Approximating Complex Polynomial Zeros: Modified Weyl's Quadtree Construction and Improved Newton's Iteration

Pan, Victor Y.

doi:10.1006/jcom.1999.0532

Cited by 55 publications

(66 citation statements)

References 49 publications

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“…Our root-finder remains nearly optimal (up to polylogarithmic factors) even for the more limited tasks of approximating a single root or a few roots of a polynomial, but in these cases the computational cost is slightly lower and the implementation is simpler in our distinct approaches, which use no splitting (Pan, 1987(Pan, , 2000.…”

Section: Recent Progress Our Results and Techniques And Some Furthementioning

confidence: 97%

“…(Schönhage, 1982b) (Cf. also Pan, 2000) O(n log 2 n) ops performed with O(n)-bit precision are sufficient to approximate within the relative error bound c/n d (for any fixed pair of c > 0 and d ≥ 0) all root radii r j of a polynomial p(x), j = 1, . .…”

Section: Some Basic Definitions and Resultsmentioning

confidence: 99%

“…Smale (1981) and then Schönhage (1982b) raised the question of devising optimal numerical polynomial root-finders, which would use smaller computational time, and proposed some effective algorithms. In the vast literature on root-finding (McNamee, 1993(McNamee, , 1997, including thousands of publications, not many items are devoted to this important problem, however see Bini and Pan (to appear), Kim and Sutherland (1994), Kirrinnis (1998), Neff andReif (1994, 1996), Pan (1987Pan ( , 1995aPan ( ,b, 1996Pan ( , 2000, Renegar (1987), Schönhage (1982b) and Smale (1981Smale ( , 1985. The decisive steps to yield optimality were made quite recently.…”

Section: Some History Of Polynomial Root-findingmentioning

confidence: 97%

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Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Pan

2002

Journal of Symbolic Computation

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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zero-free annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new root-finder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and root-finding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of well-conditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be ill-conditioned, forming clusters. (The worst case complexity bounds for root-finding are supported by our previous algorithms as well.) All algorithms allow processor efficient acceleration to achieve solution in polylogarithmic parallel time.

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Section: Recent Progress Our Results and Techniques And Some Furthementioning

confidence: 97%

Section: Some Basic Definitions and Resultsmentioning

confidence: 99%

Section: Some History Of Polynomial Root-findingmentioning

confidence: 97%

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Univariate Polynomials: Nearly Optimal Algorithms for Numerical Factorization and Root-finding

Pan

2002

Journal of Symbolic Computation

Self Cite

159

175

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“…Indeed, many alternative ideas exist: the Durand-Kerner [31], the Ehrlich-Aberth [11], or the JenkinsTraub algorithms [29], and Weyl's method [42] to name but a few.…”

Section: Introductionmentioning

confidence: 99%

On the stability of computing polynomial roots via confederate linearizations

Nakatsukasa

Noferini

2015

Math. Comp.

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A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems to be known for other polynomial bases. This paper studies the stability of algorithms that compute the roots via linearization in nonmonomial bases, and has three goals. First we prove normwise stability when the polynomial is properly scaled and the QZ algorithm (as opposed to the more commonly used QR algorithm) is applied to a comrade pencil associated with a Jacobi orthogonal polynomial. Second, we extend a result by Arnold that leads to a first-order expansion of the backward error when the eigenvalues are computed via QR, which shows that the method can be unstable. Finally, we focus on the special case of Chebyshev basis, in particular the Chebfun rootfinder: we discuss its stability and describe an optional functionality, made available for improved stability, for computing the roots of a general continuous function f (x), implemented in the recently updated version 5. The main message is that to guarantee backward stability QZ applied to a properly scaled pencil is necessary.

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“…The literature on the computation of polynomial zeros and bounds on such zeros is extensive, and we refer to [1], [4], [5], [6], [11], [12], [13], [15], [16], [20], [21], [22], [25], [26], [27], [28], [31], [32], [34], [37], [38], and the references therein, to name but a few. Most of these take a linear algebra approach, but some do not.…”

Section: Introductionmentioning

confidence: 99%