3,926 research outputs found
Trading Order for Degree in Creative Telescoping
We analyze the differential equations produced by the method of creative
telescoping applied to a hyperexponential term in two variables. We show that
equations of low order have high degree, and that higher order equations have
lower degree. More precisely, we derive degree bounding formulas which allow to
estimate the degree of the output equations from creative telescoping as a
function of the order. As an application, we show how the knowledge of these
formulas can be used to improve, at least in principle, the performance of
creative telescoping implementations, and we deduce bounds on the asymptotic
complexity of creative telescoping for hyperexponential terms
Reduction-Based Creative Telescoping for Definite Summation of D-finite Functions
Creative telescoping is an algorithmic method initiated by Zeilberger to
compute definite sums by synthesizing summands that telescope, called
certificates. We describe a creative telescoping algorithm that computes
telescopers for definite sums of D-finite functions as well as the associated
certificates in a compact form. The algorithm relies on a discrete analogue of
the generalized Hermite reduction, or equivalently, a generalization of the
Abramov-Petkov\v sek reduction. We provide a Maple implementation with good
timings on a variety of examples.Comment: 15 page
A Fast Approach to Creative Telescoping
In this note we reinvestigate the task of computing creative telescoping
relations in differential-difference operator algebras. Our approach is based
on an ansatz that explicitly includes the denominators of the delta parts. We
contribute several ideas of how to make an implementation of this approach
reasonably fast and provide such an implementation. A selection of examples
shows that it can be superior to existing methods by a large factor.Comment: 9 pages, 1 table, final version as it appeared in the journa
Recurrence and Polya number of general one-dimensional random walks
The recurrence properties of random walks can be characterized by P\'{o}lya
number, i.e., the probability that the walker has returned to the origin at
least once. In this paper, we consider recurrence properties for a general 1D
random walk on a line, in which at each time step the walker can move to the
left or right with probabilities and , or remain at the same position
with probability (). We calculate P\'{o}lya number of this
model and find a simple expression for as, , where is
the absolute difference of and (). We prove this rigorous
expression by the method of creative telescoping, and our result suggests that
the walk is recurrent if and only if the left-moving probability equals to
the right-moving probability .Comment: 3 page short pape
Constructing minimal telescopers for rational functions in three discrete variables
We present a new algorithm for constructing minimal telescopers for rational
functions in three discrete variables. This is the first discrete
reduction-based algorithm that goes beyond the bivariate case. The termination
of the algorithm is guaranteed by a known existence criterion of telescopers.
Our approach has the important feature that it avoids the potentially costly
computation of certificates. Computational experiments are also provided so as
to illustrate the efficiency of our approach
Efficient Algorithms for Mixed Creative Telescoping
Creative telescoping is a powerful computer algebra paradigm -initiated by
Doron Zeilberger in the 90's- for dealing with definite integrals and sums with
parameters. We address the mixed continuous-discrete case, and focus on the
integration of bivariate hypergeometric-hyperexponential terms. We design a new
creative telescoping algorithm operating on this class of inputs, based on a
Hermite-like reduction procedure. The new algorithm has two nice features: it
is efficient and it delivers, for a suitable representation of the input, a
minimal-order telescoper. Its analysis reveals tight bounds on the sizes of the
telescoper it produces.Comment: To be published in the proceedings of ISSAC'1
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