1,022 research outputs found
Solving Sparse Integer Linear Systems
We propose a new algorithm to solve sparse linear systems of equations over
the integers. This algorithm is based on a -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
Towards an exact adaptive algorithm for the determinant of a rational matrix
In this paper we propose several strategies for the exact computation of the
determinant of a rational matrix. First, we use the Chinese Remaindering
Theorem and the rational reconstruction to recover the rational determinant
from its modular images. Then we show a preconditioning for the determinant
which allows us to skip the rational reconstruction process and reconstruct an
integer result. We compare those approaches with matrix preconditioning which
allow us to treat integer instead of rational matrices. This allows us to
introduce integer determinant algorithms to the rational determinant problem.
In particular, we discuss the applicability of the adaptive determinant
algorithm of [9] and compare it with the integer Chinese Remaindering scheme.
We present an analysis of the complexity of the strategies and evaluate their
experimental performance on numerous examples. This experience allows us to
develop an adaptive strategy which would choose the best solution at the run
time, depending on matrix properties. All strategies have been implemented in
LinBox linear algebra library
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
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
An introspective algorithm for the integer determinant
We present an algorithm computing the determinant of an integer matrix A. The
algorithm is introspective in the sense that it uses several distinct
algorithms that run in a concurrent manner. During the course of the algorithm
partial results coming from distinct methods can be combined. Then, depending
on the current running time of each method, the algorithm can emphasize a
particular variant. With the use of very fast modular routines for linear
algebra, our implementation is an order of magnitude faster than other existing
implementations. Moreover, we prove that the expected complexity of our
algorithm is only O(n^3 log^{2.5}(n ||A||)) bit operations in the dense case
and O(Omega n^{1.5} log^2(n ||A||) + n^{2.5}log^3(n||A||)) in the sparse case,
where ||A|| is the largest entry in absolute value of the matrix and Omega is
the cost of matrix-vector multiplication in the case of a sparse matrix.Comment: Published in Transgressive Computing 2006, Grenade : Espagne (2006
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