Safe and Effective Determinant Evaluation

Kenneth L. Clarkson

Abstract: The problem of evaluating the sign of the determinant of a small matrix aries in many geometric algorithms. Given an n*n matrix A with integer entries, whose columns are all smaller than M in Euclidean norm, the algorithm given evaluates the sign of the determinant det A exactly. The algorithm requires an arithmetic precision of less than 1.5n+2lgM bits. The number of arithmetic operations needed is O(n/sup 3/)+O(n/sup 2/) log OD(A)/ beta , where OD(A) mod det A mod is the product of the lengths of the columns of A, and beta is the number of 'extra' bits of precision, min(lg(1/u)-1.1n-2lgn-2,lgN-lgM-1.5n-1), where u is the roundoff error in approximate arithmetic, and N is the largest representable integer. Since OD(A)<or=M/sup n/, the algorithm requires O(n/sup 3/lgM) time, and O(n/sup 3/) time when beta = Omega (logM).<