The Euler method uses the derivative at
one point to extrapolate to the value at the next point. If the
derivative, in fact, varies significantly over that range, the extrapolated
point may also differ significantly from the true value.
There are a number methods more sophisticated methods than the
Euler approach to reduce these discrepancies.
We will discuss the first order Predictor-Corrector method
here, which is slightly more reliable than the Euler method, and
in later chapters discuss some more powerful higher order methods.
The Predictor-Corrector improves on the Euler method, by first
estimating, or predicting, the next value using the Euler approach:
Then this value xp
value helps to obtain an improved value:
We see that the corrector uses the average of the derivatives at
the old position and at the predicted position to get a better estimate
of the next value.
This demonstration program uses the Predictor-Corrector
method to obtain the position and velocity of a mass attached to
Latest update: Dec.12.2003