Using 1 3 we can estimate the error in using and

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Using𝐼(𝑓)βˆ’π‘…(𝑓)β‰ˆ13�𝑇(𝑓)βˆ’ 𝑅(𝑓)οΏ½, we canestimate the error in𝑅(𝑓)using𝑇(𝑓)and𝑅(𝑓).Similarly,𝐼(𝑓)βˆ’π‘‡(𝑓)β‰ˆ23�𝑅(𝑓)βˆ’ 𝑇(𝑓)οΏ½can beused to estimate the error in𝑇(𝑓)(but they areapproximations, it is possible that𝑅(𝑓)βˆ’ 𝑇(𝑓) =0where𝐼(𝑓)βˆ’ 𝑅(𝑓)β‰ 0).When eachβ„Žπ‘–is cut in half,𝐼(𝑓)βˆ’ 𝑅12(𝑓)β‰ˆ14�𝐼(𝑓)βˆ’ 𝑅(𝑓)οΏ½. Doubling the number of panels ineither the rectangle rule or the trapezoid rule, itcan be expected to roughly quadruple theaccuracy.Why? Conrad’s Note: Error in each panel isbounded byπΈπ‘–β‰€β„Ž324𝑓′′(πœ‰)whereπœ‰πœ–[π‘Ž,𝑏](localerror is of order𝑂(β„Ž3). Summing this with npanels, this amounts to a total bounded error of𝐸 ≀ π‘›β„Ž324𝑓′′(πœ‰). As we know𝑛 βˆ— β„Ž= (𝑏 βˆ’ π‘Ž),the global error is bounded by𝐸 ≀(𝑏 βˆ’π‘Ž)β„Ž224𝑓′′(𝑦𝑖), of order𝑂(β„Ž2). Thus, half theinterval size results in(1/2)2the error, or 1/4the error, as Qiao suggests.This can be used to estimate the error as well asimprove the accuracy. How?Conrad’s Note:𝐼(𝑓)βˆ’ 𝑅12(𝑓)β‰ˆ14οΏ½ �𝐼(𝑓)βˆ’π‘…(𝑓)οΏ½can be rearranged to show𝐼(𝑓)βˆ’π‘…12(𝑓)β‰ˆ13οΏ½ �𝑅12βˆ’ 𝑅�. Computing both𝑅12and𝑅allows us to then estimate the error, usingthe derived equation.Simpson’s ruleRecall the rectangle rule𝑅(𝑓) =𝐼(𝑓)βˆ’124β„Žπ‘–3𝑓′′(𝑦𝑖)βˆ’11920β„Žπ‘–5𝑓𝑖𝑣(𝑦𝑖)+β‹―And the trapezoid rule𝑇(𝑓) =𝐼(𝑓) +112οΏ½ β„Žπ‘–3𝑓′′(𝑦𝑖) +1480β„Žπ‘–5𝑓𝑖𝑣(𝑦𝑖)𝑛𝑖=1+β‹―Combining the above two equations (canceling theO(β„Žπ‘–3) term), we get a more accurate method(Simpson’s rule):𝑆(𝑓) =23𝑅(𝑓) +13𝑇(𝑓)=𝐼(𝑓) +12880οΏ½ β„Žπ‘–5𝑓𝑖𝑣(𝑦𝑖) +⋯𝑛𝑖=1Simpson’s rule (a weighted average of therectangle and trapezoid rules):𝐼𝑖=23β„Žπ‘–π‘“ οΏ½π‘₯𝑖+π‘₯𝑖+12οΏ½+13β„Žπ‘–π‘“(π‘₯𝑖) +𝑓(π‘₯𝑖+1)2Composite Simpson’s rule:𝑆(𝑓) =οΏ½16β„Žπ‘–π‘›π‘–=1�𝑓(π‘₯𝑖) + 4𝑓 οΏ½π‘₯𝑖+π‘₯𝑖+12οΏ½+𝑓(π‘₯𝑖+1)οΏ½Function evaluations:2𝑛+ 1Error𝐼(𝑓)βˆ’ 𝑆(𝑓) =βˆ’12880οΏ½ β„Žπ‘–5𝑓𝑖𝑣(𝑦𝑖) +⋯𝑛𝑖=1Remarks-Simpson’s rule can also be derived by usingpiecewise quadratic (degree two) approximation.-Simpson’s rule is exact for cubic function (oneextra order of accuracy) since the error terminvolves fourth derivatives.-Doubling the number of panels in Simpson’s rulecan be expected to reduce the error by roughlythe factor of 1/16 (Simpson’s rule has a globalorder of accuracy ofβ„Žπ‘–4)A general technique: Richardson’sextrapolationIdea: Combine two approximations (e.g.,𝑅(𝑓)and𝑇(𝑓)) which have similar error terms to achieve amore accurate approximation (e.g.,𝑆(𝑓)).

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