K-optimal preconditioners based on approximations of inverse matrices

Abstract

The problem of constructing preconditioners of a special kind for solving systems of linear algebraic equations is considered. A new approach to the construction of preconditioners based on minimizing the K-number of conditionality for the A P matrix is proposed, where A is the initial matrix of the system, P is the preconditioner. It is proved that for circulant matrices, this approach is equivalent to constructing an optimal Chen circulant for the inverse matrix. Numerical experiments have been carried out on a series of test problems with Toeplitz matrices, showing that the proposed approach makes it possible to significantly reduce the number of iterations of the conjugate gradient method compared with the classical approach. The results obtained open up new possibilities for constructing effective preconditioners in other classes of matrices.