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

Chein Shan Liu, Hong Hua Dai, Satya N. Atluri

Research output: Contribution to journalArticle

16 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)195-227
Number of pages33
JournalCMES - Computer Modeling in Engineering and Sciences
Volume81
Issue number2
StatePublished - 2011

Keywords

  • Duffing equation
  • Fictitious time integration method (FTIM)
  • Nonlinear algebraic equations
  • Optimal descent vector (ODV)
  • Optimal iterative algorithm (OIA)
  • Optimal vector driven algorithm (OVDA)
  • Post-conditioned harmonic balance method (PCHB)
  • Residual-norm based algorithm (RNBA)

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