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].