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We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They…
Gaussian processes (GPs) are crucial in machine learning for quantifying uncertainty in predictions. However, their associated covariance matrices, defined by kernel functions, are typically dense and large-scale, posing significant…
Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In…
We introduce a subexponential algorithm for geometric solving of multivariate polynomial equation systems whose bit complexity depends mainly on intrinsic geometric invariants of the solution set. From this algorithm, we derive a new…
We give a new algorithm for performing the distinct-degree factorization of a polynomial P(x) over GF(2), using a multi-level blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by…
We study the inversion analog of the well-known Gauss algorithm for multiplying complex matrices. A simple version is $(A + iB)^{-1} = (A + BA^{-1}B)^{-1} - i A^{-1}B(A+BA^{-1} B)^{-1}$ when $A$ is invertible, which may be traced back to…
We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational…
We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
The prevalence of convolution in applications within signal processing, deep neural networks, and numerical solvers has motivated the development of numerous fast convolution algorithms. In many of these problems, convolution is performed…
We discuss efficient conversion algorithms for orthogonal polynomials. We describe a known conversion algorithm from an arbitrary orthogonal basis to the monomial basis, and deduce a new algorithm of the same complexity for the converse…
In the present paper, we deal with Fourier-transformation of Frobenius-Euler polynomials. We shall give its applications by using infinite series. Our applications possess interesting properties which we state in this paper.
We present fast and highly parallelized versions of Shor's algorithm. With a sizable quantum computer it would then be possible to factor numbers with millions of digits. The main algorithm presented here uses FFT-based fast integer…
We introduce efficient $(1+\varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem, where the inputs are a matrix $\mathbf{A}\in\{0,1\}^{n\times d}$, a rank parameter $k>0$, as well as an accuracy parameter…
In this paper, due to the important value in practical applications, we consider the coded distributed matrix multiplication problem of computing $AA^\top$ in a distributed computing system with $N$ worker nodes and a master node, where the…
We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Sch\"{o}nhage-Strassen. Both these algorithms use modular arithmetic.…
The reason why Cooley-Tukey Fast Fourier Transform (FFT) over $\mathbb{Q}$ can be efficiently implemented using complex roots of unity is that the cyclotomic extensions of the completion $\mathbb{R}$ of $\mathbb{Q}$ are at most quadratic,…
A modification of the standard Boris algorithm, called filtered Boris algorithm, is proposed for the numerical integration of the equations of motion of charged particles in a strong non-uniform magnetic field in the asymptotic scaling…
In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric…
We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a…