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We analyse the forward error in the floating point summation of real numbers, from algorithms that do not require recourse to higher precision or better hardware. We derive informative explicit expressions, and new deterministic and…
We describe, for the first time, a completely rigorous homotopy (path--following) algorithm (in the Turing machine model) to find approximate zeros of systems of polynomial equations. If the coordinates of the input systems and the initial…
Mechanism design is addressed in the context of fair allocations of indivisible goods with monetary compensation. Motivated by a real-world social choice problem, mechanisms with verification are considered in a setting where (i) agents'…
Computer algebra systems are really good at factoring polynomials, i.e. writing f as a product of irreducible factors. It is relatively easy to verify that we have a factorisation, but verifying that these factors are irreducible is a much…
Researchers have developed neural network verification algorithms motivated by the need to characterize the robustness of deep neural networks. The verifiers aspire to answer whether a neural network guarantees certain properties with…
This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential…
We present a detailed study of roundoff errors in probabilistic floating-point computations. We derive closed-form expressions for the distribution of roundoff errors associated with a random variable, and we prove that roundoff errors are…
In this paper, we derive an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and…
Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…
Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329--366] and the references therein, we propose a novel $h$-adaptive algorithm for a fixed polynomial…
The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient-difference (qd) algorithm can be numerically unstable. The Compqd algorithm…
The reliable fraction of information is an attractive score for quantifying (functional) dependencies in high-dimensional data. In this paper, we systematically explore the algorithmic implications of using this measure for optimization. We…
As a partial answer to a question of Rao, a deterministic and customizable efficient algorithm is presented to test whether an arbitrary graphical degree sequence has a bipartite realization. The algorithm can be configured to run in…
Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear…
Conformal prediction is a model-free machine learning method for constructing prediction regions at a guaranteed coverage probability level. However, a data scientist often faces three challenges in practice: (i) the determination of a…
FOLD-RM is an explainable machine learning classification algorithm that uses training data to create a set of classification rules. In this paper we introduce CON-FOLD which extends FOLD-RM in several ways. CON-FOLD assigns…
In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving…
Iterative numerical algorithms are typically equipped with a stopping criterion, where the iteration process is terminated when some error or misfit measure is deemed to be below a given tolerance. This is a useful setting for comparing…
Programs with floating-point computations are often derived from mathematical models or designed with the semantics of the real numbers in mind. However, for a given input, the computed path with floating-point numbers may differ from the…
The aim of this paper is to show that there exists a deterministic algorithm that can be applied to compute the factors of a polynomial of degree 2, defined over a finite field, given certain conditions.