  We develop algorithms and techniques to compute rigorous bounds for finite
pieces of orbits of the critical points, for intervals of parameter values, in
the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We illustrate
the effectiveness of our approach by constructing a dynamically defined
partition $\mathcal P$ of the parameter interval $\Omega=[1.4, 2]$ into almost
4 million subintervals, for each of which we compute to high precision the
orbits of the critical points up to some time $N$ and other dynamically
relevant quantities, several of which can vary greatly, possibly spanning
several orders of magnitude. We also subdivide $\mathcal P$ into a family
$\mathcal P^{+}$ of intervals which we call stochastic intervals and a family
$\mathcal P^{-}$ of intervals which we call regular intervals. We numerically
prove that each interval $\omega \in \mathcal P^{+}$ has an escape time, which
roughly means that some iterate of the critical point taken over all the
parameters in $\omega$ has considerable width in the phase space. This
suggests, in turn, that most parameters belonging to the intervals in $\mathcal
P^{+}$ are stochastic and most parameters belonging to the intervals in
$\mathcal P^{-}$ are regular, thus the names. We prove that the intervals in
$\mathcal P^{+}$ occupy almost 90% of the total measure of $\Omega$. The
software and the data is freely available at
http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying
out the calculations. The ideas and procedures can be easily generalized to
apply to other parametrized families of dynamical systems.
