Algebraic and Numerical Techniques for the Computat ion of Matrix Determinants

K e y w o r d s E v a l u a t i o n of the determinant, Sign of the determinant, Matrix singularity test, Modular (residue) arithmetic, Rounding error analysis. 1. I N T R O D U C T I O N 1.1. T h e S u b j e c t a n d S o m e B a c k g r o u n d We s tudy the classical problems of the computat ion of the determinant of a matr ix or testing if the determinant vanishes, tha t is, if the matr ix is singular. These problems have a long history (see, for instance, [1-11]) and have recently received a new major motivation, due to their impor tant applications to geometric computations, such as computat ion of convex hulls and Voronoi diagrams, and testing if the line intervals of a given family have a nonempty common intersection. In such applications, one needs sign or singularity tests, tha t is, one needs either to test if det A > 0, det A = 0, or det A < 0, for an n × n matr ix A, or just to test whether det A = 0 or not. In one group of these applications, n is relatively small [12,13], ranging from 2 to 10, but *Supported by NSF Grants CCR 9020690 and CCI:t 9625344, and by PSC CUNY Awards 665301, 666327, and 667340. The importance of the problems studied in this paper was brought to my at tent ion by K. Clarkson and F. Preparata . Clarkson and J. Hobby also described the customary requirements to the input matrices encountered in the applications to computational geometry, and Clarkson made several useful comments to the first draft of this paper. B. Braams and R. Pollack gave me some useful information about the computat ional geometry applications in the case of matrices of larger sizes. The discussions with H. Brbnnimann, and more briefly, with O. Devillers on implementation and applications of determinant sign algorithms and Brbnnimann comments on the works [12,13] were very informative. Typeset by A h~IS-TEX