Safe and Effective Determinant Evaluation

The problem of evaluating the sign of the determinant of a small matrix arises 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 here evaluates the sign of the determinant detA exactly. The algorithm requires an arithmetic precision of less than 1.5n+2 lgM bits. The number of arithmetic operations needed is O(n) + O(n) logOD(A)/β, where OD(A)|detA| is the product of the lengths of the columns of A, and β is the number of “extra” bits of precision, min{lg(1/u)− 1.1n− 2 lgn− 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) ≤M, the algorithm requires O(n lgM) time, and O(n) time when β = Ω(logM).