On the convergence of complex Jacobi methods.

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 ′) ≤ γ S (A) , where A ′ is obtained from A by applying one or more cycles of the Jacobi method and S (⋅) stands for the off-diagonal 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. [ABSTRACT FROM AUTHOR]