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Root-Refining for a Polynomial Equation

  • Victor Y. Pan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

Polynomial root-finding usually consists of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iterations. The efficiency of computing an initial approximation resists formal study, and the users employ empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q 1/α where q is the convergence order and α is the number of function evaluations per iteration. To cover iterations not reduced to function evaluations alone, e.g., ones simultaneously refining n approximations to all n roots of a degree n polynomial, we let d denote the number of arithmetic operations involved in an iteration divided by 2n because we can evaluate such a polynomial at a point by using 2n operations. For this task we employ recursive polynomial factorization to yield refinement with the efficiency \(2^{cn/\log^2 n}\) for a positive constant c. For large n this is a dramatic increase versus the record efficiency 2 of refining an approximation to a single root of a polynomial. The advance could motivate practical use of the proposed root-refiners.

Keywords

Root-refining Efficiency Polynomial factorization 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Victor Y. Pan
    • 1
  1. 1.Department of Mathematics and Computer ScienceLehman College and the Graduate Center of the City University of New YorkBronxUSA

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