On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies
Provides a rigorous convergence guarantee for a class of parallel Jacobi methods, addressing a known gap in the analysis of cyclic pivot strategies for small symmetric matrices.
The paper proves global convergence of parallel Jacobi methods for symmetric 4x4 matrices, showing that after two cycles the Frobenius norm of off-diagonal elements decreases by at least a factor of 1-10^{-5}. It also shows that for larger matrices, a single cycle can achieve arbitrarily slow convergence.
The paper analyzes special cyclic Jacobi methods for symmetric matrices of order $4$. Only those cyclic pivot strategies that enable full parallelization of the method are considered. These strategies, unlike the serial pivot strategies, 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 one has to consider two or three adjacent cycles. It is proved that for any symmetric matrix $A$ of order~$4$ the inequality $S(A^{[2]})\leq(1-10^{-5})S(A)$ holds, where $A^{[2]}$ results from $A$ by applying two cycles of a particular parallel method. Here $S(A)$ stands for the Frobenius norm of the strictly upper-triangular part of $A$. The result holds for two special parallel strategies and implies the global convergence of the method under all possible fully parallel strategies. It is also proved that for every $ε>0$ and $n\geq4$ there exist a symmetric matrix $A(ε)$ of order $n$ and a cyclic strategy, such that upon completion of the first cycle of the appropriate Jacobi method the inequality $S(A^{[1]})> (1-ε)S(A(ε))$ holds.