@Article{cmes.2012.084.575, AUTHOR = {Chein-Shan Liu, Satya N. Atluri}, TITLE = {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}, JOURNAL = {Computer Modeling in Engineering \& Sciences}, VOLUME = {84}, YEAR = {2012}, NUMBER = {6}, PAGES = {575--602}, URL = {http://www.techscience.com/CMES/v84n6/25829}, ISSN = {1526-1506}, 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.}, DOI = {10.3970/cmes.2012.084.575} }