Computing the sign or the value of the determinant of an integer matrix, a complexity survey

Authors:
Erich Kaltofen;Gilles Villard
Affiliations:
Department of Mathematics, North Carolina State University, Raleigh, NC;CNRS, Laboratoire LIP, École Normale Supérieure de Lyon 46, Allée d'Italie, 69364 Lyon Cedex 07, France
Venue:
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Year:
2004

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Abstract

Computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods. We survey the complexity of existing methods to solve these problems when the input is an n × n matrix A with integer entries. We study the bit-complexities of the algorithms asymptotically in n and the norm of A. Existing approaches rely on numerical approximate computations, on exact computations, or on both types of arithmetic in combination.