Convergence of the Eberlein diagonalization method under the generalized serial pivot strategies

TL;DR

The paper proves global convergence of the Eberlein diagonalization method under a broad class of pivot strategies called generalized serial pivot strategies with permutations, extending prior results beyond cyclic and wavefront schemes. It shows that for any starting matrix \\A^{(0)} \\in \\mathbb{C}^{n\\times n}, the iterates satisfy \\lim_{k\\to\\infty} extup{off}(B^{(k)})=0 and \\lim_{k\\to\\infty}C(A^{(k)})=0, with \\lim_{k\\to\\infty}B^{(k)}=\\diag(\

Abstract

The Eberlein method is a Jacobi-type process for solving the eigenvalue problem of an arbitrary matrix. In each iteration two transformations are applied on the underlying matrix, a plane rotation and a non-unitary elementary transformation. The paper studies the method under the broad class of generalized serial pivot strategies. We prove the global convergence of the Eberlein method under the generalized serial pivot strategies with permutations and present several numerical examples.

Figures (3)

• Figure 1: Change in $\textup{off}(A^{(k)})$, $\textup{off}(B^{(k)})$ and $\|C(A^{(k)})\|_F$.
• Figure 2: Progress of the off-norms of $A^{(k)}$ and $B^{(k)}$, in logarithmic scale, for a unitarily diagonalizable matrix.
• Figure 3: Block diagonal structure.

Theorems & Definitions (4)

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