Related papers: Toward accurate polynomial evaluation in rounded a…
Let $\RR$ be a real closed field (e.g. the field of real numbers) and $\mathscr{S} \subset \RR^n$ be a semi-algebraic set defined as the set of points in $\RR^n$ satisfying a system of $s$ equalities and inequalities of multivariate…
Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we…
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.
Model counting, or counting the satisfying assignments of a Boolean formula, is a fundamental problem with diverse applications. Given #P-hardness of the problem, developing algorithms for approximate counting is an important research area.…
Modern computer architectures support low-precision arithmetic, which present opportunities for the adoption of mixed-precision algorithms to achieve high computational throughput and reduce energy consumption. As a growing number of…
The problem of maximizing the $p$-th power of a $p$-norm over a halfspace-presented polytope in $\R^d$ is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in 1986 that this problem…
We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational…
Dynamic programming (DP) is an algorithmic design paradigm for the efficient, exact solution of otherwise intractable, combinatorial problems. However, DP algorithm design is often presented in an ad-hoc manner. It is sometimes difficult to…
Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the…
Algorithms operating on real numbers are implemented as floating-point computations in practice, but floating-point operations introduce roundoff errors that can degrade the accuracy of the result. We propose $\Lambda_{num}$, a functional…
Model checking undiscounted reachability and expected-reward properties on Markov decision processes (MDPs) is key for the verification of systems that act under uncertainty. Popular algorithms are policy iteration and variants of value…
The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the…
We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given…
This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of…
The XL algorithm is an algorithm for solving overdetermined systems of multivariate polynomial equations, which was initially introduced for quadratic equations. However, the algorithm works for polynomials of any degree, and in this paper…
We present a new algorithm for solving a polynomial program P based on the recent "joint + marginal" approach of the first author for, parametric optimization. The idea is to first consider the variable x1 as a parameter and solve the…
We give an algorithm for computing all roots of polynomials over a univariate power series ring over an exact field $\mathbb{K}$. More precisely, given a precision $d$, and a polynomial $Q$ whose coefficients are power series in $x$, the…
We present an algorithm for computing approximate $\ell_p$ Lewis weights to high precision. Given a full-rank $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $m \geq n$ and a scalar $p>2$, our algorithm computes $\epsilon$-approximate…