This article is cited in 3 papers

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

I. E. Kaporin, O. Yu. Milyukova


Abstract: 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.

Keywords: onjugate gradient method, factorized preconditioner, the K-condition number.