An optimized iterative method for numerical solution of large systems of equations based on the extremal property of zeroes of Chebyshev polynomials

An iterative method suitable for numerical solution of large systems of equations is presented. An extremal property of the Chebyshev polynomials is established, providing a logical foundation for the proposed procedure. A modification of the method is applicable for evaluation of the maximal eigenvalue of a matrix with real eigenvalues and of the associated eigenvector.