Direct algorithms of arithmetic complexity 
for
performing Phase 1 of a Toeplitz solver
have been presented in
[14,9,15,34,5,6].
These algorithms all rely on Toeplitz inversion formulas for their
second phase, and use doubling procedures and
fast Fourier transform techniques to perform the first phase.
We refer to these as algorithms superfast Toeplitz solvers. However,
these algorithms will involve fewer arithmetic operations than
a fast algorithm only for sufficiently large n.
It is shown in
[5] that the superfast Toeplitz solver
presented independently by Musicus [34] and de Hoog
[15] can be interpreted in terms of a doubling
strategy applied to the continued fraction generated
by Schur's algorithm.
This algorithm can
be viewed as a hybrid algorithm that uses the results of
a fast implementation of Schur's algorithm
to construct the Szego polynomial 
.