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  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.