Abstract
We describe some major recent progress in exact and symbolic linear algebra. These advances concern the improvement of complexity estimates for fundamental problems such as linear system solution, determinant, inversion and computation of canonical forms. The matrices are over a finite field, the integers, or univariate polynomials. We show how selected techniques are key ingredients for the new solutions: randomization and algebraic conditioning, lifting, subspace approach, divide-double and conquer, minimum matrix polynomial, matrix approximants. These algorithmic progress allow the design of new generation high performance libraries such as LinBox, and open various research directions.We refer to [3] for an overview of methods in exact linear algebra, see also [37], [1] (in French), and [7, x2.3]. For fundamentals of computer algebra we refer to [16, 7].
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