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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.

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APA Style
Liu, C., Dai, H., Atluri, S.N. (2011). Iterative solution of a system of nonlinear algebraic equations f(x) = 0, using x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>r + <i>β</i>p] or x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>f + <i>β</i>p<sup>∗</sup>] R is a normal to a hyper-surface function of F, P normal to R, and P* normal to F. Computer Modeling in Engineering & Sciences, 81(3&4), 335-363. https://doi.org/10.3970/cmes.2011.081.335
Vancouver Style
Liu C, Dai H, Atluri SN. Iterative solution of a system of nonlinear algebraic equations f(x) = 0, using x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>r + <i>β</i>p] or x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>f + <i>β</i>p<sup>∗</sup>] R is a normal to a hyper-surface function of F, P normal to R, and P* normal to F. Comput Model Eng Sci. 2011;81(3&4):335-363 https://doi.org/10.3970/cmes.2011.081.335
IEEE Style
C. Liu, H. Dai, and S.N. Atluri "Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>R + <i>β</i>P] or x<sup style='margin-left:-6.5px'>·</sup> = λ[<i>α</i>F + <i>β</i>P<sup>∗</sup>] R is a Normal to a Hyper-Surface Function of F, P Normal to R, and P* Normal to F," Comput. Model. Eng. Sci., vol. 81, no. 3&4, pp. 335-363. 2011. https://doi.org/10.3970/cmes.2011.081.335



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