相关论文: Analytic Integration of the Newton Potential over …
The challenge of explicitly evaluating, in elementary closed form, the weakly singular sixfold integrals for potentials and forces between two cubes has been taken up at various places in the mathematics and physics literature. It created…
We present a fast, direct and adaptive Poisson solver for complex two-dimensional geometries based on potential theory and fast multipole acceleration. More precisely, the solver relies on the standard decomposition of the solution as the…
We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…
We consider a superintegrable Hamiltonian system in a two-dimensional space with a scalar potential that allows one quadratic and one cubic integral of motion. We construct the most general associative cubic algebra and we present specific…
Multiple polylogarithms appear in analytic calculations of higher order corrections in quantum field theory. In this article we study the numerical evaluation of multiple polylogarithms. We provide algorithms, which allow the evaluation for…
We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…
An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…
We recently introduced the Fast Newton Transform (FNT), an hierarchical algorithm for performing multivariate Newton interpolation in arbitrary downward closed polynomial spaces of spatial dimension $m$. Here, we analyze the FNT in the…
This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the…
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
We present the Fast Newton Transform (FNT), an algorithm for performing $m$-variate Newton interpolation in downward closed polynomial spaces with time complexity $\mathcal{O}(|A|m\overline{n})$. Here, $A$ is a downward closed set of…
This paper presents a simplified method of expressing the solution to cubic equations in terms of function evaluation only. The method eliminates the need to manipulate the original coefficients of the cubic polynomial and makes the…
A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of…
Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial…
A C++ software design is presented that can be used to interpolate data in any number of dimensions. The design is based on a combination of templates of functional collections of elements and so-called type lists. The design allows for…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
With the aid of Hensel Lemma, we refine the 2-adic Newton polygon algorithm proposed by Magron, Koprowski, and Vaccon at ISSAC 2023 to express computationally a given positive univariate polynomial with rational coefficients as a sum of…
Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to…
It is well known that, using fast algorithms for polynomial multiplication and division, evaluation of a polynomial $F \in \mathbb{C}[x]$ of degree $n$ at $n$ complex-valued points can be done with $\tilde{O}(n)$ exact field operations in…