Convergence to diagonal form of block Jacobi-type methods

Abstract

We provide sufficient conditions for the general sequential block Jacobi-type method to converge to the diagonal form for cyclic pivot strategies which are weakly equivalent to the column-cyclic strategy. Given a block-matrix partition $$(A_{ij})$$ ( A i j ) of a square matrix $$\mathbf {A}$$ A , the paper analyzes the iterative process of the form $$\mathbf {A}^{(k+1)} = [\mathbf {P}^{(k)}]^*\,\mathbf {A}^{(k)}\,\mathbf {Q}^{(k)}$$ A ( k + 1 ) = [ P ( k ) ] ? A ( k ) Q ( k ) , $$k\ge 0$$ k ? 0 , $$\mathbf {A}^{(0)}=\mathbf {A}$$ A ( 0 ) = A , where $$\mathbf {P}^{(k)}$$ P ( k ) and $$\mathbf {Q}^{(k)}$$ Q ( k ) are elementary block matrices which differ from the identity matrix in four blocks, two diagonal and the two corresponding off-diagonal blocks. In our analysis of convergence a promising new tool is used, namely, the theory of block Jacobi operators. Typical applications lie in proving the global convergence of block Jacobi-type methods for solving standard and generalized eigenvalue and singular value problems.