The refined projection procedure is mathematically different from the orthogonal and oblique projection methods described above, because it no longer uses Ritz vectors. In particular, the refined vector is not a Ritz vector if the new residual . The refined residual vector is then not orthogonal to either itself or a left subspace such as .
If a conventional SVD algorithm is used to solve the least squares problem (3.11), the computational cost is floating point operations, which is too expensive. Fortunately, if is the Ritz value obtained by orthogonal projection methods, the refined approximate eigenvector can be computed in floating point operations. Recall that a Rayleigh-Ritz procedure typically requires floating point operations to produce and a complete set of Ritz or harmonic Ritz vectors. Therefore, the cost of computing is negligible. Hence, the refined approximate eigenvectors provide a viable alternative for approximating eigenvectors.
Several refined projection algorithms have been developed in [244,245]. Analyses of these methods are presented in [247,246].