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The paper analyzes special cyclic Jacobi methods for symmetric matrices oforder $4$. Only those cyclic pivot strategies that enable full parallelizationof the method are considered. These strategies, unlike the serial pivotstrategies, can force the method to be very slow or very fast within one cycle,depending on the underlying matrix. Hence, for the global convergence proof onehas to consider two or three adjacent cycles. It is proved that for anysymmetric matrix $A$ of order~$4$ the inequality$S(A^{[2]})\leq(1-10^{-5})S(A)$ holds, where $A^{[2]}$ results from $A$ byapplying two cycles of a particular parallel method. Here $S(A)$ stands for theFrobenius norm of the strictly upper-triangular part of $A$. The result holdsfor two special parallel strategies and implies the global convergence of themethod under all possible fully parallel strategies. It is also proved that forevery $\epsilon>0$ and $n\geq4$ there exist a symmetric matrix $A(\epsilon)$ oforder $n$ and a cyclic strategy, such that upon completion of the first cycleof the appropriate Jacobi method the inequality $S(A^{[1]})>(1-\epsilon)S(A(\epsilon))$ holds.

On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies