@Article{cmes.2011.081.335,
AUTHOR = {Chein-Shan  Liu, Hong-Hua  Dai, Satya N.  Atluri},
TITLE = {Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>R + <i>β</i>P] or x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>F + <i>β</i>P<sup>∗</sup>] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {81},
YEAR = {2011},
NUMBER = {3&4},
PAGES = {335--363},
URL = {http://www.techscience.com/CMES/v81n3&4/25769},
ISSN = {1526-1506},
ABSTRACT = {To solve an ill- (or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): <b>F(x)</b> = <b>0</b>, we define a scalar hyper-surface <i>h</i>(x,<i>t</i>) = 0 in terms of x, and a monotonically increasing scalar function <i>Q(t)</i> where t is a time-like variable. We define a vector <b>R</b> which is related to ∂h / ∂x, and a vector <b>P</b> which is normal to <b>R</b>. We define an Optimal Descent Vector (ODV): <b>u = <i>α</i>R + <i>β</i>P</b>  where <i>α</i> and <i>β</i> are optimized for fastest convergence. Using this ODV [<b>x<sup style="margin-left:-5px">·</sup> = λu</b>], we derive an Optimal Iterative Algorithm (OIA) to solve <b>F(x)</b> = <b>0</b>. We also propose an alternative Optimal Descent Vector [<b>u = <i>α</i>F + <i>β</i>P*</b>] where <b>P*</b> is normal to <b>F</b>. We demonstrate the superior performance of these two alternative OIAs with ODVs to solve NAEs arising out of the weak-solutions for several ODEs and PDEs. More importantly, we demonstrate the applicability of solving simply, most efficiently, and highly accurately, the Duffing equation, using 8 harmonics in the Harmonic Balance Method to find a weak-solution in time.},
DOI = {10.3970/cmes.2011.081.335}
}