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

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 problems such as three-dimensional convex hull or Delaunay triangulation of points typically require determinants of order three or four, non-linear problems involving algebraic curves and surfaces involve larger matrices. Furthermore, the matrix entries often exceed machine precision, while existing approaches focus on machine-precision matrices. In this paper, we outline the earlier methods for computing exact determinant signs, and evaluate their effectiveness on moderate-sized matrices. We describe a hybrid method for computing the sign of the determinant. The stages include a floating-point filter based on the singular value decomposition of a matrix, an adaptiveprecision implementation of Gaussian elimination, and a modular arithmetic determinant algorithm. We demonstrate our method on a number of examples encountered while solving polynomial systems