  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.
