Home | Quick Search | Advanced Search | Bibliography submission | Bibliography submission using bibtex | Bibliography submission using bibtex file | Links | Help | Internal


TitleAlgorithms for solving linear systems over cyclotomic fields
Author(s) Liang Chen, Michael Monagan
TypeArticle in Journal
AbstractWe consider the problem of solving a linear system A x = b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m ( z ) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear p -adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo m ( z ) using Chinese remaindering only. Because this representation is d = deg m ( z ) times more compact in general, we can compute it the fastest. We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients.
KeywordsLinear systems, Modular algorithms, Cyclotomic fields, Cyclotomic polynomials
URL http://www.sciencedirect.com/science/article/pii/S0747717110000714
JournalJournal of Symbolic Computation
Pages902 - 917
Translation No
Refereed No