TY - JOUR
T1 - Full linear multistep methods as root-finders
AU - van Lith, B.S.
AU - ten Thije Boonkkamp, J.H.M.
AU - IJzerman, W.L.
PY - 2018/3/1
Y1 - 2018/3/1
N2 - 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.
AB - 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.
KW - Convergence rate
KW - Iterative methods
KW - Linear multistep methods
KW - Nonlinear equation
KW - Root-finder
UR - http://www.scopus.com/inward/record.url?scp=85030852642&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2017.09.003
DO - 10.1016/j.amc.2017.09.003
M3 - Article
AN - SCOPUS:85030852642
VL - 320
SP - 190
EP - 201
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
SN - 0096-3003
ER -