Optimization of the factorized preconditioners of conjugate gradient method for solving the linear algebraic systems with symmetric positive definite matrix

In the paper we consider the iterative solution of linear system $Ax=b$ by the conjugate gradient method using the factorized preconditioner $B=(I+LZ)Y(I+ZU)$, where $A=D+L+U$ is the additive splitting of the coefficient matrix into the strictly lower triangular, the diagonal, and the strictly upper triangular parts. For an arbitrary symmetric positive definite matrix $A$, the diagonal matrices $Y > 0$ and $Z$ are constructed as the minimizers of a certain upper bound for the K-condition number of the inverse preconditioned matrix. The main advantages of the new method are as follows: wide range of applicability, low operation number count per iteration, good parallelizability for all the stages of computation, and sufficient reduction of the iteration number (for a properly chosen preconditioning parameters). Numerical results are given for several test problems.