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 $\gamma <1$ depending on n, such that $S\left(A^{\mathrm{\prime }}\right)\le \gamma S\left(A\right)$, where $A^{\mathrm{\prime }}$ is obtained from A by applying one or more cycles of the Jacobi method and $S(\cdot )$ 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.

Keywords:

• Complex Jacobi method
• complex Jacobi operators
• global convergence
• generalized eigenvalue problem
• Cholesky–Jacobi method

Acknowledgements

The authors are thankful to the anonymous reviewers for their suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been fully supported by Croatian Science Foundation under the project IP-2014-09-3670.