A HYBRID APPROACH FOR DETERMINANT SIGNS OF MODERATE-SIZED MATRICES

Many geometric computations have at their core the evaluation of the sign of the determinant of a matrix. A fast, failsafe determinant sign operation is often a key part of a robust implementation. While linear problems from 3D computational geometry usually require determinants no larger than six, non-linear problems involving algebraic curves and surfaces produce larger matrices. Furthermore, the matrix entries often exceed machine precision, while existing approaches focus on machine-precision matrices. In this paper, we describe a practical hybrid method for computing the sign of the determinant of matrices of order up to 60. The stages include a floating-point filter based on the singular value decomposition of a matrix, an adaptive-precision implementation of Gaussian elimination, and a standard modular arithmetic determinant algorithm. We demonstrate our method on a number of examples encountered while solving polynomial systems

Certified Numerical Computation of the Sign of a Matrix Determinant

Certified computation of the sign of a matrix determinant is a central problem in computational geometry. The certification by the known methods is practically difficult because the magnitude of the determinant of an integer input matrix A may vary dramatically, from 1 to jjAjj n , and the roundoff error bound of the determinant computation varies proportionally. Because of such a variation, high precision computation of det A is required to ensure that the error bound is smaller than the magnitude of the determinant. We observe, however, that our certification task of determining only a single bit of det A, that is, the bit carrying the sign, does not require to estimate the latter roundoff error. Instead, we solve a much simpler task of computing numerically the factorizaion of a matrix by Gaussian elimination with pivoting and of estimating the minimum distance 1=jjA \Gamma1 jj from A to a singular matrix. Such an estimate gives us a desired range for the roundoff error of the f..