Related papers: On a fast and nearly division-free algorithm for t…
In this paper, we present a probabilistic algorithm to multiply two sparse polynomials almost as efficiently as two dense univariate polynomials with a result of approximately the same size. The algorithm depends on unproven heuristics that…
We describe an algorithm to compute the essentially different factorizations of a given image primitive integer-valued polynomial $f(X)=g(X)/d\in\Q[X]$, where $g\in\Z[X]$ and $d\in\N$ is square-free, assuming that the factorization of…
We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…
We introduce a novel algorithm that computes the $k$-sparse principal component of a positive semidefinite matrix $A$. Our algorithm is combinatorial and operates by examining a discrete set of special vectors lying in a low-dimensional…
We show that there is a polynomial space algorithm that counts the number of perfect matchings in an $n$-vertex graph in $O^*(2^{n/2})\subset O(1.415^n)$ time. ($O^*(f(n))$ suppresses functions polylogarithmic in $f(n)$).The previously…
The purpose of this text is to provide an accessible introduction to a set of recently developed algorithms for factorizing matrices. These new algorithms attain high practical speed by reducing the dimensionality of intermediate…
We consider the precedence-constrained scheduling problem to minimize the total weighted completion time. For a single machine several $2$-approximation algorithms are known, which are based on linear programming and network flows. We show…
It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of…
We describe an algorithm which computes the ring laws for Witt vectors of finite length over a polynomial ring with coefficients in a finite field. This algorithm uses an isomorphism of Illusie in order to compute in an adequate polynomial…
Let $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the…
By establishing an interesting connection between ordinary Bell polynomials and rational convolution powers, some composition and inverse relations of Bell polynomials as well as explicit expressions for convolution roots of sequences are…
Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…
We introduce a framework for quasi-Newton forward--backward splitting algorithms (proximal quasi-Newton methods) with a metric induced by diagonal $\pm$ rank-$r$ symmetric positive definite matrices. This special type of metric allows for a…
We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an…
Efficient and fast predictor-corrector methods are proposed to deal with nonlinear Caputo-Fabrizio fractional differential equations, where Caputo-Fabrizio operator is a new proposed fractional derivative with a smooth kernel. The proposed…
We consider the Partition problem and propose a deterministic FPTAS (Fully Polynomial-Time Approximation Scheme) that runs in $\widetilde{O}(n + 1/\varepsilon)$-time. This is the best possible (up to a polylogarithmic factor) assuming the…
We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems…
The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on…
We describe an algorithm that, given any full-rank matrix A having fewer rows than columns, can rapidly compute the orthogonal projection of any vector onto the null space of A, as well as the orthogonal projection onto the row space of A,…
We study the bit complexity of two related fundamental computational problems in linear algebra and control theory. Our results are: (1) An $\tilde{O}(n^{\omega+3}a+n^4a^2+n^\omega\log(1/\epsilon))$ time algorithm for finding an…