Root refinement for real polynomials using quadratic interval refinement

Journal of Symbolic Computation

Sharma

et al. 2018

Self Cite

143

We describe a subdivision algorithm for isolating the complex roots of a polynomial F ∈ C [x]. Given an oracle that provides approximations of each of the coefficients of F to any absolute error bound and given an arbitrary square B in the complex plane containing only simple roots of F , our algorithm returns disjoint isolating disks for the roots of F in B.Our complexity analysis bounds the absolute error to which the coefficients of F have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square B, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the benchmark problem, namely, to isolate all the roots of a polynomial of degree n with integer coefficients of bit size less than τ , our algorithm needsÕ(n 3 + n 2 τ ) bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique.Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.

Section: Bit Complexitymentioning

confidence: 99%

A near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iteration

Becker

Journal of Symbolic Computation

Sharma

et al. 2018

Self Cite

143

“…They combine a fast convergence method (i.e., the secant method and Newton iteration, respectively) with approximate arithmetic and efficient multipoint evaluation; however, there are details missing in [31] when using multipoint evaluation. In order to achieve complexity bounds comparable to the one stated in Theorem 3, the methods from [18,31] need as input isolating intervals whose size is comparable to the separation of the corresponding root, that is, the roots must be "well isolated". This is typically achieved by using a fast method, such as Pan's method, for complex root isolation first.…”

Section: Related Workmentioning

confidence: 99%

“…, 2 · n/2 } is a multipoint of size 2 · n/2 + 1. (B) execution of the 0-Test/1-Test for an interval (a , b ) ⊂ (a, b), where a and b are admissiblepoints of corresponding multipoints contained in I 18.…”

mentioning

confidence: 99%

Computing real roots of real polynomials

Journal of Symbolic Computation

Mehlhorn

2016

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Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 1 and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity.In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted ANewDsc, which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. ANewDsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task.

“…Also, exact evaluation of a sparse polynomial at a rational point (even of small bitsize) is expensive as the output has bitsize linear in n. Instead, we consider approximate evaluation, which allows us to evaluate a sparse polynomial f as in (1.1) at an arbitrary point x ∈ (0, 1 + 1/n) to an absolute error less than 2 −L in a time that is polynomial 2 in log n, k, τ, and L. More precisely, we derive the following result: Proof. In essence, we follow the same approach as in [7]. That is, for a fixed non-negative integer K, we perform each occurring operation • (i.e.…”

Section: Polynomial Arithmeticmentioning

confidence: 99%

“…The additional cost for refining isolating intervals to a size less than 2 −τ , and thus for computing L-bit approximations of all real roots, is Õ(nτ ); e.g. see [7,10,13,16]. Notice that, for k−nomials with integer coefficients, the above bounds are not polynomial in the size of the sparse input representation of f , which is bounded by O(k(log n + τ )) as we need log n bits to store each exponent ei and τ + 1 bits to store each fi.…”

Section: Introduction 11 Problem Definition and Contributionmentioning

confidence: 99%

Efficiently Computing Real Roots of Sparse Polynomials

Jindal

Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation

2017

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We propose an efficient algorithm to compute the real roots of a sparse polynomial f ∈ R[x] having k non-zero realvalued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer L, our algorithm returns disjoint disks ∆1, . . . , ∆s ⊂ C, with s < 2k, centered at the real axis and of radius less than 2 −L together with positive integers µ1, . . . , µs such that each disk ∆i contains exactly µi roots of f counted with multiplicity. In addition, it is ensured that each real root of f is contained in one of the disks. If f has only simple real roots, our algorithm can also be used to isolate all real roots. The bit complexity of our algorithm is polynomial in k and log n, and near-linear in L and τ , where 2 −τ and 2 τ constitute lower and upper bounds on the absolute values of the non-zero coefficients of f , and n is the degree of f . For root isolation, the bit complexity is polynomial in k and log n, and near-linear in τ and log σ −1 , where σ denotes the separation of the real roots.