8,296 research outputs found
Fast Computation of Smith Forms of Sparse Matrices Over Local Rings
We present algorithms to compute the Smith Normal Form of matrices over two
families of local rings.
The algorithms use the \emph{black-box} model which is suitable for sparse
and structured matrices. The algorithms depend on a number of tools, such as
matrix rank computation over finite fields, for which the best-known time- and
memory-efficient algorithms are probabilistic.
For an \nxn matrix over the ring \Fzfe, where is a power of an
irreducible polynomial f \in \Fz of degree , our algorithm requires
\bigO(\eta de^2n) operations in \F, where our black-box is assumed to
require \bigO(\eta) operations in \F to compute a matrix-vector product by
a vector over \Fzfe (and is assumed greater than \Pden). The
algorithm only requires additional storage for \bigO(\Pden) elements of \F.
In particular, if \eta=\softO(\Pden), then our algorithm requires only
\softO(n^2d^2e^3) operations in \F, which is an improvement on known dense
methods for small and .
For the ring \ZZ/p^e\ZZ, where is a prime, we give an algorithm which
is time- and memory-efficient when the number of nontrivial invariant factors
is small. We describe a method for dimension reduction while preserving the
invariant factors. The time complexity is essentially linear in where is the number of operations in \ZZ/p\ZZ to evaluate the
black-box (assumed greater than ) and is the total number of non-zero
invariant factors.
To avoid the practical cost of conditioning, we give a Monte Carlo
certificate, which at low cost, provides either a high probability of success
or a proof of failure. The quest for a time- and memory-efficient solution
without restrictions on the number of nontrivial invariant factors remains
open. We offer a conjecture which may contribute toward that end.Comment: Preliminary version to appear at ISSAC 201
Efficient Computation of the Characteristic Polynomial
This article deals with the computation of the characteristic polynomial of
dense matrices over small finite fields and over the integers. We first present
two algorithms for the finite fields: one is based on Krylov iterates and
Gaussian elimination. We compare it to an improvement of the second algorithm
of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third
algorithm could improve both complexity and computational time. We use these
results as a basis for the computation of the characteristic polynomial of
integer matrices. We first use early termination and Chinese remaindering for
dense matrices. Then a probabilistic approach, based on integer minimal
polynomial and Hensel factorization, is particularly well suited to sparse
and/or structured matrices
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
Generic design of Chinese remaindering schemes
We propose a generic design for Chinese remainder algorithms. A Chinese
remainder computation consists in reconstructing an integer value from its
residues modulo non coprime integers. We also propose an efficient linear data
structure, a radix ladder, for the intermediate storage and computations. Our
design is structured into three main modules: a black box residue computation
in charge of computing each residue; a Chinese remaindering controller in
charge of launching the computation and of the termination decision; an integer
builder in charge of the reconstruction computation. We then show that this
design enables many different forms of Chinese remaindering (e.g.
deterministic, early terminated, distributed, etc.), easy comparisons between
these forms and e.g. user-transparent parallelism at different parallel grains
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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