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In the preface to his book Linear Algebra and Its Applications (2013), Peter Lax says

It is with genuine regret that I omit a chapter on the numerical calculation of eigenvalues of self-adjoint matrices. Astonishing connections have been discovered recently between this important subject and other seemingly unrelated topics.

Can anyone comment on Lax's "astonishing connections", or suggest papers or course notes?

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I believe a strong candidate for the "astonishing connections" are the connections explored by Deift, Demmel, and coauthors between eigenvalue algorithms for symmetric matrices and Hamiltonian dynamics. (See also the earlier work of Chu.) I'm no expert in this work, but I believe the gist is that certain $QR$-step based algorithms for eigenvalue problems can be seen as discretizations of certain isospectral Hamiltonian flows on the space of matrices. This means that these algorithms can compute the eigenvalues to much higher relative accuracy than elementary perturbation theory would suggest. The paper of Deift et al. focuses on the bidiagonal SVD, but there are very close connections between the bidiagonal SVD and the symmetric tridiagonal eigenproblem--they are so closed related that a statement about one is almost always true about the other after appropriate modification.

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