New Practical Advances in Polynomial Root Clustering

Imbach, Rémi; Pan, Victor Y.

doi:10.1007/978-3-030-43120-4_11

Cited by 8 publications

(11 citation statements)

References 17 publications

(12 reference statements)

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“…Several approaches in the literature guarantee that a neighborhood of a point contains a unique root of a given polynomial. We may cite notably Kantorovich criterion [12, §3.2], Smale's alpha theorem [12, §3.3], Newton interval method [34, Theorem 5.1.7], Pellet's test [31], Pellet's test combined with Graeffe iteration [3,25], Cauchy's integral theorem [24,23], and others [38], . .…”

Section: Certification Of the Rootsmentioning

confidence: 99%

New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

Moroz

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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We present a new data structure to approximate accurately and efficiently a polynomial f of degree d given as a list of coefficients f i . Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than 1/2 or greater than 2.Given a polynomial f of degree d with f 1 = |f i | ≤ 2 τ for τ ≥ 1, isolating all its complex roots or evaluating it at d points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least d 3/2 bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer m, we can compute our new data structure and evaluate f at d points in the unit disk with an absolute error less than 2 −m in O(d(τ + m)) bit operations, where O(•) means that we omit logarithmic factors. We also show that if κ is the absolute condition number of the zeros of f , then we can isolate all the roots of f in O(d(τ + log κ)) bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in Python/NumPy that is an order of magnitude faster than the state-of-the-art solver MPSolve for high degree polynomials with random coefficients.

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Section: Certification Of the Rootsmentioning

confidence: 99%

New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

Moroz

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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“…By following [7,13,20], we compute approximation s * 0 to s 0 by means of the evaluation of p ′ /p on q points of ∂∆. We give an effective 2 (i.e.…”

Section: Root Clustering Problem (Rcp) Is a Global Version Of Lcpmentioning

confidence: 99%

“…[15,16,19] achieve a nearly optimal complexity in the real case; [9] implements the algorithm of [19]. Much more rudimentary variants of our algorithms and of their implementation appeared in [14] and [7], respectively. In Remark 8 we comment on a technical link to [20].…”

Section: Previous Workmentioning

confidence: 99%

“…However, we cannot ensure its success unless we know that ∆ is ρ-isolated for ρ noticeably exceeding 1, and this disqualifies its use as an exclusion test within a subdivision framework of [1]. Our alternative version of the P * -test in [7] works in the case where ρ is not known, and we used it as an exclusion test while confirming its output with the T * -test.…”

Section: An Almost Sure Exclusion Testmentioning

confidence: 99%

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New progress in univariate polynomial root finding

Imbach

Pan

2020

Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation

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The recent advanced sub-division algorithm is nearly optimal for the approximation of the roots of a dense polynomial given in monomial basis; moreover, it works locally and slightly outperforms the user's choice MPSolve when the initial region of interest contains a small number of roots. Its basic and bottleneck block is counting the roots in a given disc on the complex plain based on Pellet's theorem, which requires the coefficients of the polynomial and expensive shift of the variable. We implement a novel method for both root-counting and exclusion test, which is faster, avoids the above requirements, and remains efficient for sparse input polynomials. It relies on approximation of the power sums of the roots lying in the disc rather than on Pellet's theorem. Such approximation was used by Schönhage in 1982 for the different task of deflation of a factor of a polynomial provided that the boundary circle of the disc is sufficiently well isolated from the roots. We implement a faster version of root-counting and exclusion test where we do not verify isolation and significantly improve performance of subdivision algorithms, particularly strongly in the case of sparse inputs. We present our implementation as heuristic and cite some relevant results on its formal support presented elsewhere.

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“…For real roots, the criteria that one may use are notably the Descartes' rule of signs ([29] and references therein), the Budan's theorem [17,31,36], or the Sturm's theorem [1] among others. For complex roots, one may use Pellet's test [2] or Cauchy's integral theorem [20,21] among others. Combining subdivision approaches with Newton iterations allows to match the complexity bound of Pan's algorithm for real [30].…”

Section: Introductionmentioning

confidence: 99%

Fast real and complex root-finding methods for well-conditioned polynomials

Moroz¹

2021

Preprint

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Given a polynomial of degree and a bound on a condition number of , we present the first root-finding algorithms that return all its real and complex roots with a number of bit operations quasi-linear in log 2 ( ). More precisely, several condition numbers can be defined depending on the norm chosen on the coefficients of the polynomial. Let ( )We call the condition number associated with a perturbation of the the hyperbolic condition number ℎ , and the one associated with a perturbation of the the elliptic condition number . For each of these condition numbers, we present algorithms that find the real and the complex roots of in log 2 ( ) polylog(log( )) bit operations.Our algorithms are well suited for random polynomials since ℎ (resp. ) is bounded by a polynomial in with high probability if the (resp. the ) are independent, centered Gaussian variables of variance 1.

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New Practical Advances in Polynomial Root Clustering

Cited by 8 publications

References 17 publications

New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

New data structure for univariate polynomial approximation and applications to root isolation, numerical multipoint evaluation, and other problems

New progress in univariate polynomial root finding

Fast real and complex root-finding methods for well-conditioned polynomials

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