Faster Inversion and Other Black Box Matrix Computations Using Efficient Block Projections

Block projections have been used, in [Eberly et al. 2006], to obtain an efficient algorithm to find solutions for sparse systems of linear equations. A bound of softO(n^(2.5)) machine operations is obtained assuming that the input matrix can be multiplied by a vector with constant-sized entries in softO(n) machine operations. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections, and this has been conjectured. In this paper we establish the correctness of the algorithm from [Eberly et al. 2006] by proving the existence of efficient block projections over sufficiently large fields. We demonstrate the usefulness of these projections by deriving improved bounds for the cost of several matrix problems, considering, in particular, ``sparse'' matrices that can be be multiplied by a vector using softO(n) field operations. We show how to compute the inverse of a sparse matrix over a field F using an expected number of softO(n^(2.27)) operations in F. A basis for the null space of a sparse matrix, and a certification of its rank, are obtained at the same cost. An application to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix yields algorithms requiring softO(n^(2.66)) machine operations. The derived algorithms are all probabilistic of the Las Vegas type

Efficient Parallel Solution of Sparse Systems of Linear Diophantine Equations

. An efficient new algorithm is presented for solving large sparse systems of linear Diophantine equations which is substantially and provably faster than those previously known in both a sequential and parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very small number of rational solutions of Toeplitz perturbations of the original system. We then employ the Block-Wiedemann algorithm to solve these perturbed systems efficiently in parallel. On an input matrix A 2 Z n\Thetan of rank r and w 2 Z n\Theta1 , the algorithm finds a v 2 Z n\Theta1 such that Av = w with about O(r(r log kAk \Delta + log kwk \Delta )=N) matrix-vector products by A modulo single-word primes, on N r(r log kAk \Delta +log kwk \Delta ) processors. Here kAk \Delta = max ij jA ij j and kwk \Delta = max i jw i j. Additionally, about O ` r 2 + rn(r log kAk \Delta + log kwk \Delta ) N + n(r log kAk \Delta + log kwk \Delta ) min(n; N)..

Efficient parallel solution of sparse systems of linear diophantine equations

We present a new iterative algorithm for solving large sparse systems of linear Diophantine equations which is fast, provably exploits sparsity, and allows an efficient parallel implementation. This is accomplished by reducing the problem of finding an integer solution to that of finding a very small number of rational solutions of random Toeplitz preconditionings of the original system. We then employ the Block-Wiedemann algorithm to solve these preconditioned systems efficiently in parallel. Solutions produced are small and space required is essentially linear in the output size