## 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 language | English |
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Pages (from-to) | 195-227 |

Number of pages | 33 |

Journal | CMES - Computer Modeling in Engineering and Sciences |

Volume | 81 |

Issue number | 2 |

State | Published - 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)