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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

APA Style
Liu, C., Atluri, S.N. (2012). A globally optimal iterative algorithm using the best descent vector x<sup style='margin-left:-6.5px'>· = λ[αcf + btf], with the critical value αc, for solving a system of nonlinear algebraic equations f(x) = 0. Computer Modeling in Engineering & Sciences, 84(6), 575-602. https://doi.org/10.3970/cmes.2012.084.575
Vancouver Style
Liu C, Atluri SN. A globally optimal iterative algorithm using the best descent vector x<sup style='margin-left:-6.5px'>· = λ[αcf + btf], with the critical value αc, for solving a system of nonlinear algebraic equations f(x) = 0. Comput Model Eng Sci. 2012;84(6):575-602 https://doi.org/10.3970/cmes.2012.084.575
IEEE Style
C. Liu and S.N. Atluri, "A Globally Optimal Iterative Algorithm Using the Best Descent Vector x<sup style='margin-left:-6.5px'>· = λ[αcF + BTF], with the Critical Value αc, for Solving a System of Nonlinear Algebraic Equations F(x) = 0," Comput. Model. Eng. Sci., vol. 84, no. 6, pp. 575-602. 2012. https://doi.org/10.3970/cmes.2012.084.575

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