Reliable Krylov-based Algorithms for Matrix Null Space
Date
2004-12-20
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Abstract
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 F 2 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 a matrix A E F m x n over a
small finite field F. The expected number of matrix-vector products by A or
A t used by this algorithm is in O (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 polylog 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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Computer Science