Using Selected Rates of Residual Vector Decrease for Solving Iteratively Large Linear Systems

Cycles of iteration with an imposed rate of decrease of the residual vector at the beginning of each cycle are used to derive an approximate solution of linear systems of algebraic equations. When the rate of decrease of the residual vector deteriorates significantly a new cycle of iteration is performed with the starting value of the unknown being the value computed in the preceding cycle. The latter is improved by bringing it as close as possible to the exact solution of the system by applying a formula derived in a previous work. The method is straightforward and of high efficiency for computing an approximate solution of large systems.