Vjeran Hari

Vjeran Hari, Erna Begovic

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.

Vjeran Hari, Erna Begovic

In this paper we prove the global convergence of the complex Jacobi method for Hermitian matrices for a large class of generalized serial pivot strategies. For a given Hermitian matrix $A$ of order $n$ we find a constant $γ<1$ depending on $n$, such that $S(A')\leqγ{S(A)}$, where $A'$ is obtained from $A$ by applying one or more cycles of the Jacobi method and $S(\cdot)$ stands for the off-norm. Using the theory of complex Jacobi operators, the result is generalized so it can be used for proving convergence of more general Jacobi-type processes. In particular, we use it to prove the global convergence of Cholesky-Jacobi method for solving the positive definite generalized eigenvalue problem.

Erna Begovic, Vjeran Hari

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.

Erna Begovic, Vjeran Hari

The paper studies the global convergence of the Jacobi method for symmetric matrices of size $4$. We prove global convergence for all $720$ cyclic pivot strategies. Precisely, we show that inequality $S(A^{[t+3]})\leqγS(A^{[t]})$, $t\geq1$, holds with the constant $γ<1$ that depends neither on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the matrix obtained from $A$ after $t$ full cycles of the Jacobi method and $S(A)$ is the off-diagonal norm of $A$. We show why three consecutive cycles have to be considered. The result has a direct application on the $J$-Jacobi method.