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Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x· = λ[αR + βP] or x· = λ[αF + βP] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F

Chein-Shan Liu1,2, Hong-Hua Dai1, Satya N. Atluri1
Center for Aerospace Research & Education, University of California, Irvine
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw

Computer Modeling in Engineering & Sciences 2011, 81(3&4), 335-363. https://doi.org/10.3970/cmes.2011.081.335

Abstract

To solve an ill- (or well-) conditioned system of Nonlinear Algebraic Equations (NAEs): F(x) = 0, we define a scalar hyper-surface h(x,t) = 0 in terms of x, and a monotonically increasing scalar function Q(t) where t is a time-like variable. We define a vector R which is related to ∂h / ∂x, and a vector P which is normal to R. We define an Optimal Descent Vector (ODV): u = αR + βP where α and β are optimized for fastest convergence. Using this ODV [x· = λu], we derive an Optimal Iterative Algorithm (OIA) to solve F(x) = 0. We also propose an alternative Optimal Descent Vector [u = αF + βP*] where P* is normal to F. We demonstrate the superior performance of these two alternative OIAs with ODVs to solve NAEs arising out of the weak-solutions for several ODEs and PDEs. More importantly, we demonstrate the applicability of solving simply, most efficiently, and highly accurately, the Duffing equation, using 8 harmonics in the Harmonic Balance Method to find a weak-solution in time.

Keywords

Nonlinear algebraic equations, Perpendicular operator, Optimal Descent Vector (ODV), Optimal Iterative Algorithm (OIA), Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV), Invariant-manifold

Cite This Article

Liu, C., Dai, H., Atluri, S. N. (2011). Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x· = λ[αR + βP] or x· = λ[αF + βP] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F. CMES-Computer Modeling in Engineering & Sciences, 81(3&4), 335–363.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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