Trapezoid rule to Simpson's by Richardson extrapolation
The trapezoid formula (for one interval) is given by,
a) The trapezoid method has a global error of . In the above equation we use . Simpson's 1/3-rule can be derived by performing Richardson extrapolation on the trapoezoid formula. Do this.
b) What happens with the error?
First of all I divide the interval into two intervals,
This approximation has global error and so I am tempted to write that,
but I've read that it can be shown that for the composite trapezoid formula we have,
The book I have does not show how it can be shown, it merely points out that it cannot be shown very easily..I will go with it and use that result for now.
I let to get,
Here's the part that kind of confuses me. The trapezoid rule has local error . I am then tempted to say that,
but then I do not see how Richardson extrapolation can do any good...
If I instead use the global error, I can perform Richardson extrapolation and get,
b) The global error is .
Like usual, I write too much to say too little!