differential equations.
• Software systems for the symbolic manipulation of mathematical expressions.
• Special topics

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	Yes
Teaching Methods	Activity Semester Workload Lectures 39 Independent study 78 Exercises 70.5 Course total 187.5
Student Performance Evaluation	Final exam (40%) comprised of: Questions on the representation of mathematical data and the use of algorithms for the symbolic processing of mathematical expressions Questions requiring critical thinking Exercises - problem solution, programming using computer algebra software (30%). Presentations of related topics (30%).

Attached Bibliography

• Joel S. Cohen, "Computer Algebra and Symbolic Computation: Elementary Algorithms" Publisher: A K Peters/CRC Press, 2002
• Joel S. Cohen, "Computer Algebra and Symbolic Computation: Mathematical Methods" Publisher: A K Peters/CRC Press, 2003
• Keith O. Geddes, Stephen R. Czapor, George Labahn, “Algorithms for Computer Algebra”, Springer, 1992
• Davenport, J.H. and Siret, Y. and Tournier, E., Copmuter Algebra: Systems and Algorithms for Algebraic Computation, Academic Press, 1988.
• Akritas, A., Elements of Computer Algebra with Applications, Jhon Wiley, 1989,
• Modern Computer Algebra, Second Edition Joachim Von Zur Gathen, Juergen Gerhard Cambridge University Press, Cambridge, 2003.
• Computer algebra handbook. Foundations. Applications. Systems. Edited by Johannes Grabmeier, Erich Kaltofen and Volker Weispfenning. Springer-Verlag, Berlin, 2003.
• http://www.journals.elsevier.com/journal-of-symbolic-computation/