Chee Yap

Amortized Bound for Root Isolation via Sturm Sequences

This paper presents two results on the complexity of root isolation via Sturm sequences. Both results exploit amortization arguments.

Explicit Mesh Surfaces for Particle Based Fluids

We introduce the idea of using an explicit triangle mesh to track the air/fluid interface in a smoothed particle hydrodynamics (SPH) simulator. Once an initial surface mesh is created, this mesh is carried forward in time using nearby particle velocities to advect the mesh vertices. The mesh connectivity remains mostly unchanged across time‐steps; it is only modified locally for topology change events or for the improvement of triangle quality. In order to ensure that the surface mesh does not diverge from the underlying particle simulation, we periodically project the mesh surface onto an implicit surface defined by the physics simulation. The mesh surface gives us several advantages over previous SPH surface tracking techniques. We demonstrate a new method for surface tension calculations that clearly outperforms the state of the art in SPH surface tension for computer graphics. We also demonstrate a method for tracking detailed surface information (like colors) that is less susceptible to numerical diffusion than competing techniques. Finally, our temporally‐coherent surface mesh allows us to simulate high‐resolution surface wave dynamics without being limited by the particle resolution of the SPH simulation.

The design of core 2: a library for exact numeric computation in geometry and algebra

There is a growing interest in numeric-algebraic techniques in the computer algebra community as such techniques can speed up many applications. This paper is concerned with one such approach called Exact Numeric Computation (ENC). The ENC approach to algebraic number computation is based on iterative verified approximations, combined with constructive zero bounds. This paper describes Core 2, the latest version of the Core Library, a package designed for applications such as non-linear computational geometry. The adaptive complexity of ENC combined with filters makes such libraries practical. Core 2 smoothly integrates our algebraic ENC subsystem with transcendental functions with e-accurate comparisons. This paper describes how the design of Core 2 addresses key software issues such as modularity, extensibility, efficiency in a setting that combines algebraic and transcendental elements. Our redesign preserves the original goals of the Core Library, namely, to provide a simple and natural interface for ENC computation to support rapid prototyping and exploration. We present examples, experimental results, and timings for our new system, released as Core Library 2.0.

Complexity Analysis of Root Clustering for a Complex Polynomial

Let F(z) be an arbitrary complex polynomial. We introduce the {local root clustering problem}, to compute a set of natural epsilon-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters. Our computational model assumes that arbitrarily good approximations of the coefficients of F(z) are provided by means of an oracle at the cost of reading the coefficients. Our algorithmic techniques come from a companion paper [3] and are based on the Pellet test, Graeffe and Newton iterations, and are independent of Schonhages splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

Planar minimization diagrams via subdivision with applications to anisotropic voronoi diagrams

Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic.

Certified computation of planar Morse-Smale complexes

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds (Edelsbrunner et al., 2003a). However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse-Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse-Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse-Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse-Smale complexes.

An Approach for Certifying Homotopy Continuation Paths: Univariate Case

Homotopy continuation is a well-known method in numerical root-finding. Recently, certified algorithms for homotopy continuation based on Smales alpha-theory have been developed. This approach enforces very strong requirements at each step, leading to small step sizes. In this paper, we propose an approach that is independent of alpha-theory. It is based on the weaker notion of well-isolated approximations to the roots. We apply it to univariate polynomials and provide experimental evidence of its feasibility.

On \(\mu \)-Symmetric Polynomials and D-Plus

We study functions of the roots of a univariate polynomial of degree \(n\ge 1\) in which the roots have a given multiplicity structure \({\varvec{\mu }}\), denoted by a partition of n. For this purpose, we introduce a theory of \({\varvec{\mu }}\)-symmetric polynomials which generalizes the classic theory of symmetric polynomials. We designed three algorithms for checking if a given root function is \({\varvec{\mu }}\)-symmetric: one based on Grobner bases, another based on preprocessing and reduction, and the third based on solving linear equations. Experiments show that the latter two algorithms are significantly faster. We were originally motivated by a conjecture about the \({\varvec{\mu }}\)-symmetry of a certain root function \(D^+({\varvec{\mu }})\) called D-plus. This conjecture is proved to be true. But prior to the proof, we studied the conjecture experimentally using our algorithms.

Implementation of a Near-Optimal Complex Root Clustering Algorithm

We describe Ccluster, a software for computing natural \(\varepsilon \)-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.

Isotopic Arrangement of Simple Curves: an Exact Numerical Approach based on Subdivision

We present a purely numerical (i.e., non-algebraic) subdivision algorithm for computing an isotopic approximation of a simple arrangement of curves. The arrangement is “simple” in the sense that any three curves have no common intersection, any two curves intersect transversally, and each curve is non-singular. A curve is given as the zero set of an analytic function on the plane, along with effective interval forms of the function and its partial derivatives. Our solution generalizes the isotopic curve approximation algorithms of Plantinga-Vegter (2004) and Lin-Yap (2009). We use certified numerical primitives based on interval methods. Such algorithms have many favorable properties: they are practical, easy to implement, suffer no implementation gaps, integrate topological with geometric computation, and have adaptive as well as local complexity. A preliminary implementation is available in Core Library.