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 depending on n, such that , where is obtained from A by applying one or more cycles...

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Bibliographic Details
Published in:Linear & multilinear algebra 2021-02, Vol.69 (3), p.489-514
Main Authors: Hari, Vjeran, Kovač, Erna Begović
Format: Article
Language:English
Subjects:
Cholesky-Jacobi method
Complex Jacobi method
complex Jacobi operators
Convergence
Eigenvalues
generalized eigenvalue problem
global convergence
Operators (mathematics)
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Summary: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 depending on n, such that , where is obtained from A by applying one or more cycles of the Jacobi method and 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.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2019.1604622