Convergence of the Cyclic and Quasi-cyclic Block Jacobi Methods

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.