Common Factors in Fraction-Free Matrix Reduction

We consider LU factoring of matrices in the context of exact and symbolic computation, as opposed to floating-point computation. Although initially developed for Gaussian elimination, fraction-free methods have been extended to LU factoring and related forms. We present surprising evidence that the rows and columns of the three matrices in the fraction-free form contain more common factors than one would expect. We describe and analyze experimental evidence for the existence of common factors, both in the case of integer matrices and matrices containing polynomials. The factors discovered grow linearly in the size of the matrix being factored. The common factors allow the entries in the factored form to be decreased in size.