Convergence to diagonal form of block Jacobi-type methods

Abstract

We provide sufficient conditions for the general sequential block Jacobitype 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 i j ) of a square matrix A, the paper analyzes the iterative process of the formwhere P (k) and 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.

On the convergence of complex Jacobi methods

“…In this subsection we formulate and prove the theorem on the convergence of element-wise Jacobi-type processes for Hermitian matrices. After we have proved the bounds for the complex Jacobi operators in previous subsections, the proof of this more general result becomes similar to the corresponding proofs in [14,Corollary 5.8] and [16,Corollary 5.3].…”

“…They appear when the iteration matrices are represented by vectors. They were introduced in [24,17] and later were generalized to work with complex and block methods (see [9,13,14,16]). As we will see later, Jacobi annihilators and operators are crucial in proving the convergence of the method for some cyclic pivot strategies.…”

On the global convergence of the Jacobi method for symmetric matrices of order 4 under parallel strategies

“…Note that Theorem 2.1 also covers the cases of equivalent and shift-equivalent strategies. Another important result regarding the convergence under two weakly equivalent strategies is proved in [11,Lemma 4.8].…”

“…[12,17,19,10]). In that case the proof is valid for a more general iterative process used in the global convergence analysis of Jacobi-type processes which use nonorthogonal transformation matrices [11,1].…”

“…Its global (asymptotic) convergence has been considered in [8,9,3,12,17] ( [19,10]) and its accuracy in [4,5,6,15]. A one-sided version of the method has been studied in [13,18] and the block versions in [7,11,1]. There are many papers on Jacobi methods, and further references can be found within the bibliographies of the papers cited above.…”