A tutorial introduction to Maple
Journal of Symbolic Computation
Algorithms for computer algebra
Algorithms for computer algebra
Efficient rational number reconstruction
Journal of Symbolic Computation
Approximate algorithms to derive exact solutions to systems of linear equations
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Exact solution of linear equations
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
A p-adic algorithm for univariate partial fractions
SYMSAC '81 Proceedings of the fourth ACM symposium on Symbolic and algebraic computation
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Maximal quotient rational reconstruction: an almost optimal algorithm for rational reconstruction
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
A BLAS based C library for exact linear algebra on integer matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Faster inversion and other black box matrix computations using efficient block projections
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Solving Very Sparse Rational Systems of Equations
ACM Transactions on Mathematical Software (TOMS)
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We consider the problem of solving a linear system Ax=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=degm(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.