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A Further Study on Using x· = λ[αR + βP] (P = F − R(F·R) / ||R||2) and x· = λ[αF + βP] (P = R − F(F·R) / ||F||2) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0

Chein-Shan Liu1,2, Hong-Hua Dai1, Satya N. Atluri1
Center for Aerospace Research & Education, University of California, Irvine
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw

Computer Modeling in Engineering & Sciences 2011, 81(2), 195-228. https://doi.org/10.3970/cmes.2011.081.195

Abstract

In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x· = λ[αR + βP], where P = F − R(F·R) / ||R||2 such that P·R = 0; or x· = λ[αF + βP], where P = R − F(F·R) / ||F||2 such that P*·F = 0. From these evolution equations, we derive Optimal Iterative Algorithms (OIAs) with Optimal Descent Vectors (ODVs), abbreviated as ODV(R) and ODV(F), by deriving optimal values of α and β for fastest convergence. Several numerical examples illustrate that the present algorithms converge very fast. We also provide a solution of the nonlinear Duffing oscillator, by using a harmonic balance method and a post-conditioner, when very high-order harmonics are considered.

Keywords

Nonlinear algebraic equations, Optimal iterative algorithm (OIA), Optimal descent vector (ODV), Optimal vector driven algorithm (OVDA), Fictitious time integration method (FTIM), Residual-norm based algorithm (RNBA), Duffing equation, Post-conditioned harmonic balance method (PCHB)

Cite This Article

Liu, C., Dai, H., Atluri, S. N. (2011). A Further Study on Using x· = λ[αR + βP] (P = F − R(F·R) / ||R||2) and x· = λ[αF + βP] (P = R − F(F·R) / ||F||2) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0. CMES-Computer Modeling in Engineering & Sciences, 81(2), 195–228.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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