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

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 mono-tonically 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 dynam-ics evolving on the invariant manifold, we arrive at a purely iterative algorithm for solving x, which we label as an Optimal Iterative Algorithm (OIA) involv-ing 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 Gradi-ent 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)