@Article{cmes.2011.073.395,
AUTHOR = {Chein-Shan  Liu, Satya N.  Atluri},
TITLE = {An Iterative Algorithm for Solving a System of Nonlinear Algebraic Equations, F(x) = 0, Using the System of ODEs with an Optimum α in x<sup style='margin-left:-6.5px'>·</sup> = λ[αF + (1−α)B<sup>T</sup>F]; <i>B<sub>ij</sub> = ∂F<sub>i</sub>/∂x<sub>j</sub></i>},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {73},
YEAR = {2011},
NUMBER = {4},
PAGES = {395--432},
URL = {http://www.techscience.com/CMES/v73n4/25673},
ISSN = {1526-1506},
ABSTRACT = {In this paper we solve a system of nonlinear algebraic equations (NAEs) of a vector-form: F(x) = 0. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we derive a system of nonlinear ordinary differential equations (ODEs) with a fictitious time-like variable t as an independent variable: x<sup style="margin-left:-4.5px">·</sup> = λ[αF + (1−α)B<sup>T</sup>F], where λ and α are scalars and <i>B<sub>ij</sub> = ∂F<sub>i</sub>/∂x<sub>j</sub></i>. From this set of nonlinear ODEs, we derive a purely iterative algorithm for finding the solution vector x, without having to invert the Jacobian (tangent stiffness matrix) B. Here, we introduce three new concepts of <i>attracting set, bifurcation and optimal combination</i>, which are controlled by two parameters γ and α. Because we have derived all the related quantities explicitly in terms of <b>F</b> and its differentials, the attracting set, and an optimal α can be derived exactly. When γ changes from zero to a positive value the present algorithms undergo a Hopf bifurcation, such that the convergence speed is much faster than that by using γ = 0. Moreover, when the optimal α is used we can further accelerate the convergence speed several times. Some numerical examples are used to validate the performance of the present algorithms, which reveal a very fast convergence rate in finding the solution, and which display great efficiencies and accuracies than achieved before.},
DOI = {10.3970/cmes.2011.073.395}
}