166,527 research outputs found
NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions
This article describes the implementation in the software package NumGfun of
classical algorithms that operate on solutions of linear differential equations
or recurrence relations with polynomial coefficients, including what seems to
be the first general implementation of the fast high-precision numerical
evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our
descriptions contain improvements over existing algorithms. We also provide
references to relevant ideas not currently used in NumGfun
Fast systematic encoding of multiplicity codes
We present quasi-linear time systematic encoding algorithms for multiplicity
codes. The algorithms have their origins in the fast multivariate interpolation
and evaluation algorithms of van der Hoeven and Schost (2013), which we
generalise to address certain Hermite-type interpolation and evaluation
problems. By providing fast encoding algorithms for multiplicity codes, we
remove an obstruction on the road to the practical application of the private
information retrieval protocol of Augot, Levy-dit-Vehel and Shikfa (2014)
On the evaluation of modular polynomials
We present two algorithms that, given a prime ell and an elliptic curve E/Fq,
directly compute the polynomial Phi_ell(j(E),Y) in Fq[Y] whose roots are the
j-invariants of the elliptic curves that are ell-isogenous to E. We do not
assume that the modular polynomial Phi_ell(X,Y) is given. The algorithms may be
adapted to handle other types of modular polynomials, and we consider
applications to point counting and the computation of endomorphism rings. We
demonstrate the practical efficiency of the algorithms by setting a new
point-counting record, modulo a prime q with more than 5,000 decimal digits,
and by evaluating a modular polynomial of level ell = 100,019.Comment: 19 pages, corrected a typo in equation (8) and added equation (9
Optimal, scalable forward models for computing gravity anomalies
We describe three approaches for computing a gravity signal from a density
anomaly. The first approach consists of the classical "summation" technique,
whilst the remaining two methods solve the Poisson problem for the
gravitational potential using either a Finite Element (FE) discretization
employing a multilevel preconditioner, or a Green's function evaluated with the
Fast Multipole Method (FMM). The methods utilizing the PDE formulation
described here differ from previously published approaches used in gravity
modeling in that they are optimal, implying that both the memory and
computational time required scale linearly with respect to the number of
unknowns in the potential field. Additionally, all of the implementations
presented here are developed such that the computations can be performed in a
massively parallel, distributed memory computing environment. Through numerical
experiments, we compare the methods on the basis of their discretization error,
CPU time and parallel scalability. We demonstrate the parallel scalability of
all these techniques by running forward models with up to voxels on
1000's of cores.Comment: 38 pages, 13 figures; accepted by Geophysical Journal Internationa
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