Numerische Mathematik

, Volume 14, Issue 3, pp 252–263 | Cite as

A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration

  • M. A. Jenkins
  • J. F. Traub


We introduce a new three-stage process for calculating the zeros of a polynomial with complex coefficients. The algorithm is similar in spirit to the two stage algorithms studied by Traub in a series of papers. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. To obtain additional insight we recast the problem and algorithm into matrix form. The third stage is inverse iteration with the companion matrix, followed by generalized Rayleigh iteration.


Mathematical Method Matrix Form Additional Insight Companion Matrix Mathematical Algorithm 
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Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • M. A. Jenkins
    • 1
    • 2
  • J. F. Traub
    • 3
  1. 1.Stanford UniversityStanford
  2. 2.Queens UniversityKingstonCanada
  3. 3.Bell Telephone Laboratories, IncorporatedMurray HillUSA

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