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# Iterative Solution of a System of Nonlinear Algebraic Equations F(x) = 0, Using x^{·} = λ[*α*R + *β*P] or x^{·} = λ[*α*F + *β*P^{∗}] 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 Dai^{1}, Satya N. Atluri^{1}

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 =**where

*α*R +*β*P*α*and

*β*are optimized for fastest convergence. Using this ODV [

**x**], we derive an Optimal Iterative Algorithm (OIA) to solve

^{·}= λu**F(x)**=

**0**. We also propose an alternative Optimal Descent Vector [

**u =**] where

*α*F +*β*P***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

### 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[

*α*R +

*β*P] or x

^{·}= λ[

*α*F +

*β*P

^{∗}] 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.