On the Complexity of Reliable Root Approximation

Kerber, Michael

doi:10.1007/978-3-642-04103-7_15

Cited by 10 publications

(32 citation statements)

References 11 publications

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“…Abbott's QIR method [1,12] is a hybrid of the simple (but inefficient) bisection method with a quadratically converging variant of the secant method. We refer to this method as EQIR, where "E" stands for "exact" in order to distinguish from the variant presented in Section 3.…”

Section: Review Of Exact Qirmentioning

confidence: 99%

“…In [12], the root refinement problem is analyzed using the just described EQIR method for the case of integer coefficients and exact arithmetic with rational numbers. For that, a sequence of EQIR steps is performed with N = 4 initially.…”

Section: Review Of Exact Qirmentioning

confidence: 99%

“…After a successful EQIR step, N is squared for the next step; after a failing step, N is set to √ N. If N drops to 2, a bisection step is performed, and N is set to 4 for the next step. In [12], a bound on the size of the interval is given, where every EQIR step will be successful proving that the method converges quadratically from this point on.…”

Section: Review Of Exact Qirmentioning

confidence: 99%

“…For the analysis, we divide the sequence of QIR steps in the refinement process into a linear sequence where the method behaves like bisection in the worst case, and a quadratic sequence where the interval is converging quadratically towards the root, following the approach in [12]. We do not require any conditions on the initial intervals except that they are disjoint and cover all real roots of f ; an initial normalization phase modifies the intervals to guarantee the efficiency of our refinement strategy.…”

Section: Introductionmentioning

confidence: 99%

“…The case of integer coefficients is often of special interest, and the problem has been investigated in previous work [12] for this restricted case. In that work, the complexity of root refinement was bounded byÕ(d 4 τ 2 + d 3 L).…”

Section: Introductionmentioning

confidence: 99%

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Efficient real root approximation

Kerber

Sagraloff

2011

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

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We consider the problem of approximating all real roots of a squarefree polynomial f . Given isolating intervals, our algorithm refines each of them to a width at most 2 −L , that is, each of the roots is approximated to L bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only requiring finite approximations of the polynomial coefficient and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating f , without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of degree, coefficient size and discriminant. Our bound improves previous work on integer polynomials by a factor of deg f and essentially matches best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms.

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Section: Review Of Exact Qirmentioning

confidence: 99%

Section: Review Of Exact Qirmentioning

confidence: 99%

Section: Review Of Exact Qirmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficient real root approximation

Kerber

Sagraloff

2011

Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation

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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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Nearly optimal refinement of real roots of a univariate polynomial

Pan

Tsigaridas

2016

Journal of Symbolic Computation

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On the Complexity of Reliable Root Approximation

Cited by 10 publications

References 11 publications

Efficient real root approximation

Efficient real root approximation

Root refinement for real polynomials using quadratic interval refinement

Nearly optimal refinement of real roots of a univariate polynomial

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