On the convergence of complex Jacobi methods
Provides a rigorous convergence guarantee for complex Jacobi methods, which is a theoretical contribution for numerical linear algebra researchers.
Proved global convergence of complex Jacobi methods for Hermitian matrices under generalized serial pivot strategies, establishing a contraction factor γ<1 for the off-norm after each cycle. Extended the result to prove convergence of the Cholesky-Jacobi method for generalized eigenvalue problems.
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 $S(A')\leqγ{S(A)}$, where $A'$ 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.