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I'm looking at methods to find the eigenpairs of symmetric block tridiagonal matrices, with sparse blocks on the main diagonal and diagonal blocks on the outer diagonals. Has any research been done on similar matrices ?

$$ M = S+D = \\ \begin{pmatrix}S_1&&&\\&.&&\\&&.&\\&&&S_n\end{pmatrix}+\begin{pmatrix}&D_1&&\\D_1&&.&\\&.&&D_n\\&&D_n&\end{pmatrix}\\ =\begin{pmatrix}S_1&D_1&&\\D_1&.&.&\\&.&.&D_n\\&&D_n&S_n\end{pmatrix}\\ \text{$S_i$ : sparse symmetric blocks}\\\text{$D_i$ : diagonal blocks} $$

I've been looking at eigensolvers for the more general problem of block tridiagonal matrices : either tridiagonalization based (DSBEV, DSBEVD, DSBEVX in LAPACK) or working directly on the block matrix (Block Divide & Conquer and Twisted Block Factorization).

I'm interested in methods that take advantage of the matrix structure, maybe relating the eigenpairs of M to those of S and D, or getting rid of the $S_i$ through some similarity transformation ? Given my math background i'm not sure i'd be able to devise such methods by myself, which is why i'm asking for help here.

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