engleski

On the convergence of the Jacobi methods under certain periodic pivot strategies

Jacobi method for computing the eigensystem of a symmetric matrix A is the iterative process of the form A^(k+1)=U_k*A^(k)U_k, k>=0, A^(0)=A, where U_k, k>=0, are orthogonal matrices generated by the method. It is known that this process converges under some classes of cyclic pivot strategies and among them the class of weakly wavefront strategies is best known and understood. We consider several new classes of periodic pivot strategies. They include certain cyclic and quasi–cyclic strategies. The convergence proofs use different tools and one of them is the theory of Jacobi operators introduced by Henrici and Zimmermann. The new strategies can also be used with block Jacobi methods, but then a new tool, the theory of block Jacobi operators, is used. The obtained results and the new tools can be used for proving convergence of standard and block Jacobi–type methods for solving other eigenvalue and singular value problems.

Jacobi methods ; pivot strategies ; convergence

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