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 Liu^{1,2}, Hong-Hua Dai^{1}, Satya N. Atluri^{1}
Center for Aerospace Research & Education, University of California, Irvine
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw
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.
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.
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