An iterative method using an optimal descent vector, for solving an Ill-conditioned system Bx = b, better and faster than the conjugate gradient method

To solve an ill-conditioned system of linear algebraic equations (LAEs): Bxb = 0, we define an invariant-manifold in terms of r := Bxb, 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.