On the convergence of complex Jacobi methods
Vjeran Hari, Erna Begović Kovač · Linear and Multilinear Algebra · 2019
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′)≤γS(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.