TY - EJOU
AU - Liu, Chein-Shan
AU - Dai, Hong-Hua
AU - Atluri, Satya N.
TI - 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
T2 - Computer Modeling in Engineering \& Sciences
PY - 2011
VL - 81
IS - 2
SN - 1526-1506
AB - 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.
KW - Nonlinear algebraic equations
KW - Optimal iterative algorithm (OIA)
KW - Optimal descent vector (ODV)
KW - Optimal vector driven algorithm (OVDA)
KW - Fictitious time integration method (FTIM)
KW - Residual-norm based algorithm (RNBA)
KW - Duffing equation
KW - Post-conditioned harmonic balance method (PCHB)
DO - 10.3970/cmes.2011.081.195