Full linear multistep methods as root-finders

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Root-finders based on full linear multistep methods (LMMs) use previous function values, derivatives and root estimates to iteratively find a root of a nonlinear function. As ODE solvers, full LMMs are typically not zero-stable. However, used as root-finders, the interpolation points are convergent so that such stability issues are circumvented. A general analysis is provided based on inverse polynomial interpolation, which is used to prove a fundamental barrier on the convergence rate of any LMM-based method. We show, using numerical examples, that full LMM-based methods perform excellently. Finally, we also provide a robust implementation based on Brent's method that is guaranteed to converge.

Original languageEnglish
Pages (from-to)190-201
Number of pages12
JournalApplied Mathematics and Computation
Publication statusPublished - 1 Mar 2018


  • Convergence rate
  • Iterative methods
  • Linear multistep methods
  • Nonlinear equation
  • Root-finder


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