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The implicit (Francis) QR iteration, with several additional tricks, is the standard algorithm. I suggest you to check a book such as Watkins' The Matrix Eigenvalue Problem for details; this is a clas …

Given an LU decomposition of $A\in \mathbb{R}^{n\times n}$, is there a way to compute $\operatorname{trace}(A^{-1})$ with lower complexity than that of the inversion ($O(n^3)$ in practice)? This quest …

Very quick answer... The exponential of a Hamiltonian matrix is symplectic, a property that you probably wish to preserve, otherwise you would simply use a non-structure-preserving method. Indeed, th …

It can be lowered; it is called packed storage and Lapack has some functions to deal with it, e.g., ?PPSVX, ?SPSVX. As this storage scheme is somewhat uncommon, I don't think you can use it easily in …

A thing you might try: 1) perform a few iterations of the matrix sign iteration $A\mapsto \frac{1}{2}(A+A^{-1})$; the eigenvectors are unchanged, while the eigenvalues converge quadratically to $\pm …

Have you considered working with the Cholesky (or low-rank) factor of $\rho(t_0)$ rather than with the matrix itself? This might reduce the number of products that you need to make, and it has the add …

This is what I wrote in the comments, formulated as an answer. If $\hat{\beta}_j=0$, then $\hat{\alpha}_j\neq 0$ otherwise the matrix wouldn't be invertible. So you can swap column $j$ with column $n …