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ARTICLE

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

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 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 α_c in the best descent vector u = α_cF + B^TF, 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 = α_cF + B^TF; 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.

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