The entire bootstrap process is essentially a sequence of steps, where [w.sub.[bar]a+1](x) is constructed from [w.sub.[bar]a](x) in the first bootstrap step, then [w.sub.[bar]a+2](x) is constructed from [w.sub.[bar]a+1](x) in the second bootstrap step and so on until [w.sub.a](x) is finally constructed.
--the bootstrap process requires 2(a - [bar]a)N fewer evaluations of z([w.sub.a],(x)) and 2(a - [bar]a)N - (N + 1) fewer evaluations of the differential equation than would otherwise be the case.
This requirement reduces the number of evaluations of z([w.sub.a](x)) in each step of the bootstrap process by reusing as many of these values as possible.
First, we have to choose an initial bootstrap approximation, [w.sub.[bar]a](x), that will be used as a starting point for the bootstrap process. An obvious candidate for it is COLNEW's approximate solution (3), which corresponds to a [w.sub.k](x) in our notation.
There is also a theoretical result which justifies reusing the [[Phi].sub.j, i, s] values in the first step of the bootstrap process. In Ascher et al.
There is also another way to achieve savings in computation at the beginning of the bootstrap process. We will describe this alternative for the case where [m.sub.1] = ...
Therefore, those components that have reached the O([h.sup.2k]) error bound cannot be improved further by the bootstrap process. However, those components which have not yet reached the O([h.sup.2k]) error bound can still be improved by the bootstrap process, provided that we can somehow generate sufficiently accurate derivative data values.
However, when different orders are present in the system of ODEs, it carries on the bootstrap process beyond the (k - mmax)th bootstrap step while the standard bootstrap approach does not do so.
Therefore the bootstrap process can proceed as long as (12) is satisfied.
Suppose that we allowed this bootstrap process to always perform k - mmin bootstrap steps, as if (13) were satisfied.