Nearly optimal refinement of real roots of a univariate polynomial

Pan, Victor Y.; Tsigaridas, Elias

doi:10.1016/j.jsc.2015.06.009

Cited by 19 publications

(35 citation statements)

References 40 publications

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“…We are confident, however, that also other hybrid methods can be modified in a way to yield comparable complexity bounds. For polynomials with integer coefficients, a recently proposed method [35,36] combines Newton iteration and bisection, achieving similar complexity bounds. In comparison to our approach, their method requires the initial isolating interval I = (a, b) to be small enough such that, except for the isolated root, there is no other (real or non-real) root of f with a distance less than (1 + ε) · b−a 2 to the center mid(I) of I, where (1 + ε) n > 2 for some n = O(log d).…”

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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“…Recall that |p q (ω i )| ≤ τ + 2 lg d + lg lg d + 2 and |p q (ω i ) − p q (ω i )| ≤ −λ + τ lg(2d) + 3/2 lg 2 d + 5/2 lg d + lg lg d + 5 for all i, [11,Lemma 16], and similar bounds hold for pq(ω i ).…”

Section: Boolean Cost Boundsmentioning

confidence: 90%

“…This would be too costly, and so instead we employ the Moenck-Borodin algorithm, which still enables us to obtain a nearly optimal root-refiner. Technically, in a relatively minor change of our algorithm, we replace the matrix Ω = [ω j(k+1) ] j,k in (3) The Moenck-Borodin algorithm uses nearly linear arithmetic time, and Kirrinnis in [2] proved that this algorithm supports multipoint polynomial evaluation at a low Boolean cost as well (see also [14], [10], [3], [11], [8], [9] for alternative proofs). Consequently our algorithm supporting Corollary 4 can be extended to support a nearly optimal Boolean cost bound for refining all simple isolated roots of a polynomial.…”

Section: Complex Root Refinementmentioning

confidence: 99%

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Accelerated approximation of the complex roots of a univariate polynomial

Pan

Tsigaridas

2014

Proceedings of the 2014 Symposium on Symbolic-Numeric Computation

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Highly efficient and even nearly optimal algorithms have been developed for the classical problem of univariate polynomial root-finding (see, e.g., [6], [7], [4], and the bibliography therein), but this is still an area of active research. By combining some powerful techniques developed in this area we devise new algorithm for refinement of isolated complex roots that have nearly optimal Boolean complexity.

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“…The cost of approximating one root of A up to a desired precision is the same as the cost of approximating all the roots [31,33]. It is O B (m 2 n(σ + τ )).…”

Section: An Application: Solving All the Polynomialsmentioning

confidence: 99%

Univariate real root isolation in an extension field and applications

Strzeboński

Tsigaridas

2019

Journal of Symbolic Computation

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We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in B α ∈ L[y], where L = Q(α) is a simple algebraic extension of the rational numbers.We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of O B (N 8 ) for isolating the real roots of B α , where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. The bound becomes O B (N 7 ) if we use Pan's algorithm for isolating the real roots. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of descartes' algorithm. For the former we prove a Boolean complexity bound of O B (N 12 ) and for the latter a bound of O B (N 5 ). We present aggregate separation bounds and complexity results for isolating the real roots of all polynomials B α k , when α k runs over all the real conjugates of α. We show that we can isolate the real roots of all polynomials in O B (N 5 ). Finally, we implemented the algorithms in C as part of the core library of MATHEMATICA and we illustrate their efficiency over various data sets.

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Nearly optimal refinement of real roots of a univariate polynomial

Cited by 19 publications

References 40 publications

Root refinement for real polynomials using quadratic interval refinement

Root refinement for real polynomials using quadratic interval refinement

Accelerated approximation of the complex roots of a univariate polynomial

Univariate real root isolation in an extension field and applications

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