Time:    Tuesdays, 10:15 - 11:45
Rooms:  S3 047

In many applications of symbolic computation (e.g., summation, integration, solving difference/differential equations) one has to solve systems of linear equations that are not defined over floating-point numbers, but for instance over rational function fields or over principle ideal domains. In this lecture we discuss how one can generalize and/or optimize the well-known linear algebra methods in order to solve such systems. The application of these methods are illustrated by various examples.

In the first part of the lecture we revisit the theory of linear algebra in a more general setting: We do not work over fields but general rings. In this way the underlying structures of vector spaces generalize to modules. Here we require more and more ring properties (general rings with 1, commutative rings -> noetherian rings -> principle ideal rings -> principle ideal domains -> Euclidean domains -> fields) and can derive more and more intersting (algorithmic) results of the corresponding modules. In particular, we will derive the Smith normal form and illustrate it by various examples (including, e.g., the cyclic decomposition of finitely generated groups).

The second part of the lecture deals with advanced computer algebra methods to solve linear systems over a computable field. Among them we will focus on black box linear algebra. Here one gains speed ups in linear system solving provided that one can carry out matrix multiplications efficiently.

Some slides of the lecture are collected in the following file: SymbLA.pdf

The exercises appearing within the lecture are also collected in the following file: Exercises.pdf

Grading: The grades are either given by oral exams (30 minutes) that are schelduled in July on an indiviual basis. Alternatively, the students may elaborate on the exercises posed in the lecture. Then based on the quality and number of solved problems the grades are given accordingly.