A globally optimal tri-vector method to solve an ill-posed linear system

Abstract

In the present paper, a future cone in the Minkowski space defined in terms of the square-norm of the residual vector for an ill-posed linear system to be solved, is used to derive a nonlinear system of ordinary differential equations. Then the forward Euler scheme is used to generate an iterative algorithm. Two critical values in the critical descent tri-vector are derived, which lead to the largest convergence rate of the resultant iterative algorithm, namely the globally optimal tri-vector method (GOTVM). Some numerical examples are used to reveal the superior performance of the GOTVM than the famous methods of conjugate gradient (CGM) and generalized minimal residual (GMRES). Through the numerical tests we also set forth the rationale by assuming the tri-vector as being a better descent direction.