On Asymptotic Convergence of Nonsymmetric Jacobi Algorithms

The asymptotic convergence behavior of cyclic versions of the nonsymmetric Jacobi algorithm for the computation of the Schur form of a general complex matrix is investigated. Similar to the symmetric case, the nonsymmetric Jacobi algorithm proceeds by applying a sequence of rotations that annihilate a pivot element in the strict lower triangular part of the matrix until convergence to the Schur form of the matrix is achieved. In this paper, it is shown that the cyclic nonsymmetric Jacobi method converges locally and asymptotically quadratically under mild hypotheses if special ordering schemes are chosen, namely, ordering schemes that lead to so-called northeast directed sweeps. The theory is illustrated by the help of numerical experiments. In particular, it is shown that there are ordering schemes that lead to asymptotic quadratic convergence for the cyclic symmetric Jacobi method, but only to asymptotic linear convergence for the cyclic nonsymmetric Jacobi method. Finally, a generalization of the nonsymmetric Jacobi method to the computation of the Hamiltonian Schur form for Hamiltonian matrices is introduced and investigated.