A further study on using ẋ = λ [αR + βP] (P = F-R(F · R)/∥R∥ ²) and ẋ = λ[αF + βP ^*] (P ^* = R - F (F · R)/∥F∥ ²) in iteratively solving the nonlinear system of algebraic equations F(x) = 0

In this continuation of a series of our earlier papers, we define a hypersurface 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 dh/dx, we consider the evolution equation x = λ [αR + βP], where P = F - R(F · R)/∥R∥ ² such that P · R = 0; or ẋ = 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.