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Trapezoidal, simpsons rule, and higher order approximations

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okkvlt
#1
Dec4-09, 09:25 AM
P: 53
hi. i was able to prove the trapezoidal rule and simpsons rule. (basically i used matrices to determine the coefficients m and b for mx+b when proving the trapezoidal rule and a,b,c for ax^2+bx+c such that the points coincide, then i integrated the approximating polynomial) the amount of number-crunching and expanding products of linear terms was probably the most work ive ever done, but i was amazed to see terms cancel out to yeild vastly simplified formulas. but i was wondering, suppose i approximated each step with suppose, a cubic equation. i know how to do this, but after setting up the 4x4 matrix i quickly realized that this would be extremely mundane due to all the algebraic manipulations. so before i try deriving a numerical method that uses cubics to approximate the curve pieces, i wanna know whether it will simplify. will it simplify?
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arildno
#2
Dec4-09, 09:36 AM
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P: 12,016
You can look upon Newton-Cotes formulae, and Gaussian quadrature to see how these numerical techniques have been developed:
http://mathworld.wolfram.com/Newton-CotesFormulas.html


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