Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main diagonal. Each 1 by 1 block corresponds to a real eigenvalue of $A$ and each 2 by 2 block corresponds to a pair of complex conjugate eigenvalues of $A$.
The eigenvalue reordering problem consists of finding an orthogonal similarity transformation $V$ such that the user's selection of eigenvalues of $A$ appears along the diagonal of the upper left corner of $S = V^T T V$.
In LAPACK the relevant routine double precision routine is called DTRSEN. Daniel Kressner has written a blocked version by the name BDTRSEN. The ScaLAPACK routine is PDTRSEN.
I am looking for applications and algorithms where advances in solving the eigenvalue reordering problem would have real benefits.
We can easily generate test matrices in quasi triangular form, but we are having trouble deciding the shape of a realistic distribution of the user's selection of eigenvalues.
From my perspective, subspace iteration with Ritz acceleration is an ideal algorithm for testing improvements to the reordering algorithm. It needs (sparse) matrix vector multiplication, a tall QR algorithm and a reordering algorithm.
However, is hard for me to find real life problems where it is clear that a particular set of eigenpairs is physically interesting.
We can do eigenvalue reordering for dense matrices of dimension 40,000 using a shared memory machine. The best performance is achieved when the user selects about 50% of all eigenvalues.