23,658 research outputs found
Computational linear algebra over finite fields
We present here algorithms for efficient computation of linear algebra
problems over finite fields
A fast solver for linear systems with displacement structure
We describe a fast solver for linear systems with reconstructable Cauchy-like
structure, which requires O(rn^2) floating point operations and O(rn) memory
locations, where n is the size of the matrix and r its displacement rank. The
solver is based on the application of the generalized Schur algorithm to a
suitable augmented matrix, under some assumptions on the knots of the
Cauchy-like matrix. It includes various pivoting strategies, already discussed
in the literature, and a new algorithm, which only requires reconstructability.
We have developed a software package, written in Matlab and C-MEX, which
provides a robust implementation of the above method. Our package also includes
solvers for Toeplitz(+Hankel)-like and Vandermonde-like linear systems, as
these structures can be reduced to Cauchy-like by fast and stable transforms.
Numerical experiments demonstrate the effectiveness of the software.Comment: 27 pages, 6 figure
Fast computation of the matrix exponential for a Toeplitz matrix
The computation of the matrix exponential is a ubiquitous operation in
numerical mathematics, and for a general, unstructured matrix it
can be computed in operations. An interesting problem arises
if the input matrix is a Toeplitz matrix, for example as the result of
discretizing integral equations with a time invariant kernel. In this case it
is not obvious how to take advantage of the Toeplitz structure, as the
exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself.
The main contribution of this work are fast algorithms for the computation of
the Toeplitz matrix exponential. The algorithms have provable quadratic
complexity if the spectrum is real, or sectorial, or more generally, if the
imaginary parts of the rightmost eigenvalues do not vary too much. They may be
efficient even outside these spectral constraints. They are based on the
scaling and squaring framework, and their analysis connects classical results
from rational approximation theory to matrices of low displacement rank. As an
example, the developed methods are applied to Merton's jump-diffusion model for
option pricing
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
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