A Grassmann--Rayleigh quotient iteration for computing invariant subspaces (Q2780627)

This is a paper with the typical SIAM Review quality. The classical Rayleigh quotient iteration (RQI) computes a 1-dimensional invariant subspace using a Rayleigh quotient to estimate the associated eigenvalue. In this paper, this is generalized to an iterative algorithm to find a \(p\)-dimensional invariant subspace. Thereto the iteration is in the Grassmannian manifold of matrices with \(p\) columns. Whereas the core of the iteration for \(p = 1\) is to solve \((A - \rho I) z = x\) for \(z\) where \(\rho = x^T A x\), in the Grassmannian version this is replaced by the solution of a Sylvester equation \(AZ- ZX^T AX = X\) for \(Z\). The algorithm is gradually introduced and built up from the case \(p = 1\). Cubic and global convergence properties are proved, computational complexity, numerical stability and comparison with other Grassmannian methods are carefully explained.