3,183 research outputs found
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
Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues
The Erlangian approximation of Markovian fluid queues leads to the problem of
computing the matrix exponential of a subgenerator having a block-triangular,
block-Toeplitz structure. To this end, we propose some algorithms which exploit
the Toeplitz structure and the properties of generators. Such algorithms allow
to compute the exponential of very large matrices, which would otherwise be
untreatable with standard methods. We also prove interesting decay properties
of the exponential of a generator having a block-triangular, block-Toeplitz
structure
On Functions of quasi Toeplitz matrices
Let be a complex valued continuous
function, defined for , such that
. Consider the semi-infinite Toeplitz
matrix associated with the symbol
such that . A quasi-Toeplitz matrix associated with the
continuous symbol is a matrix of the form where
, , and is called a
CQT-matrix. Given a function and a CQT matrix , we provide conditions
under which is well defined and is a CQT matrix. Moreover, we introduce
a parametrization of CQT matrices and algorithms for the computation of .
We treat the case where is assigned in terms of power series and the
case where is defined in terms of a Cauchy integral. This analysis is
applied also to finite matrices which can be written as the sum of a Toeplitz
matrix and of a low rank correction
Novel Structured Low-rank algorithm to recover spatially smooth exponential image time series
We propose a structured low rank matrix completion algorithm to recover a
time series of images consisting of linear combination of exponential
parameters at every pixel, from under-sampled Fourier measurements. The spatial
smoothness of these parameters is exploited along with the exponential
structure of the time series at every pixel, to derive an annihilation relation
in the domain. This annihilation relation translates into a structured
low rank matrix formed from the samples. We demonstrate the algorithm in
the parameter mapping setting and show significant improvement over state of
the art methods.Comment: 4 pages, 3 figures, accepted at ISBI 2017, Melbourne, Australi
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