A Global Convergence Proof for Cyclic Jacobi Methods with Block Rotations

A Global Convergence Proof for Cyclic Jacobi Methods with Block Rotations

This paper introduces a globally convergent block (column– and row–) cyclic Jacobi method for diagonalization of Hermitian matrices and for computation of the singular value decomposition of general matrices. It is shown that a block rotation (generalization of the Jacobi’s 2× 2 rotation) must be computed and implemented in a particular way to guarantee global convergence. This solves a long st...

A global convergence proof of cyclic Jacobi methods with block rotations LAPACK Working Note 196

This paper introduces a globally convergent block (column– and row–) cyclic Jacobi method for diagonalization of Hermitian matrices and for computation of the singular value decomposition of general matrices. It is shown that a block rotation (generalization of the Jacobi’s 2× 2 rotation) must be computed and implemented in a particular way to guarantee global convergence. This solves a long st...

A Unified Proof for the Convergence of Jacobi and Gauss-Seidel Methods

We present a new unified proof for the convergence of both the Jacobi and the Gauss–Seidel methods for solving systems of linear equations under the criterion of either (a) strict diagonal dominance of the matrix, or (b) diagonal dominance and irreducibility of the matrix. These results are well known. The proof for criterion (a) makes use of Geršgorin’s theorem, while the proof for criterion (...

Quadratic Convergence of a Special Quasi-cyclic Jacobi Method

The paper considers the ultimate asymptotic convergence of a block-oriented, quasi-cyclic Jacobi method for symmetric matrices. The conclusion applies to the new one-sided Jacobi method for computing the singular value decomposition, recently proposed by Drmač and Veselić. Using the simple qualitative analysis, the discussion indicates that the quadratic off-norm reduction per quasi-sweep is to...

Two modified Jacobi methods for M-matrices