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I understand that trapezoidal or midpoint rule's error bound needs the second derivative,

but I just don't get why the fourth derivative in simpson's rule

please help me :)

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    Really, we use derivatives that go up to any order that is needed. For midpoint and trapezoidal rule, we only need to go to the second derivative to get an upper bound on the error. With Simpsons rule, all of the third derivatives cancel each other out and the 4th derivative is what provides the upper bound on the error.2012-08-03
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    Have you seen how Simpson's rule is derived in the first place?2012-08-03

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Simpson's Rule is exact for polynomials of degree 3 or less, so the error term can't depend on any derivative less than the 4th. Put another way: the 3rd derivative of a 3rd degree polynomial is nonzero, so it can't be involved in the error term for Simpson, since the error for 3rd degree polynomials is zero.

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    Thank you very much. Thanks to the others for answering my question.. :)2012-08-03
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Not need. In Sendov and Popov, The Averaged Moduli of Smoothness, Wiley,1988, chapter 3, Numerical integration, gives a "complete" answer for your question.

In the recent paper of I. A. Parvanova and P. E. Parvanov (2005) Exact Constants in Estimations of the Error of the Quadrature Formulae of Simpson with the Averaged Moduli of Smoothness www.fmi.uni-sofia.bg/lecturers/ma/pparvan/QF.pdf there is a very detailed discussion.

Of course, this is a very special topic, so if you want to have some taste of it, the first section, about one and a half page, is enough. I recommend as a prerequisite to investigate the (usual) moduli of smoothness with relation to Jackson(type) theorems in basic approximation/numerical methods books. Hopefully I could help you.

This is the first time to reedit my earlier answer, sorry for me if I made something wrong.

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    Cna you maybe give a quick summary of the Sendov/Popov argument, for the benefit of those of us who don't have the book?2012-08-03
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    The basic idea in approximation theory is to replace the n-th derivative of a function with its n-th order of modulus of continuity.2012-08-03
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    "In this book the authors (Bulgarian mathematicians) "describes a new method of estimating the error associated with commonly used numerical methods (such as interpolation, approximation of functions by means of operators, quadrature formulas, and network methods) for solution of integral and differential equations. The method is based on a new characteristic of functions (first used in the theory of Hausdorff approximations) called averaged moduli of smoothness.2012-08-03
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    The authors show, by many examples, the methods and results of applying the averaged moduli of smoothness--the advantage of this method is that it allows error estimation without making any assumptions about the function involved beyond those imposed by the problem itself."2012-08-03
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    I was asking about the part of the book that discusses Simpson's. :) Also, that should be edited into the answer, not put in the comments.2012-08-03
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    Here there are some formula http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/11835559262012-08-03
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Simpson's rule is identical with Taylor expansion up to order 4 i.e. O(h^4) that is the term includes the fourth derivative.