Open Access

ARTICLE

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 Liu^1,2, Hong-Hua Dai¹, Satya N. Atluri¹

1 Center for Aerospace Research & Education, University of California, Irvine
2 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.

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