An effective implementation of a modified Laguerre method for the roots of a polynomial

Abstract

Two common strategies for computing all roots of a polynomial with Laguerre’s method are explicit deflation and Maehly’s procedure. The former is only a semi-stable process and is not suitable for solving large degree polynomial equations. In contrast, the latter implicitly deflates the polynomial using previously accepted roots and is, therefore, a more practical strategy for solving large degree polynomial equations. However, since the roots of a polynomial are computed sequentially, this method cannot take advantage of parallel systems. In this article, we present an implementation of a modified Laguerre method for the simultaneous approximation of all roots of a polynomial. We provide a derivation of this method along with a detailed analysis of our algorithm’s initial estimates, stopping criterion, and stability. Finally, the results of several numerical experiments are provided to verify our analysis and the effectiveness of our algorithm.

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Acknowledgements

We are grateful to the NAG for providing a free trial that was used in the testing of our work. In addition, we are thankful for numerous private conversations with Dario Bini and David Watkins throughout our research and for the referees’ comments and suggestions that significantly improved this manuscript.

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Correspondence to Thomas R. Cameron.

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Cameron, T.R. An effective implementation of a modified Laguerre method for the roots of a polynomial. Numer Algor 82, 1065–1084 (2019). https://doi.org/10.1007/s11075-018-0641-9

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Keywords

  • Laguerre’s method
  • Polynomial roots
  • Mathematical software

Mathematics Subject Classification (2010)

  • 26C10
  • 65H04
  • 65Y20