 Award ID(s):
 1764071
 NSFPAR ID:
 10382084
 Date Published:
 Journal Name:
 Proceedings of the ACM on Computer Graphics and Interactive Techniques
 Volume:
 5
 Issue:
 3
 ISSN:
 25776193
 Page Range / eLocation ID:
 1 to 15
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the rootfinding method of Weierstrass, also known as the Durand–Kernermethod: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130 years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the wellknown Newton method, is not generally convergent: there are open sets of polynomials p p of every degree d ≥ 3 d \ge 3 such that the dynamics of the Weierstrass method applied to p p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z 3 + Z + 175 Z^3 + Z + 175 have attracting cycles of period 4 4 . Here, period 4 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 2 or 3 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d ≥ 3 d\ge 3 there are orbits that are defined for all iterates but converge to ∞ \infty ; this is a problem that does not occur for Newton’s method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higherdimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems.more » « less

Abstract This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This tradeoff between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPUbased numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation.more » « less

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Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots a_n!\left(d_i\sum^n_{j=1}a_j \right)!}$. Recent progress on Smale's 17$\thth$ Problem by Lairez  building upon seminal work of Shub, Beltran, Pardo, B\"{u}rgisser, and Cucker  has resulted in a deterministic algorithm that finds a single (complex) approximate root of $F$ using just $N^{O(1)}$ arithmetic operations on average, where $N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$ ($=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$) is the maximum possible total number of monomial terms for such an $F$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain averagecase polynomialtime with more general probability measures? We show the answer is yes when $F$ is instead a binomial system  a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $O(n^3\log^2(n\max_i d_i))$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining averagecase time polynomial in $n\log \max_i d_i$ when $F$ has more terms.more » « less