A bibliography on roots of polynomials

McNamee, John

doi:10.1016/0377-0427(93)90064-i

Cited by 97 publications

(54 citation statements)

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“…For g = 3, 4, it is more convenient to use a polynomial root finding algorithm rather than to solve (13) algebraically, and for g ≥ 5 it is only possible to deduce Ξ in this way; see McNamee (1993) and Pan (1997) for details regarding polynomial root finding. Unfortunately, (14) is not in the usual monomial basis form that is required by most algorithms.…”

Section: Theoremmentioning

confidence: 99%

A globally convergent algorithm for lasso-penalized mixture of linear regression models

Lloyd‐Jones

Nguyen

McLachlan

2018

Computational Statistics & Data Analysis

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Variable selection is an old and pervasive problem in regression analysis. One solution is to impose a lasso penalty to shrink parameter estimates toward zero and perform continuous model selection. The lasso-penalized mixture of linear regressions model (L-MLR) is a class of regularization methods for the model selection problem in the fixed number of variables setting. In this article, we propose a new algorithm for the maximum penalized-likelihood estimation of the L-MLR model. This algorithm is constructed via the minorization-maximization algorithm paradigm. Such a construction allows for coordinate-wise updates of the parameter components, and produces globally convergent sequences of estimates that generate monotonic sequences of penalized log-likelihood values. These three features are missing in the previously presented approximate expectation-maximization algorithms. The previous difficulty in producing a globally convergent algorithm for the maximum penalized-likelihood estimation of the L-MLR model is due to the intractability of finding exact updates for the mixture model mixing proportions in the maximization-step. In our algorithm, we solve this issue by showing that it can be converted into a polynomial root finding problem. Our solution to this problem involves a polynomial basis conversion that is interesting in its own right. The method is tested in simulation and with an application to Major League Baseball salary data from the 1990s and the present day. We explore the concept of whether player salaries are associated with batting performance.

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Section: Theoremmentioning

confidence: 99%

A globally convergent algorithm for lasso-penalized mixture of linear regression models

Lloyd‐Jones

Nguyen

McLachlan

2018

Computational Statistics & Data Analysis

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“…This is typical for multivariate polynomial root-finding: symbolic techniques are used at the first stage, which can be viewed as preconditioning, followed by numerical approximation techniques at the final stage. Let us consider the case of a hidden variable x nþ1 : If we have computed the coefficients of Rðx nþ1 Þ with respect to the monomial basis, as in the algorithm supporting Corollary 6.2, we may compute all of its roots by a variety of available numerical methods (e.g., [McN93,McN97,Pan97,Pan02c,PH01] and their references). However, practically, one may prefer to rely on computing polynomial values rather than the coefficients; this shall be Problem 7.3.…”

Section: Algebraic System Solvingmentioning

confidence: 99%

“…Our assumption about at least quadratic convergence is more realistic than it may seem to be. It is proved for the input polynomials with no multiple roots (compare the effective treatment of multiple roots in [Yun76,Zen03]), provided that the initial approximations are either close enough to the roots ( [McN93,McN97,PH01] and their references) or satisfy some other readily available conditions [Bin96,Kim88,PPI03,Sma86,ZW95]. Furthermore, immense statistics and extensive experiments show very rapid convergence of these algorithms (when properly implemented) for all input polynomials (including specially devised ''hard'' polynomials) even under the primitive customary choices of the initial approximations, e.g.…”

Section: Algebraic System Solvingmentioning

confidence: 99%

Improved algorithms for computing determinants and resultants

Emiris

2004

Journal of Complexity

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Our first contribution is a substantial acceleration of randomized computation of scalar, univariate, and multivariate matrix determinants, in terms of the output-sensitive bit operation complexity bounds, including computation modulo a product of random primes from a fixed range. This acceleration is dramatic in a critical application, namely solving polynomial systems and related studies, via computing the resultant. This is achieved by combining our techniques with the primitive-element method, which leads to an effective implicit representation of the roots. We systematically examine quotient formulae of Sylvestertype resultant matrices, including matrix polynomials and the u-resultant. We reduce the known bit operation complexity bounds by almost an order of magnitude, in terms of the resultant matrix dimension. Our theoretical and practical improvements cover the highly important cases of sparse and degenerate systems.

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“…Iterative methods for the simultaneous determination of simple or multiple zeros of a polynomial are efficient tool for solving algebraic equations, see the bibliography [6] and the corresponding updating site www.elsevier.com/homepage/sac/cam/ mcnamee. More details on simultaneous methods, their convergence properties, computational efficiency, and parallel implementation may be found in, e.g., [2], [5], [8], [9], [10] and references cited there.…”

Section: Introductionmentioning

confidence: 99%

On the fourth order zero-finding methods for polynomials

Ilic

Rančić

2003

Filomat

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Abstract. The fourth order methods for the simultaneous approximation of simple complex zeros of a polynomial are considered. The main attention is devoted to a new method that may be regarded as a modification of the well known cubically convergent Ehrlich-Aberth method. It is proved that this method has the order of convergence equals four. Two numerical examples are given to demonstrate the convergence behavior of the studied methods.

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A bibliography on roots of polynomials

Cited by 97 publications

References 0 publications

A globally convergent algorithm for lasso-penalized mixture of linear regression models

A globally convergent algorithm for lasso-penalized mixture of linear regression models

Improved algorithms for computing determinants and resultants

On the fourth order zero-finding methods for polynomials

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