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Full linear multistep methods as root-finders

Author

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  • van Lith, Bart S.
  • ten Thije Boonkkamp, Jan H.M.
  • IJzerman, Wilbert L.

Abstract

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.

Suggested Citation

  • van Lith, Bart S. & ten Thije Boonkkamp, Jan H.M. & IJzerman, Wilbert L., 2018. "Full linear multistep methods as root-finders," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 190-201.
    • Handle: RePEc:eee:apmaco:v:320:y:2018:i:c:p:190-201
    DOI: 10.1016/j.amc.2017.09.003
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    References listed on IDEAS

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    1. Behl, Ramandeep & Cordero, Alicia & Motsa, Sandile S. & Torregrosa, Juan R., 2015. "Construction of fourth-order optimal families of iterative methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 89-101.
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    Cited by:

    1. Mohit Garg & Suneel Sarswat, 2022. "The Design and Regulation of Exchanges: A Formal Approach," Papers 2210.05447, arXiv.org.

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