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.

Originele taal-2Engels
Pagina's (van-tot)190-201
Aantal pagina's12
TijdschriftApplied Mathematics and Computation
Volume320
DOI's
StatusGepubliceerd - 1 mrt 2018

Vingerafdruk

Linear multistep Methods
Interpolation
Roots
Convergence of numerical methods
Polynomials
Derivatives
Polynomial Interpolation
Nonlinear Function
Value Function
Rate of Convergence
Interpolate
Converge
Derivative
Numerical Examples
Zero
Estimate

Citeer dit

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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.",
keywords = "Convergence rate, Iterative methods, Linear multistep methods, Nonlinear equation, Root-finder",
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Full linear multistep methods as root-finders. / van Lith, B.S.; ten Thije Boonkkamp, J.H.M.; IJzerman, W.L.

In: Applied Mathematics and Computation, Vol. 320, 01.03.2018, blz. 190-201.

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

TY - JOUR

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