10,437 research outputs found
Products of Ordinary Differential Operators by Evaluation and Interpolation
It is known that multiplication of linear differential operators over ground
fields of characteristic zero can be reduced to a constant number of matrix
products. We give a new algorithm by evaluation and interpolation which is
faster than the previously-known one by a constant factor, and prove that in
characteristic zero, multiplication of differential operators and of matrices
are computationally equivalent problems. In positive characteristic, we show
that differential operators can be multiplied in nearly optimal time.
Theoretical results are validated by intensive experiments
Quasi-optimal multiplication of linear differential operators
We show that linear differential operators with polynomial coefficients over
a field of characteristic zero can be multiplied in quasi-optimal time. This
answers an open question raised by van der Hoeven.Comment: To appear in the Proceedings of the 53rd Annual IEEE Symposium on
Foundations of Computer Science (FOCS'12
Exponential Integrators on Graphic Processing Units
In this paper we revisit stencil methods on GPUs in the context of
exponential integrators. We further discuss boundary conditions, in the same
context, and show that simple boundary conditions (for example, homogeneous
Dirichlet or homogeneous Neumann boundary conditions) do not affect the
performance if implemented directly into the CUDA kernel. In addition, we show
that stencil methods with position-dependent coefficients can be implemented
efficiently as well.
As an application, we discuss the implementation of exponential integrators
for different classes of problems in a single and multi GPU setup (up to 4
GPUs). We further show that for stencil based methods such parallelization can
be done very efficiently, while for some unstructured matrices the
parallelization to multiple GPUs is severely limited by the throughput of the
PCIe bus.Comment: To appear in: Proceedings of the 2013 International Conference on
High Performance Computing Simulation (HPCS 2013), IEEE (2013
Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups
We wish to use graded structures [KrVu87], [Vu01] on dffierential operators
and quasimodular forms on classical groups and show that these structures
provide a tool to construct p-adic measures and p-adic L-functions on the
corresponding non-archimedean weight spaces. An approach to constructions of
automorphic L-functions on uni-tary groups and their p-adic avatars is
presented. For an algebraic group G over a number eld K these L functions are
certain Euler products L(s, , r, ). In particular, our constructions
cover the L-functions in [Shi00] via the doubling method of Piatetski-Shapiro
and Rallis. A p-adic analogue of L(s, , r, ) is a p-adic analytic
function L p (s, , r, ) of p-adic arguments s Z p , mod
p rPresented in a talk for the INTERNATIONAL SCIENTIFIC CONFERENCE "GRADED
STRUCTURES IN ALGEBRA AND THEIR APPLICATIONS" dedicated to the memory of Prof.
Marc Krasner on Friday, September 23, 2016, International University Centre
(IUC), Dubrovnik, Croatia
Fast Computation of Common Left Multiples of Linear Ordinary Differential Operators
We study tight bounds and fast algorithms for LCLMs of several linear
differential operators with polynomial coefficients. We analyze the arithmetic
complexity of existing algorithms for LCLMs, as well as the size of their
outputs. We propose a new algorithm that recasts the LCLM computation in a
linear algebra problem on a polynomial matrix. This algorithm yields sharp
bounds on the coefficient degrees of the LCLM, improving by one order of
magnitude the best bounds obtained using previous algorithms. The complexity of
the new algorithm is almost optimal, in the sense that it nearly matches the
arithmetic size of the output.Comment: The final version will appear in Proceedings of ISSAC 201
An Integro-Differential Structure for Dirac Distributions
We develop a new algebraic setting for treating piecewise functions and
distributions together with suitable differential and Rota-Baxter structures.
Our treatment aims to provide the algebraic underpinning for symbolic
computation systems handling such objects. In particular, we show that the
Green's function of regular boundary problems (for linear ordinary differential
equations) can be expressed naturally in the new setting and that it is
characterized by the corresponding distributional differential equation known
from analysis.Comment: 38 page
Formulas for Continued Fractions. An Automated Guess and Prove Approach
We describe a simple method that produces automatically closed forms for the
coefficients of continued fractions expansions of a large number of special
functions. The function is specified by a non-linear differential equation and
initial conditions. This is used to generate the first few coefficients and
from there a conjectured formula. This formula is then proved automatically
thanks to a linear recurrence satisfied by some remainder terms. Extensive
experiments show that this simple approach and its straightforward
generalization to difference and -difference equations capture a large part
of the formulas in the literature on continued fractions.Comment: Maple worksheet attache
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