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## Reverse Engineering

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
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