In this paper we present an inscribed ellipsoid (IE) algorithm for
fixed point problems that enjoys O($n^2(\ln\frac{1}{\e}+\ln\frac{1}{1-q}
+ \ln n)$) bound. Therefore the IE algorithm has almost the same (modulo
multiplicative constant) number of function evaluations as the
(nonconstructive) centroid method. We conjecture that this bound is the
best possible for mildly contractive functions ($q \approx 1$) in
moderate dimensional case. Affirmative solution of this conjecture would
imply that the IE algorithm and the centroid algorithms are almost
optimal in the worst case. In particular they are much faster than the
simple iteration method, that requires $\left\lceil \frac{\ln(1/\e)}
{\ln (1/q)} \right\rceil$ function evaluations to solve the problem.