Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.Comment: Research report V6: description of the main algorithm update

Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities

Isolating the real roots of the piecewise algebraic variety

AbstractThe piecewise algebraic variety, as a set of the common zeros of multivariate splines, is a kind of generalization of the classical algebraic variety. In this paper, we present an algorithm for isolating the zeros of the zero-dimensional piecewise algebraic variety which is primarily based on the interval zeros of univariate interval polynomials. Numerical example illustrates that the proposed algorithm is flexible

An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of points. First, the amount of purely symbolic operations is significantly reduced, that is, only resultant computation and square-free factorization is still needed. Second, our algorithm neither assumes generic position of the input system nor demands for any change of the coordinate system. The latter is due to a novel inclusion predicate to certify that a certain region is isolating for a solution. Our implementation exploits graphics hardware to expedite the resultant computation. Furthermore, we integrate a number of filtering techniques to improve the overall performance. Efficiency of the proposed method is proven by a comparison of our implementation with two state-of-the-art implementations, that is, LPG and Maple's isolate. For a series of challenging benchmark instances, experiments show that our implementation outperforms both contestants.Comment: 16 pages with appendix, 1 figure, submitted to ALENEX 201

Computation Sequences for Series and Polynomials

Approximation to the solutions of non-linear differential systems is very useful when the exact solutions are unattainable. Perturbation expansion replaces the system with a sequences of smaller problems, only the first of which is typically nonlinear. This works well by hand for the first few terms, but higher order computations are typically too demanding for all but the most persistent. Symbolic computation is thus attractive; however, symbolic computation of the expansions almost always encounters intermediate expression swell, by which we mean exponential growth in subexpression size or repetitions. A successful management of spatial complexity is vital to compute meaningful results. This thesis contains two parts. In the first part, we investigate a heat transfer problem where two-dimensional buoyancy-induced flow between two concentric cylinders is studied. Series expansion with respect to Rayleigh number is used to compute an approximation of a solution, using a symbolic- numerical algorithm. Computation sequences are used to help reduce the size of intermediate expressions. Up to 30th order solutions are computed. Accuracy, validity and stability of the computed series solution are studied. In the second part, Hilbert’s 16th problem is investigated to find the maximum number of limit cycles of certain systems. Focus values of the systems are computed using perturbation theory, which form multivariate polynomial sys- tems. The real roots of such systems leads to possible limit cycle conditions. A modular regular chains approach is used to triangularize the polynomial systems and help to compute the real roots. A system with 9 limit cycles is constructed using the computed real roots