63,290 research outputs found
Matrix Shanks Transformations
Shanks' transformation is a well know sequence transformation for accelerating the convergence of scalar sequences. It has been extended to the case of sequences of vectors and sequences of square matrices satisfying a linear difference equation with scalar coefficients. In this paper, a more general extension to the matrix case where the matrices can be rectangular and satisfy a difference equation with matrix coefficients is proposed and studied. In the particular case of square matrices, the new transformation can be recursively implemented by the matrix -algorithm of Wynn. Then, the transformation is related to matrix Pad\ue9-type and Pad\ue9 approximants. Numerical experiments showing the interest of this transformation end the paper
Abstract State Machines 1988-1998: Commented ASM Bibliography
An annotated bibliography of papers which deal with or use Abstract State
Machines (ASMs), as of January 1998.Comment: Also maintained as a BibTeX file at http://www.eecs.umich.edu/gasm
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
On the Complexity of Solving Quadratic Boolean Systems
A fundamental problem in computer science is to find all the common zeroes of
quadratic polynomials in unknowns over . The
cryptanalysis of several modern ciphers reduces to this problem. Up to now, the
best complexity bound was reached by an exhaustive search in
operations. We give an algorithm that reduces the problem to a combination of
exhaustive search and sparse linear algebra. This algorithm has several
variants depending on the method used for the linear algebra step. Under
precise algebraic assumptions on the input system, we show that the
deterministic variant of our algorithm has complexity bounded by
when , while a probabilistic variant of the Las Vegas type
has expected complexity . Experiments on random systems show
that the algebraic assumptions are satisfied with probability very close to~1.
We also give a rough estimate for the actual threshold between our method and
exhaustive search, which is as low as~200, and thus very relevant for
cryptographic applications.Comment: 25 page
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
- …