Simpson's rule, was Re: Integrating a Taylor series?

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   Date: Mon, 13 May 2013 09:51:04 +1000
   From: Alasdair McAndrew <amca01 at gmail.com>
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   Yes, I know that in general Newton-Cotes rules of order 2n are exact for
   polynomials of order 2n+1.  However, I was interested in some more general
   rules (including Newton-Cotes with derivative end-corrections), some of
   which have errors of a very high order of h, and working using a Taylor
   series seemed the best way to go.

   Thanks,
   Alasdair

No, I think you are mistaken. Let me expand upon RJF's remark. Let's
look at the problem from a more abstract perspective. You are looking
to approximate a linear map I : P -> P, where I is the operator

I(f) := integrate(f,x,a,a+h);

and P is the linear space of polynomials in a single variable (f is a
polynomial in x, g:I(f) is a polynomial in h). You are using a linear
operator J : P -> P to approximate I. Your notion of approximation is
that

I == J mod P_n  ---(*)

where P_n is the subspace of degree at most n polynomials. Then

R[n] : I(x^{n+1}) - J(x^{n+1}) ;

is the lowest-order remainder term in your approximation. At the point
where you find n, you can employ Taylor's theorem, etc. to
characterize the remainder term for your algorithm in terms of the
n+1-th derivative, etc.

But, and this simplifies things greatly, you never need to work
directly with a Taylor polynomial to do any of this analysis. For
example, to check (*), you can compute R[0], ..., R[n-1]. 

Leo