On the block Eberlein diagonalization method

TL;DR

The paper addresses the eigenvalue problem for general complex matrices and introduces a block variant of the Eberlein diagonalization method. It formulates a Jacobi-type iteration with block unitary rotations $R_k$ and norm-reducing blocks $S_k$, combining to $\mathbf{T}_k = \mathbf{R}_k \mathbf{S}_k$ and the update $\mathbf{A}^{(k+1)} = \mathbf{T}_k^{-1} \mathbf{A}^{(k)} \mathbf{T}_k$, and shows that the Hermitian part $\mathbf{B}^{(k)}$ converges to a diagonal and $C(\mathbf{A}^{(k)})$ tends to zero, so $\mathbf{A}^{(k)}$ tends to a normal matrix. The convergence is established for a broad class of pivot orderings $\mathcal{B}_{sg}^{(m)}$ using uniformly bounded cosine (UBC) block unitary transformations. Numerical experiments on random and structured $n\times n$ matrices demonstrate rapid decay of off-norms and high accuracy of eigenvalues/eigenvectors, with preconditioning recommended when eigenvalues share real parts. The results enable cache-efficient, BLAS3-accelerated Jacobi-type eigensolvers for general matrices, extending classical Eberlein theory to the block setting.

Abstract

The Eberlein diagonalization method is an iterative Jacobi-type method for solving the eigenvalue problem of a general complex matrix. In this paper we develop the block version of the Eberlein method. We prove the global convergence of our block method and present several numerical examples.

Figures (4)

• Figure 1: Results for the test matrix $\textbf{A}_1$ with $n=200$.
• Figure 2: Results for the test matrix $\textbf{A}_1$ with $n=200$.
• Figure 3: Results for the test matrix $\textbf{A}_2$ with $n=200$, $m_1=40$, $m_2=m_3=m_4=m_5=20$.
• Figure 4: Results for the test matrix $\textbf{A}_2$ with $n=200$, $m_1=40$, $m_2=m_3=m_4=m_5=20$, with preconditioning.

Theorems & Definitions (4)

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