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ARTICLE

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

by Chein-Shan Liu¹, Hong-Hua Dai¹, Satya N. Atluri¹

1 Center for Aerospace Research & Education, University of California, Irvine
2 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.

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