@Article{cmes.2012.084.575,
AUTHOR = {Chein-Shan  Liu, Satya N.  Atluri},
TITLE = {A Globally Optimal Iterative Algorithm Using the Best Descent Vector x<sup style='margin-left:-6.5px'>·</sup> = λ[α<sub>c</sub>F + B<sup>T</sup>F], with the Critical Value α<sub>c</sub>, 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 <i>best descent vector</i> <b>u</b> in <b>x<sup style="margin-left:-5px">·</sup> = λu</b> is proposed to solve a system of nonlinear algebraic equations (NAEs): <b>F(x) = 0</b>. In terms of the residual vector <b>F</b> and a monotonically increasing positive function <i>Q(t)</i> of a time-like variable <i>t</i>, we define a <b>future cone</b> in the Minkowski space, wherein the discrete dynamics of the proposed algorithm evolves. A new method to approximate the <b>best descent vector</b> is developed, and we find a critical value of the weighting parameter α<sub>c</sub> in the best descent vector <b>u = α<sub>c</sub>F + B<sup>T</sup>F</b>, where <b>B = ∂F/∂x</b> is the Jacobian matrix. We can prove that such an algorithm leads to the largest convergence rate with the descent vector given by <b>u = α<sub>c</sub>F + B<sup>T</sup>F</b>; hence we label the present algorithm as a <b>globally optimal iterative algorithm</b> (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}
}