On the boolean complexity of real root refinement

Pan, Victor Y.; Tsigaridas, Elias

doi:10.1145/2465506.2465938

Cited by 22 publications

(17 citation statements)

References 35 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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“…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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“…By lg(·) we denote the logarithm with base 2. To estimate the Boolean complexity of the algorithms supported by Corollaries 5 and 6 we apply some results from Pan and Tsigaridas (2013) [26] and Pan and Tsigaridas (2015) [27], which hold in the general case where the coefficients of the polynomials are known up to an arbitrary precision. In this section we assume that the polynomial p = p(x) has Gaussian (that is, complex integer) coefficients known exactly; the parameter λ, to be specified in the sequel, should be considered as the working precision.…”

Section: Boolean Cost Boundsmentioning

confidence: 99%

Accelerated approximation of the complex roots and factors of a univariate polynomial

Pan

Tsigaridas

2017

Theoretical Computer Science

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The algorithms of Pan (1995) [20] and Pan (2002) [22]) approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time but require a 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.1

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On the boolean complexity of real root refinement

Cited by 22 publications

References 35 publications

Root refinement for real polynomials using quadratic interval refinement

Root refinement for real polynomials using quadratic interval refinement

Univariate real root isolation in an extension field and applications

Accelerated approximation of the complex roots and factors of a univariate polynomial

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