Abstract

No abstract

We propose a new algorithm to find a rational solution to a sparse system of linear equations over the integers. This algorithm is based on a p-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox -based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art.

Efficient block projections of non-singular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound of O˜(n 2.5 ) machine operations is presented for this computation assuming that the input matrix can be multiplied by a vector with constantsized entries using O˜(n) machine operations. Somewhat more general bounds for black-box matrix computations are also derived. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections of non-singular matrices, and this was only conjectured.In this paper we establish the correctness of the algorithm from [10] by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields. We further demonstrate the usefulness of these projections by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems. We consider, in particular, matrices that can be multiplied by a vector using O˜(n) field operations: We show how to compute the inverse of any such non-singular matrix over any field using an expected number of O˜(n 2.27 ) operations in that field. A basis for the null space of such a matrix, and a certification of its *

Las Vegas algorithms that are basedon Lanczos's method for solving symmetric linear systems are presented and analyzed. These arecompared toasimilar randomized Lanczos algorithm that has been used for integer factorization, and to the (provably reliable) algorithm of Wiedemann.The analysis suggests that our Lanczos algorithms are preferable to several versions of Wiedemann's method for computations over large fields, especially for certain symmetric matrix computations.