arXiv subjects

Vitaly Zaderman

Publications and source records attributed to Vitaly Zaderman.

3 recordsLinked to original sources

arXiv2019

Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-finding, and would be especially usefull in application scenarios where high-precision polynomial coefficients are hard to obtain but we succeed with counting already by using polynomial evaluation with lower precision. We provide the pseudo code of the algorithm, proof of its correctness as well as estimation of its complexity.

math.NA↗

arXiv2019

Root-finding with Implicit Deflation

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to approximation of the remaining roots. Such situation is also realistic for root-finding by means of subdivision. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we contribute to the alternative approach of applying implicit deflation, which we combine with mapping the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase their local efficiency by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.

math.NA↗

arXiv2016

Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

The algorithms of Pan (1995) and(2002) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require precision of computing that exceeds the degree of the polynomial. This causes numerical stability problems when the degree is large. We observe, however, that such a difficulty disappears at the initial stage of the algorithms, and in our present paper we extend this stage to root-finding within a nearly optimal arithmetic and Boolean complexity bounds provided that some mild initial isolation of the roots of the input polynomial has been ensured. Furthermore our algorithm is nearly optimal for the approximation of the roots isolated in a fixed disc, square or another region on the complex plane rather than all complex roots of a polynomial. Moreover the algorithm can be applied to a polynomial given by a black box for its evaluation (even if its coefficients are not known); it promises to be of practical value for polynomial root-finding and factorization, the latter task being of interest on its own right. We also provide a new support for a winding number algorithm, which enables extension of our progress to obtaining mild initial approximations to the roots. We conclude with summarizing our algorithms and their extension to the approximation of isolated multiple roots and root clusters.

cs.SC↗