Chebyshev iterations based on adaptive update of the lower bound of the spectrum of the matrix

For numerical solution of symmetric systems of linear equations with a positive-definite matrix an adaptive Chebyshev iterative method is constructed. In this method, the unknown lower bound of the spectrum of the matrix is refined in cycle of the outer iterations; the upper bound is taken by the Gershgorin theorem. Such procedure ensures the convergence of the iterations with computational costs close to the costs of the Chebyshev method, which uses the exact boundaries of the spectrum of the matrix.