@Article{cmes.2011.080.275,
AUTHOR = {Chein-Shan  Liu, Satya N.  Atluri},
TITLE = {An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {80},
YEAR = {2011},
NUMBER = {3&4},
PAGES = {275--298},
URL = {http://www.techscience.com/CMES/v80n3&4/25757},
ISSN = {1526-1506},
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 <i>Q(t)</i> of a time-like variable <i>t</i>. Using this, we derive an evolution equation for <i>dx / dt</i>, which is a system of Nonlinear Ordinary Differential Equations (NODEs) for x in terms of <i>t</i>. 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 <i>Optimal Iterative Algorithm</i> (OIA) involving an <i>Optimal Descent Vector</i> (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.},
DOI = {10.3970/cmes.2011.080.275}
}