Abstract
The paper studies the global convergence of the Jacobi method for symmetric matrices of size 4. We prove global convergence for all 720 cyclic pivot strategies. Precisely, we show that inequality S(A [t+3]) ≤ γ S(A [t]), t ≥ 1, holds with the constant γ < 1 that depends neither on the matrix A nor on the pivot strategy. Here, A [t] stands for the matrix obtained from A after t full cycles of the Jacobi method and S(A) is the off-diagonal norm of A. We show why three consecutive cycles have to be considered. The result has a direct application on the J-Jacobi method.
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The authors are thankful to the anonymous reviewers for their suggestions and comments.
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This work has been fully supported by Croatian Science Foundation under the project IP-09-2014-3670.
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Begović Kovač, E., Hari, V. Jacobi method for symmetric 4 × 4 matrices converges for every cyclic pivot strategy. Numer Algor 78, 701–720 (2018). https://doi.org/10.1007/s11075-017-0396-8
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DOI: https://doi.org/10.1007/s11075-017-0396-8