Ruprecht-Karls-Universität Heidelberg
Siegel der Universität Heidelberg

Module for [Scientific Computing]

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[Computer Algebra I] - [2015 Sommer]

Module Code
Computer Algebra I
Credit Points
8 CP
240 h
1 semester
Methods Lecture 4 h + Exercise course 2 h
Objectives To have a firm command of basic notions and methods in Computer Algebra
Content This lecture attends to the theory and complexity of basic mathematical algorithms and their implementations in computer algebra systems. Main topics are: - Fast arithmetic: complexity of basic operations, discrete Fourier transformation, fast multiplication, fast Euclidean algorithm, subresultants and polynomial remainder sequences, modular algorithms, computations with algebraic numbers, fast matrix multiplication - Prime factorisation and primality tests: Solovay-Strassen and Miller-Rabin primality test, AKS primality test, RSA algorithm, elementary factorisation methods, quadratic sieve, irreducibility tests for polynomials, Berlekamp algorithm, Zassenhaus algorithm, lattice basis reduction, factorisation of multivariate polynomials - Groebner Basis algorithms: Groebner Bases and reduced Groebner Bases, Buchberger algorithm, elimination theory, algorithms for elementary operations on ideals, computation of the dimension of an ideal
Learning outcomes Ability to solve problems in Computer Algebra and to present these solutions in problem sessions, ability to work with computer algebra systems
Suggested previous knowledge Mathematics Bachelor, Algebra I (MB1)
Assessments Homework assignments, written or oral exam. Modalities for make-up exams are to be determined by the lecturer and will be announced at the beginning of the course.
Literature J. von zur Gathen, J. Gerhard: Modern Computer Algebra
O. Geddes, S. R. Czapor, G. Labahn: Algorithms for Computer Algebra
D. Cox, J. Little, D. O’Shea: Ideals, Varieties and Algorithms
B. H. Matzat: Computeralgebra (lecture notes)
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