On the Stability of the Chebyshev Iterative Method

Numerical solution of the systems of linear equations (linear system), especially in case of nonstationary problems, takes a significant part of computer time. Normally, for solving linear system the applied program packages use either Chebyshev iterative method (wave linear problems etc.), which requires setting the optimal parameter, or gradient iterative schemes, which do not require the initial setting of the optimal iterative parameter and the system matrix self-adjointness. For the systems of linear equations Chebyshev method is optimal in terms of convergence rate (theoretically not improvable) if the system matrix is self-adjoint and positively defined. The only downside of this method is the requirement to know the spectrum boundaries for the equations matrix being solved. This paper proposes the numerical test of the Chebyshev method stability both in classical variant and in the form of multistep scheme.