On the convergence of complex Jacobi methods

Abstract

In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix $A$ of order $n$ we find a constant $γ<1$ depending on $n$, such that $S(A&#39;)\leqγ{S(A)}$, where $A&#39;$ is obtained from $A$ by applying one or more cycles of the Jacobi method and $S(\cdot)$ stands for the off-norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky-Jacobi method for solving the positive definite generalized eigenvalue problem.