On the global and cubic convergence of a quasi-cyclic Jacobi method

Summary In this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.This work was supported in part by the University of Minnesota Army High Performance Computing Research Center and the U.S Army Contract DAAL03-89-C-0038. The paper was partly written while this author was a visiting faculty in the Department of Mathematics, University of Kansas, Lawrence, Kansas. The first version of this paper was made in July 1990 while this author was visiting AHPCRC.