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Sometimes you are faced with an equation that appears to be a guess
by someone. Is there a way to figure out what it approximates? This
is particularly true with difference equations when someone is
approximating an ordinary differential equation or a partial differential
equation and they state a bunch of difference equations and one looks a bit
spooky. So here is a way to reverse engineer what is happening.
Typically, the equations involve a function and a couple of points, say
Now lets expand the function, V, around
point i as

Now some preliminaries. Suppose we use
to approximate
a derivative. We see immediately form above that

which shows that we estimate the left derivative with an error that is
proportional to the 2nd derivative. To get a sort of unbiased estimator
of the derivative - subtract equation 4 from 5 and you see:

or

Now this is a nice estimator - Note that dividing each side by
gives a
perfect approximation of the derivative - i.e. the higher order terms
disappear!
Now let's say that someone uses the difference equation
and we would like to know what it estimates. So we multiply equation 5 by 2
and add them getting

so moving the to the left we have

The paper where this appeared stated that the above difference equation
approximated the first derivative - and its obvious that it approximates a bit
more than the simple first derivative.

** Next:** Algebraic Models
** Up:** Taylor Series
** Previous:** Taylor Series
** Index**
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Frank Starmer
2004-05-19