A Note on Convergence and a Posteriori Error Estimates of the Classical Jacobi Method

We consider convergence and a posteriori error estimates of the classical Jacobi method for solving symmetric eigenvalue problems. The famous convergence proof of the classical Jacobi method consists of two phases. First, it is shown that all the off-diagonal elements converge to zero. Then, from a perturbation theorem, Parlett or Wilkinson shows convergence of the diagonal elements in the textbooks. Ciarlet also gives another convergence proof based on a discussion about a bounded sequence corresponding to a diagonal element. In this paper, we simplify the Ciarlet’s convergence proof. Our proof does not use any perturbation theory. Moreover, employing this approach, we obtain a posteriori error estimates for eigenvectors.