Volume 46, pp. 107-147, 2017.
Convergence of the cyclic and quasi-cyclic block Jacobi methods
Vjeran Hari and Erna Begović Kovač
Abstract
This paper studies the global convergence of the block Jacobi method for
symmetric matrices.
Given a symmetric matrix of order , the method generates a sequence of
matrices by the rule , , where are
orthogonal elementary block matrices. A class of generalized serial pivot
strategies is introduced, significantly enlarging the known class of weak
wavefront strategies, and appropriate global convergence proofs are obtained.
The results are phrased in the stronger form: , where is
the matrix obtained from after one full cycle, is a constant, and
is the off-norm of . Hence, using the theory of block Jacobi
operators, one can apply the obtained results to prove convergence of block
Jacobi methods for other eigenvalue problems such as the generalized
eigenvalue problem. As an example, the results are applied to the block
-Jacobi method. Finally, all results are extended to the corresponding
quasi-cyclic strategies.
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Key words
eigenvalues, block Jacobi method, pivot strategies, global convergence
AMS subject classifications
65F15
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