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An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method

Chein-Shan Liu1,2, 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, 80(3&4), 275-298. https://doi.org/10.3970/cmes.2011.080.275

Abstract

To solve an ill-conditioned system of linear algebraic equations (LAEs): Bx - b = 0, we define an invariant-manifold in terms of r := Bx - b, and a monotonically increasing function Q(t) of a time-like variable t. Using this, we derive an evolution equation for dx / dt, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of t. Using the concept of discrete dynamics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involving an Optimal Descent Vector (ODV). The presently used ODV is a modification of the Descent Vector used in the well-known and widely used Conjugate Gradient Method (CGM). The presently proposed OIA/ODV is shown, through several examples, to converge faster, with better accuracy, than the CGM. The proposed method has the potential for a wide-applicability in solving the LAEs arising out of the spatial-discretization (using FEM, BEM, Trefftz, Meshless, and other methods) of Partial Differential Equations.

Keywords

Linear algebraic equations, Ill-conditioned linear system, Conjugate Gradient Method (CGM), Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV), Invariant-manifold, Linear PDE

Cite This Article

Liu, C., Atluri, S. N. (2011). An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method. CMES-Computer Modeling in Engineering & Sciences, 80(3&4), 275–298.



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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