Course syllabus - Computer algebra with applications

Objectives

The purpose of the course is to give an introduction to algorithms (and the theory underlying them) for algebraic computations on a computer, and to introduce the students to some computer algebra system that is used in practice.

Learning outcomes

After passing the course the students should be able to
1. describe data structures and algorithms that can be used for basic computations with numbers and polynomials
2. describe the pros and cons of symbolic methods as compared to numerical methods
3. read and write pseudo-code
4. describe and apply algorithms for Gröbner basis computations, and know something about the areas of application of Gröbner bases
5. apply other methods for solving polynomial equations and systems of polynomial equations in simple cases
6. describe algorithms for factoring polynomials in Z[x]
7. formalize a mathematical formula in machine-readable format
8. use some software package for computer algebra

Course content

-Symbolic representation of and computation with integers, rational numbers, complex numbers and polynomials. To some extent, also algebraic numbers. Differences between symbolic and numerical representations.
-Pseudo-code
-Algorithms for modular arithmetic with integers and polynomials
-Gröbner bases and their use in solving equations
-To some extent, other methods for solving polynomial equations and systems of polynomial equations
-Applications of polynomial equations in mechanics and robotics
-Factorization of polynomials in Z[x]
-Machine-readable encoding of mathematics. Introduction to the subset of XML that is required for that purpose.
- Use of software for computer algebra.

Specific requirements

Discrete Mathematics, 7.5 credits and Basic Vector Algebra, 7.5 credits or equivalent.

Examination

INL1, Written assignment, 2.5 credits, written assignments concerning learning outcomes 1-8, grades Fail (U) or Pass (G).
TEN1, Examination, 5 credits, written examination concerning learning outcomes 1-7, grades Fail (U), 3, 4 or 5.

A student who has a certificate from MDU regarding a disability has the opportunity to submit a request for supportive measures during written examinations or other forms of examination, in accordance with the Rules and Regulations for Examinations at First-cycle and Second-cycle Level at Mälardalen University (2020/1655). It is the examiner who takes decisions on any supportive measures, based on what kind of certificate is issued, and in that case which measures are to be applied.

Suspicions of attempting to deceive in examinations (cheating) are reported to the Vice-Chancellor, in accordance with the Higher Education Ordinance, and are examined by the University’s Disciplinary Board. If the Disciplinary Board considers the student to be guilty of a disciplinary offence, the Board will take a decision on disciplinary action, which will be a warning or suspension.

Grade

Grading scale: 5, 4, 3