The power method and beyond

For computing the dominant eigenvalue and the corresponding eigenvector of a real and symmetric matrix, inspired by the classic and powerful power method, we construct a general paradigm of nonstationary Richardson methods and gradient descent methods, called also as the parameterized power methods, and establish their convergence theory. This paradigm also includes the power method as a special case. Both theoretical analysis and numerical experiments show that the parameterized power methods can result in iteration methods that may be much more effective than the power method, provided the involved iteration parameters are chosen appropriately.