Certification of Numerical Computation of the Sign of the Determinant of a Matrix

Abstract.

Certified computation of the sign of the determinant of a matrix is a central problem in computational geometry. Certification by known methods is practically difficult because the magnitude of the determinant of an integer input matrix Amay vary dramatically, from 1 to || A || ⁿ , and the round-off error bound of the determinant computation varies proportionally. Because of such a variation, high precision computation of \det Ais 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 us to estimate the latter round-off error. Instead, we solve a much simpler task of computing numerically the factorization of a matrix by Gaussian elimination with pivoting (which is a subroutine of LAPACK and LINPACK) and of estimating the minimum distance 1/||A ^-1 ||from Ato a singular matrix. Such an estimate gives us a desired range for the round-off error of the factorization such that the invariance of the sign of det Ais ensured as long as the error varies in this range. Based on these simple but novel observations, we devise new effective arithmetic filters for the certification of the sign, compare them with the known filters, and confirm the efficiency of our techniques by some numerical tests.