@Article{cmes.2011.081.195,
AUTHOR = {Chein-Shan Liu, Hong-Hua Dai, Satya N. Atluri},
TITLE = {A Further Study on Using x· = λ[αR + βP] (P = F − R(F·R) / ||R||2) and x· = λ[αF + βP∗] (P∗ = R − F(F·R) / ||F||2) in Iteratively Solving the Nonlinear System of Algebraic Equations F(x) = 0},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {81},
YEAR = {2011},
NUMBER = {2},
PAGES = {195--228},
URL = {http://www.techscience.com/CMES/v81n2/25763},
ISSN = {1526-1506},
ABSTRACT = {In this continuation of a series of our earlier papers, we define a hyper-surface h(x,t) = 0 in terms of the unknown vector x, and a monotonically increasing function Q(t) of a time-like variable t, to solve a system of nonlinear algebraic equations F(x) = 0. If R is a vector related to ∂h / ∂x, , we consider the evolution equation x· = λ[αR + βP], where P = F − R(F·R) / ||R||2 such that P·R = 0; or x· = λ[αF + βP∗], where P∗ = R − F(F·R) / ||F||2 such that P*·F = 0. From these evolution equations, we derive Optimal Iterative Algorithms (OIAs) with Optimal Descent Vectors (ODVs), abbreviated as ODV(R) and ODV(F), by deriving optimal values of α and β for fastest convergence. Several numerical examples illustrate that the present algorithms converge very fast. We also provide a solution of the nonlinear Duffing oscillator, by using a harmonic balance method and a post-conditioner, when very high-order harmonics are considered.},
DOI = {10.3970/cmes.2011.081.195}
}