Common Factors in Fraction-Free Matrix Decompositions

Abstract

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauß–Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith–Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms that create them and give estimates of the frequency of their occurrence. Similar observations apply to the common factors in a fraction-free QR decomposition. Our conclusions are tested experimentally.

VaLiPro: Linear Programming Validator for Cluster Computing Systems

“…The LPlex system can solve medium to large LP problems to optimality. Based on exact arithmetic (integer, rational, or modular), LPlex implements a module to detect block structures in matrices [12] and supports LU-factorizations of sparse matrices, the Bareiss method [13,14], and the Wiedemann method [15]. The main drawback of the approach is that LPlex fails if the certified solution is not close enough to the optimal one.…”