1,617 research outputs found
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
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
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
The Computational Complexity of the Restricted Isometry Property, the Nullspace Property, and Related Concepts in Compressed Sensing
This paper deals with the computational complexity of conditions which
guarantee that the NP-hard problem of finding the sparsest solution to an
underdetermined linear system can be solved by efficient algorithms. In the
literature, several such conditions have been introduced. The most well-known
ones are the mutual coherence, the restricted isometry property (RIP), and the
nullspace property (NSP). While evaluating the mutual coherence of a given
matrix is easy, it has been suspected for some time that evaluating RIP and NSP
is computationally intractable in general. We confirm these conjectures by
showing that for a given matrix A and positive integer k, computing the best
constants for which the RIP or NSP hold is, in general, NP-hard. These results
are based on the fact that determining the spark of a matrix is NP-hard, which
is also established in this paper. Furthermore, we also give several complexity
statements about problems related to the above concepts.Comment: 13 pages; accepted for publication in IEEE Trans. Inf. Theor
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
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