9,373 research outputs found
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
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
Generating Polynomials and Symmetric Tensor Decompositions
This paper studies symmetric tensor decompositions. For symmetric tensors,
there exist linear relations of recursive patterns among their entries. Such a
relation can be represented by a polynomial, which is called a generating
polynomial. The homogenization of a generating polynomial belongs to the apolar
ideal of the tensor. A symmetric tensor decomposition can be determined by a
set of generating polynomials, which can be represented by a matrix. We call it
a generating matrix. Generally, a symmetric tensor decomposition can be
determined by a generating matrix satisfying certain conditions. We
characterize the sets of such generating matrices and investigate their
properties (e.g., the existence, dimensions, nondefectiveness). Using these
properties, we propose methods for computing symmetric tensor decompositions.
Extensive examples are shown to demonstrate the efficiency of proposed methods.Comment: 35 page
Faster Geometric Algorithms via Dynamic Determinant Computation
The computation of determinants or their signs is the core procedure in many
important geometric algorithms, such as convex hull, volume and point location.
As the dimension of the computation space grows, a higher percentage of the
total computation time is consumed by these computations. In this paper we
study the sequences of determinants that appear in geometric algorithms. The
computation of a single determinant is accelerated by using the information
from the previous computations in that sequence.
We propose two dynamic determinant algorithms with quadratic arithmetic
complexity when employed in convex hull and volume computations, and with
linear arithmetic complexity when used in point location problems. We implement
the proposed algorithms and perform an extensive experimental analysis. On one
hand, our analysis serves as a performance study of state-of-the-art
determinant algorithms and implementations. On the other hand, we demonstrate
the supremacy of our methods over state-of-the-art implementations of
determinant and geometric algorithms. Our experimental results include a 20 and
78 times speed-up in volume and point location computations in dimension 6 and
11 respectively.Comment: 29 pages, 8 figures, 3 table
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