The Initial Value Problem y' = f(x, y); y(c) = d 93
These results illustrate that it is not only the number off function evaluations
which are used in producing an approximation to the solution but also the
manner in which these function evaluations are combined which ultimately
determines the accuracy of the approximate solution.Stepsize Selection When the solution of an initial value problem is in-
creasing or decreasing "slowly" a "large" stepsize may be taken by a nu-
merical approximation technique. However, when the solution is increasing
or decreasing "rapidly" a "small" stepsize must be taken by the numerical
method in order to maintain accuracy. Stepsize control techniques usually
involve comparing current local error estimates with previous local error esti-
mates (absolute error or relative error estimates). Single-step methods have
no computational difficulty in selecting an appropriate stepsize except for the
initial step, since there is no previous results with which to compare. If the
initial stepsize selected is too large, then the lack of accuracy which occurs at
the first step will corrupt the accuracy of the entire numerical approximation.
Therefore, most initial stepsize selection algorithms are very conservative and
tend to select a stepsize which is somewhat smaller than actually required.
Many computer software packages permit the user to select the first stepsize.
However, most users, including experienced users, often have no real informed
idea as to what the best first stepsize should be. A software program which
depends on the user's best guess of the first stepsize to start a numerical
approximation method is not reliable. In many of our previous examples, we
selected a constant stepsize of h = .1. If the computer we are using represents
numbers internally as a power of the base 2, we have, in some respect, made
a bad choice for the stepsize, since .1 cannot be represented exactly in that
computer. Observe that
(.lho = (0.0001100110011 ... )2 = (0.0121212 ... ) 4
= (0.063146314 ... )s = (0.1999999 ... )15
where the subscript denotes the base. Rounding-off or chopping-off any of
these infinite expansions after a specified number of digits and then adding
ten of the same rounded or chopped numbers does not result in the number
1 exactly!
One technique which has been used to select stepsize, either initially or at
each step, is to take the step twice with two different order numerical approx-
imation methods and compare the results. If the results compare favorably,
then the step is accepted. If not, the stepsize is decreased, two approximations
obtained using the smaller stepsize are made and compared, etc.
A numerical approximation method which selects the stepsize to be used
at each step is called an adaptive method. One of the more popular adap-
t ive, single-step methods was published in 1968 and 1969 by Erwin Fehlberg
in two NASA Technical Reports, R287 and R315. In these reports Fehlberg
developed adaptive Runge-Kutta methods ranging in order from 2 to 9. In