Related papers: Accelerated Newton Iteration: Roots of Black Box P…

We revisit a classical problem in computational geometry: finding the largest-volume axis-aligned empty box (inside a given bounding box) amidst $n$ given points in $d$ dimensions. Previously, the best algorithms known have running time…

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if…

Given a function f: [a,b] -> R, if f(a) < 0 and f(b)> 0 and f is continuous, the Intermediate Value Theorem implies that f has a root in [a,b]. Moreover, given a value-oracle for f, an approximate root of f can be computed using the…

Finding a good approximation of the top eigenvector of a given $d\times d$ matrix $A$ is a basic and important computational problem, with many applications. We give two different quantum algorithms that, given query access to the entries…

Large-scale eigenvalue problems pose a significant challenge to classical computers. While there are efficient quantum algorithms for unitary or Hermitian matrices, eigenvalue problems for non-normal matrices remain open in quantum…

Let $f:2^{E} \rightarrow \mathbb{Z}_+$ be a submodular function on a ground set $E = [n]$, and let $P(f)$ denote its extended polymatroid. Given a direction $d \in \mathbb{Z}^n$ with at least one positive entry, the line search problem is…

Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\"o}nhage and Strassen designed an algorithm that improved the multiplication time for…

In this paper, we establish lower bounds for the oracle complexity of the first-order methods minimizing regularized convex functions. We consider the composite representation of the objective. The smooth part has H\"older continuous…

We design a fast implicit real QZ algorithm for eigenvalue computation of structured companion pencils arising from linearizations of polynomial rootfinding problems. The modified QZ algorithm computes the generalized eigenvalues of an…

Despite their remarkable performance, deep neural networks remain mostly ``black boxes'', suggesting inexplicability and hindering their wide applications in fields requiring making rational decisions. Here we introduce HOPE (High-order…

In the computation of Feynman integrals which evaluate to multiple polylogarithms one encounters quite often square roots. To express the Feynman integral in terms of multiple polylogarithms, one seeks a transformation of variables, which…

For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, it is well known that $f(X) =…

We present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the Dirichlet Laplacian in a smooth, star-shaped domain in $\mathbb{R}^d$, $d\ge 2$. Conventional boundary-based methods require a…

Given an order, a commutative ring whose additive group is free of finite rank, a natural computational question is whether a fixed univariate polynomial $f \in \mathbb{Z}[X]$ has a root in this ring. In this paper, we show that the…

In this paper we develop a new method which is a generalization of the Obreshkoff -Ehrlich method for the cases of algebraic, trigonometric and exponential polynomials. This method has a cubic rate of convergence. It is efficient from the…

We propose a novel algorithm for finding square roots modulo p. Although there exists a direct formula to calculate square root of an element modulo prime (3 mod 4), but calculating square root modulo prime (1 mod 4) is non trivial.…

We propose a new approach to the computation of the hypervolume indicator, based on partitioning the dominated region into a set of axis-parallel hyperrectangles or boxes. We present a nonincremental algorithm and an incremental algorithm,…

We provide faster algorithms and improved sample complexities for approximating the top eigenvector of a matrix. Offline Setting: Given an $n \times d$ matrix $A$, we show how to compute an $\epsilon$ approximate top eigenvector in time…

We present an algorithm for computing $F_p$, the $p$th moment of an $n$-dimensional frequency vector of a data stream, for $2 < p < \log (n) $, to within $1\pm \epsilon$ factors, $\epsilon \in [n^{-1/p},1]$ with high constant probability.…

In this article we apply a formula for the $n$-th power of a $3\times 3$ matrix (found previously by the authors) to investigate a procedure of Khovanskii's for finding the cube root of a positive integer. We show, for each positive integer…