@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· = λ[αR + βP] or x· = λ[αF + βP] 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): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h / ∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x· = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x) = 0. We also propose an alternative Optimal Descent Vector [u = αF + βP*] where P* is normal to F. 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} }