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Reliable Krylov-Based Algorithms for Matrix Null Space and Rank

W. Eberly


Krylov-based algorithms have recently been used, in combination with other methods, to solve systems of linear equations and to perform related matrix computations over finite fields. For example, large and sparse systems of linear equations over the finite field with two elements are formed during the use of the number field sieve for integer factorization, and elements of the null space of these systems are sampled. Block Lanczos algorithms have been used to perform this computation with considerable success. However, the algorithms that are currently in use do not appear to be reliable in the worst case.

This report presents a block Lanczos algorithm that is somewhat simpler than block algorithms that are presently in use and provably reliable for computations over large fields. This can be implemented, using a field extension, in order to produce several uniformly and independently selected elements from the null space at once. The amortized cost to produce each vector closely matches the cost to generate such a vector with the methods currently in use.

An algorithm is also given to compute the rank of an mxn matrix over a small finite field F. The expected number of matrix-vector products by A or its transpose used by this algorithm is at most linear in r, where r is the rank of A. The expected number of additional field operations used by this algorithm is within a polylog factor of r(n+m), and the expected storage space is within a polynog factor of n+m. This is asymptotically more efficient than existing black box algorithms to compute the rank of a matrix over a small field, assuming that the cost of matrix-vector products dominates the cost of other operations.

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