− F ( F · R ) / ‖ F ‖ 2 ) in Iteratively Solving the Nonlinear System of Algebraic Equations F ( x ) = 0
نویسندگان
چکیده
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 ∂h/∂x, we consider the evolution equation ẋ = 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.
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