The Symm-Wilkinson method for improving an approximate eigenvalue and its associated eigenvector

Here is discussed the Symm-Wilkinson method (called a relaxed algorithm in 4]) for improving an approximate simple eigenvalue of ann×n matrix and a corresponding approximate eigenvector which were obtained by some method. It is shown that their method is a Newton-like method applied to a system of nonlinear equations so that the process converges linearly under the usual assumptions. The Symm-Wilkinson method needs more multiplications than the standard Newton-like method applied to the same equations byn?1 at each step. Therefore, there does not seem to be any great advantage in using the former in place of the latter.