GDB
MTH-603
Extrapolation is an extremely powerful tool available to numerical analysts for improving the performance of a wide variety of mathematical methods. Since many "interesting" problems cannot be solved analytically, we often have to use computers to derive approximate solutions.
In order to make this process efficient, we have to think carefully about our implementations: this is where extrapolation comes in. Extrapolation allows us to achieve greater precision with fewer function evaluations than un-extrapolated methods. In some cases (as we saw with simple difference schemes), extrapolation is the only way to improve numeric behavior using the same algorithm (i.e. without switching to centered differences or something better). Using sets of poor estimates with small step-sizes, we can easily and quickly eliminate error modes, reaching results that would be otherwise unobtainable without significantly smaller step-sizes and function evaluations. Thus we turn relatively poor numeric scheme like the Trapezoid Rule into excellent performers. Thus far we have concentrated on the Richardson Extrapolation process since it is the easiest to understand, not to mention that it has plenty of applications and is extremely useful. Accuracy is more when we have taken more iteration using Richardson extrapolation method. On a closing note, we will now quickly outline how extrapolation can be used to easily derive Simpson's rule from the Trapezoid rule. More generally, extrapolating the n = i Closed Newton-Cotes Formula results in the n = i +1 version. By now, this should not be all that surprising, but without knowledge of extrapolation, this relation between the ith and i+1th formulas is not obvious at all.
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