Algebraic and Numerical Algorithms

Emiris, Ioannis Z.; Pan, Victor Y.; Tsigaridas, Elias

doi:10.1201/9781584888239-c17

Cited by 6 publications

(9 citation statements)

References 212 publications

(38 reference statements)

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“…Newton iteration, Graeffes method, discrete Fourier transforms) and fast algorithms for polynomial and integer multiplication. For polynomials with integer coefficients, this yields an algorithm with a bit complexity ofÕ(d 2 τ + dL) for approximating all roots to an accuracy of 2 −L , and, for square-free polynomials with arbitrary real coefficients, it still scales likẽ O(dL) for large L; see [29,18] for details. Our bound for AQIR matches this complexity if L is the dominant factor, but it is still inferior by a factor of dτ in the first term.…”

Section: Related Workmentioning

confidence: 99%

Root refinement for real polynomials using quadratic interval refinement

Kerber

Sagraloff

2015

Journal of Computational and Applied Mathematics

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We consider the problem of approximating all real roots of a square-free polynomial f with real coefficients. Given isolating intervals for the real roots and an arbitrary positive integer L, the task is to approximate each root to L bits after the binary point. Abbott has proposed the quadratic interval refinement method (QIR for short), which is a variant of Regula Falsi combining the secant method and bisection. We formulate a variant of QIR, denoted AQIR ("Approximate QIR"), that considers only approximations of the polynomial coefficients and chooses a suitable working precision adaptively. It returns a certified result for any given real polynomial, whose roots are all simple. In addition, we propose several techniques to improve the asymptotic complexity of QIR: We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. For integer coefficients, our variant improves, in theory and practice, the variant with exact integer arithmetic. Combining our approach with multipoint evaluation, we obtain near-optimal bounds that essentially match the best known theoretical bounds on root approximation as obtained by very sophisticated algorithms.

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Section: Related Workmentioning

confidence: 99%

Root refinement for real polynomials using quadratic interval refinement

Kerber

Sagraloff

2015

Journal of Computational and Applied Mathematics

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“…Together with Pan's result on approximate polynomial factorization, this yields a complexity of O(n 2 L) for the benchmark problem. Interestingly, the latter bound was not explicitly stated until recently ( [11,Theorem 3.1]).…”

Section: Introductionmentioning

confidence: 99%

Complexity Analysis of Root Clustering for a Complex Polynomial

Becker¹,

Sagraloff

Sharma

et al. 2016

Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation

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Let F (z) be an arbitrary complex polynomial. We introduce the local root clustering problem, to compute a set of natural ε-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 twofold: 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 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 Schönhage's splitting circle method. Our algorithm is relatively simple and promises to be efficient in practice.

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“…In some applications, e.g., to algebraic and geometric optimization, one seeks only the r real roots, which typically make up just a small fraction of all roots. 1 The design of efficient real root-finders is a well studied subject (see [19,Section 10.3.5], [47], [54], and the bibliography therein), but the most popular packages of subroutines for root-finding such as MPSolve 2000 [5], Eigensolve 2001 [21], and MPSolve 2012 [10] approximate the r real roots about as fast and as slow as all the n complex roots. It can be surprising, but we present some novel methods that accelerate the known numerical real root-finders by a factor of n/r, which is dramatic in various important applications.…”

Section: Introductionmentioning

confidence: 99%

Real polynomial root-finding by means of matrix and polynomial iterations

Pan

Zhao

2017

Theoretical Computer Science

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Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. The subject of devising efficient real root-finders has been long and intensively studied. Nevertheless, we propose some novel ideas and techniques and obtain dramatic acceleration of the known numerical algorithms. In order to achieve our progress we exploit the correlation between the computations with matrices and polynomials, randomized matrix computations, and complex plane geometry, extend the techniques of the matrix sign iterations, and use the structure of the companion matrix of the input polynomial. The results of our extensive numerical tests with benchmark polynomials and random matrices are quite encouraging. In particular in these tests we have consistently computed accurate approximations of the real roots of benchmark polynomials of degree up to 1024 by using the IEEE standard double precision. Moreover the number of iterations required for convergence of our algorithms grew very slowly (if at all) as we increased the degree of the univariate input polynomials and the dimension of the input matrices from 64 to 1024.

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Algebraic and Numerical Algorithms

Cited by 6 publications

References 212 publications

Root refinement for real polynomials using quadratic interval refinement

Root refinement for real polynomials using quadratic interval refinement

Complexity Analysis of Root Clustering for a Complex Polynomial

Real polynomial root-finding by means of matrix and polynomial iterations

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