Convergence of the Cyclic and Quasi-cyclic Block Jacobi Methods
Provides stronger convergence guarantees for block Jacobi methods, which are used in eigenvalue computations, but the contribution is incremental as it generalizes existing wavefront strategies.
The paper proves global convergence of block Jacobi methods for symmetric matrices under a new class of generalized serial pivot strategies, showing that the off-norm decreases by a constant factor per cycle. The results are extended to quasi-cyclic strategies and applied to the block J-Jacobi method.
The paper studies the global convergence of the block Jacobi me\-thod for symmetric matrices. Given a symmetric matrix $A$ of order $n$, the method generates a sequence of matrices by the rule $A^{(k+1)}=U_k^TA^{(k)}U_k$, $k\geq0$, where $U_k$ 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: $S(A')\leq c S(A)$, where $A'$ is the matrix obtained from $A$ after one full cycle, $c<1$ is a constant and $S(A)$ is the off-norm of $A$. 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 $J$-Jacobi method. Finally, all results are extended to the corresponding quasi-cyclic strategies.