On the convergence of complex Jacobi methods

Vjeran Hari; Erna Begovic

doi:10.1080/03081087.2019.1604622

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arXiv · 1810.12720

On the convergence of complex Jacobi methods

Vjeran Hari·Erna Begovic

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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 $S(A')\leq\gamma{S(A)}$, 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.

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Vjeran Hari, Erna Begovic. 2018-10-30. On the convergence of complex Jacobi methods. https://doi.org/10.1080/03081087.2019.1604622

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