Related papers: Polyarc bounded complex interval arithmetic
This paper presents a novel set-based computing method, called interval superposition arithmetic, for enclosing the image set of multivariate factorable functions on a given domain. In order to construct such enclosures, the proposed…
The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the…
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…
A new robust algorithm for the numerical computation of biarcs, i.e. $G^1$ curves composed of two arcs of circle, is presented. Many algorithms exist but are based on geometric constructions, which must consider many geometrical…
In the field of statistical learning and data analysis, estimating precision matrices (i.e., the inverse of covariance matrices) is a critical task, particularly for understanding dependency structures among variables. However, traditional…
Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. This paper proposes adaptations of these tools focusing on solutions of the real analogue:…
We study arrangements of intervals in $\mathbb{R}^2$ for which many pairs form trapezoids. We show that any set of intervals forming many trapezoids must have underlying algebraic structure, which we characterise. This leads to some…
We study a broad class of polynomial optimization problems whose constraints and objective functions exhibit sparsity patterns. We give two characterizations of the number of critical points to these problems, one as a mixed volume and one…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb…
Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying…
We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this…
Many algorithms, especially in the field of computational geometry, are based on the premise that arithmetic operations are performed exactly. Real machines are based on inexact floating-point arithmetic. Various number types have been…
Spatial join processing techniques that identify intersections between complex geometries (e.g., polygons) commonly follow a two-step filter-and-refine pipeline. The filter step evaluates the query predicate on the minimum bounding…
A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too…
We provide a technique to obtain explicit bounds for problems that can be reduced to linear forms in three complex logarithms of algebraic numbers. This technique can produce bounds significantly better than general results on lower bounds…
In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…
Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
Time series are difficult to monitor, summarize and predict. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). For scalability, we require fast…