211 research outputs found
Asymptotically fast polynomial matrix algorithms for multivariable systems
We present the asymptotically fastest known algorithms for some basic
problems on univariate polynomial matrices: rank, nullspace, determinant,
generic inverse, reduced form. We show that they essentially can be reduced to
two computer algebra techniques, minimal basis computations and matrix fraction
expansion/reconstruction, and to polynomial matrix multiplication. Such
reductions eventually imply that all these problems can be solved in about the
same amount of time as polynomial matrix multiplication
Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
We reduce the problem of computing the rank and a nullspace basis of a
univariate polynomial matrix to polynomial matrix multiplication. For an input
n x n matrix of degree d over a field K we give a rank and nullspace algorithm
using about the same number of operations as for multiplying two matrices of
dimension n and degree d. If the latter multiplication is done in
MM(n,d)=softO(n^omega d) operations, with omega the exponent of matrix
multiplication over K, then the algorithm uses softO(MM(n,d)) operations in K.
The softO notation indicates some missing logarithmic factors. The method is
randomized with Las Vegas certification. We achieve our results in part through
a combination of matrix Hensel high-order lifting and matrix minimal fraction
reconstruction, and through the computation of minimal or small degree vectors
in the nullspace seen as a K[x]-moduleComment: Research Report LIP RR2005-03, January 200
Computing the Kalman form
We present two algorithms for the computation of the Kalman form of a linear
control system. The first one is based on the technique developed by
Keller-Gehrig for the computation of the characteristic polynomial. The cost is
a logarithmic number of matrix multiplications. To our knowledge, this improves
the best previously known algebraic complexity by an order of magnitude. Then
we also present a cubic algorithm proven to more efficient in practice.Comment: 10 page
Recent progress in linear algebra and lattice basis reduction (invited)
International audienceA general goal concerning fundamental linear algebra problems is to reduce the complexity estimates to essentially the same as that of multiplying two matrices (plus possibly a cost related to the input and output sizes). Among the bottlenecks one usually finds the questions of designing a recursive approach and mastering the sizes of the intermediately computed data. In this talk we are interested in two special cases of lattice basis reduction. We consider bases given by square matrices over K[x] or Z, with, respectively, the notion of reduced form and LLL reduction. Our purpose is to introduce basic tools for understanding how to generalize the Lehmer and Knuth-Schönhage gcd algorithms for basis reduction. Over K[x] this generalization is a key ingredient for giving a basis reduction algorithm whose complexity estimate is essentially that of multiplying two polynomial matrices. Such a problem relation between integer basis reduction and integer matrix multiplication is not known. The topic receives a lot of attention, and recent results on the subject show that there might be room for progressing on the question
Fast Computation of Minimal Interpolation Bases in Popov Form for Arbitrary Shifts
We compute minimal bases of solutions for a general interpolation problem,
which encompasses Hermite-Pad\'e approximation and constrained multivariate
interpolation, and has applications in coding theory and security.
This problem asks to find univariate polynomial relations between vectors
of size ; these relations should have small degree with respect to an
input degree shift. For an arbitrary shift, we propose an algorithm for the
computation of an interpolation basis in shifted Popov normal form with a cost
of field operations, where
is the exponent of matrix multiplication and the notation
indicates that logarithmic terms are omitted.
Earlier works, in the case of Hermite-Pad\'e approximation and in the general
interpolation case, compute non-normalized bases. Since for arbitrary shifts
such bases may have size , the cost bound
was feasible only with restrictive
assumptions on the shift that ensure small output sizes. The question of
handling arbitrary shifts with the same complexity bound was left open.
To obtain the target cost for any shift, we strengthen the properties of the
output bases, and of those obtained during the course of the algorithm: all the
bases are computed in shifted Popov form, whose size is always . Then, we design a divide-and-conquer scheme. We recursively reduce
the initial interpolation problem to sub-problems with more convenient shifts
by first computing information on the degrees of the intermediate bases.Comment: 8 pages, sig-alternate class, 4 figures (problems and algorithms
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
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
Analyse numérique et réduction de réseaux
29 pagesNational audienceL'algorithmique des réseaux euclidiens est un outil fréquemment utilisé en informatique et en mathématiques. Elle repose essentiellement sur la réduction LLL qu'il est donc important de rendre aussi efficace que possible. Une approche initiée par Schnorr consiste à effectuer des calculs approchés pour estimer les orthogonalisations de Gram-Schmidt sous-jacentes. Sans approximations, ces calculs dominent le coût de la réduction. Récemment, des outils classiques d'analyse numérique ont été revisités et améliorés, pour exploiter plus systématiquement l'idée de Schnorr et réduire les coûts. Nous décrivons ces développements, notamment comment l'algorithmique en nombres flottants peut être introduite à plusieurs niveaux dans la réduction
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