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A Globally Optimal Iterative Algorithm Using the Best Descent Vector x· = λ[αcF + BTF], with the Critical Value αc, for Solving a System of Nonlinear Algebraic Equations F(x) = 0

Chein-Shan Liu1, Satya N. Atluri2
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw
Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2012, 84(6), 575-602. https://doi.org/10.3970/cmes.2012.084.575

Abstract

An iterative algorithm based on the concept of best descent vector u in x· = λu is proposed to solve a system of nonlinear algebraic equations (NAEs): F(x) = 0. In terms of the residual vector F and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm evolves. A new method to approximate the best descent vector is developed, and we find a critical value of the weighting parameter αc in the best descent vector u = αcF + BTF, where B = ∂F/∂x is the Jacobian matrix. We can prove that such an algorithm leads to the largest convergence rate with the descent vector given by u = αcF + BTF; hence we label the present algorithm as a globally optimal iterative algorithm (GOIA). Some numerical examples are used to validate the performance of the GOIA; a very fast convergence rate in finding the solution is observed.

Keywords

Nonlinear algebraic equations, Future cone, Optimal Iterative Algorithm (OIA), Globally Optimal Iterative Algorithm (GOIA)

Cite This Article

Liu, C., Atluri, S. N. (2012). A Globally Optimal Iterative Algorithm Using the Best Descent Vector x· = λ[αcF + BTF], with the Critical Value αc, for Solving a System of Nonlinear Algebraic Equations F(x) = 0. CMES-Computer Modeling in Engineering & Sciences, 84(6), 575–602.



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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