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Simple "Residual-Norm" Based Algorithms, for the Solution of a Large System of Non-Linear Algebraic Equations, which Converge Faster than the Newton’s Method

Chein-Shan Liu¹, Satya N. Atluri²

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail: liucs@ntu.edu.tw
2 Center for Aerospace Research & Education, University of California, Irvine

Computer Modeling in Engineering & Sciences 2011, 71(3), 279-304. https://doi.org/10.3970/cmes.2011.071.279

Abstract

For solving a system of nonlinear algebraic equations (NAEs) of the type: F(x)=0, or F_i(x_j) = 0, i,j = 1,...,n, a Newton-like algorithm has several drawbacks such as local convergence, being sensitive to the initial guess of solution, and the time-penalty involved in finding the inversion of the Jacobian matrix ∂F_i/∂x_j. Based-on an invariant manifold defined in the space of (x,t) in terms of the residual-norm of the vector F(x), we can derive a gradient-flow system of nonlinear ordinary differential equations (ODEs) governing the evolution of x with a fictitious time-like variable t as an independent variable. We can prove that in the present novel Residual-Norm Based Algorithms (RNBAs), the residual-error is automatically decreased to zero along the path of x(t). More importantly, we have derived three iterative algoritms which do not involve the fictitious time and its stepsize ∆t. We apply the three RNBAs to several numerical examples, revealing exponential convergences with different slopes and displaying the high efficiencies and accuracies of the present iterative algorithms. All the three presently proposed RNBAs: (i) are easy to implement numerically, (ii) converge much faster than the Newton's method, (iii) do not involve the inversion of the Jacobian ∂F_i/∂x_j, (iv) are suitable for solving a large system of NAEs, and (v) are purely iterative in nature.

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Keywords

Nonlinear algebraic equations, Non-Linear Ordinary Differential Equations, Non-Linear Partial Differential Equations, Residual-Norm Based Algorithms (RNBAs), Fictitious time integration method (FTIM), Iterative algorithm

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Cite This Article

Liu, C., Atluri, S. N. (2011). Simple "Residual-Norm" Based Algorithms, for the Solution of a Large System of Non-Linear Algebraic Equations, which Converge Faster than the Newton’s Method. CMES-Computer Modeling in Engineering & Sciences, 71(3), 279–304. https://doi.org/10.3970/cmes.2011.071.279

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