TY - EJOU
AU - Liu, Chein-Shan
AU - Atluri, Satya N.
TI - An Iterative Method Using an Optimal Descent Vector, for Solving an Ill-Conditioned System Bx=b, Better and Faster than the Conjugate Gradient Method
T2 - Computer Modeling in Engineering \& Sciences
PY - 2011
VL - 80
IS - 3&4
SN - 1526-1506
AB - 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.
KW - Linear algebraic equations
KW - Ill-conditioned linear system
KW - Conjugate Gradient Method (CGM)
KW - Optimal Iterative Algorithm with an Optimal Descent Vector (OIA/ODV)
KW - Invariant-manifold
KW - Linear PDE
DO - 10.3970/cmes.2011.080.275