It is worth noting that if , then
and this
iteration is precisely
the same as the implicitly shifted QR iteration. Even for
, the
first
columns of
and the leading
tridiagonal submatrix of
are
mathematically equivalent to the matrices that would appear in the
full implicitly shifted QR iteration using the same shifts
In this sense, the IRLM may be viewed
as a truncation of the implicitly shifted QR iteration. The
fundamental difference is that the standard implicitly shifted QR iteration
selects shifts to drive subdiagonal elements of
to zero from the
bottom up while the shift selection in the implicitly restarted Lanczos method
is made to drive subdiagonal elements of
to zero from the top down.
Of course, convergence of the implicit restart scheme here
is like a ``shifted power" method, while the full implicitly shifted
QR iteration is like an ``inverse iteration" method.
Thus the exact shift strategy can be viewed both as
a means to damp unwanted components from the starting vector
and also as directly forcing the starting vector to be a
linear combination of wanted eigenvectors.
See [419] for information on the convergence of
IRLM and [22,421] for other possible shift strategies for
Hermitian The reader is referred to [293,334] for studies
comparing implicit restart with other schemes.