Abstract
The problem of constructing preconditioners of a special type for solving systems of linear algebraic equations is considered. A new approach to constructing preconditioners based on minimizing the K-condition number for the matrix \({{A}^{{ - 1}}}P\) , where \(A\) is the initial matrix of the system and \(P\) is the preconditioner, is proposed. It is proved that for circulant matrices such an approach is equivalent to constructing the optimal Chan circulant for the inverse matrix. Numerical experiments are carried out on a series of benchmark problems with Toeplitz matrices, which show that the proposed approach allows one to significantly reduce the number of iterations of the conjugate gradient method compared to the classical approach. The obtained results open up new possibilities for constructing effective preconditioners in other classes of matrices.
