Decision schemes
In the idea to replace the human expert with an automatic expert, we must analyze, first of all, how the human expert accomplish the task of solving a problem. In particular, we are interested in initial value problems of ordinary differential equations

y'(t)=f(t,y(t)), y(t0)=y0

and an expert in numerical analysis which can select a method for approximate the exact solution of the problem.

One of the hardness task in constructing an automatic expert is to simulate the human capacity to learn from practical experience and to add new knowledge.

There are many scheme to decide the better way to solve the given problem:

Criteria Simple option/ Opposite variant/
Requirements Requirements
Method kind Constant Variable
Working with a single met. Working with a method class
Method selection All methods A class of methods
Arbitrary selection Analyze the problem
Stepsize kind Constant Variable
Unchanged until end Changing using some rules
Stepsize selection All values Limited values
Arbitrary selected Controled by the error level
Time Any time Limited time
Nonmeasured Time estimation scheme

The simplest way is to use an solving scheme with an arbitrary constant method, an arbitrary constant stepsize, and no control on the computation time (choose a method, select an arbitrary stepsize, test if the computation process reaches the interval end, otherwise take a smaller step, stop the computation process and take a greater step if the computation process is to slow, after a number of failures, try another method).

We can improve such a solving scheme, classifying first the problem in order to reduce the method class to be tested. Unpleased returns to the starting values or to many computations can be avoided which we use an error control scheme to decide the allowed stepsize. Constructing an scheme for the estimation of the computation time is not a very complicated task, since the computational effort is almost the same at each step if we use only one method in the integration process; this allowed us to reject the methods or stepsizes which are not convenient from the point of view of the time. If the problem is nonlinear, a scheme with a variable stepsize will improve the computation time and the error level (in this case is hard to estimate the overall time and error). The schemes which are using method changing formula are very complicated to control due to the instabilities which may occurs.