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ARTICLE

An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F(x) = 0, Using the System of ODEs with an Optimum α in x^· = λ[αF + (1−α)B^TF]; B_ij = ∂F_i/∂x_j

by Chein-Shan Liu¹, Satya N. Atluri²

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw
2 Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2011, 73(4), 395-432. https://doi.org/10.3970/cmes.2011.073.395

Abstract

In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x^· = λ[αF + (1−α)B^TF], where λ and α are scalars and B_ij = ∂F_i/∂x_j. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian (tangent stiffness matrix) B. Here, we introduce three new concepts of attracting set, bifurcation and optimal combination, which are controlled by two parameters γ and α. Because we have derived all the related quantities explicitly in terms of F and its differentials, the attracting set, and an optimal α can be derived exactly. When γ changes from zero to a positive value the present algorithms undergo a Hopf bifurcation, such that the convergence speed is much faster than that by using γ = 0. Moreover, when the optimal α is used we can further accelerate the convergence speed several times. Some numerical examples are used to validate the performance of the present algorithms, which reveal a very fast convergence rate in finding the solution, and which display great efficiencies and accuracies than achieved before.

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