# Using 1 3 we can estimate the error in using and

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Using𝐼(𝑓)𝑅(𝑓)13�𝑇(𝑓)− 𝑅(𝑓), we canestimate the error in𝑅(𝑓)using𝑇(𝑓)and𝑅(𝑓).Similarly,𝐼(𝑓)𝑇(𝑓)23�𝑅(𝑓)− 𝑇(𝑓)can beused to estimate the error in𝑇(𝑓)(but they areapproximations, it is possible that𝑅(𝑓)− 𝑇(𝑓) =0where𝐼(𝑓)− 𝑅(𝑓)0).When each𝑖is cut in half,𝐼(𝑓)− 𝑅12(𝑓)14�𝐼(𝑓)− 𝑅(𝑓). 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 by𝐸𝑖324𝑓′′(𝜉)where𝜉𝜖[𝑎,𝑏](localerror is of order𝑂(3). Summing this with npanels, this amounts to a total bounded error of𝐸 ≤ 𝑛324𝑓′′(𝜉). As we know𝑛 ∗ ℎ= (𝑏 − 𝑎),the global error is bounded by𝐸 ≤(𝑏 −𝑎)224𝑓′′(𝑦𝑖), of order𝑂(2). 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)− 𝑅12(𝑓)14� �𝐼(𝑓)𝑅(𝑓)can be rearranged to show𝐼(𝑓)𝑅12(𝑓)13� �𝑅12− 𝑅�. Computing both𝑅12and𝑅allows us to then estimate the error, usingthe derived equation.Simpson’s ruleRecall the rectangle rule𝑅(𝑓) =𝐼(𝑓)124𝑖3𝑓′′(𝑦𝑖)11920𝑖5𝑓𝑖𝑣(𝑦𝑖)+And the trapezoid rule𝑇(𝑓) =𝐼(𝑓) +112� ℎ𝑖3𝑓′′(𝑦𝑖) +1480𝑖5𝑓𝑖𝑣(𝑦𝑖)𝑛𝑖=1+Combining the above two equations (canceling theO(𝑖3) term), we get a more accurate method(Simpson’s rule):𝑆(𝑓) =23𝑅(𝑓) +13𝑇(𝑓)=𝐼(𝑓) +12880� ℎ𝑖5𝑓𝑖𝑣(𝑦𝑖) +𝑛𝑖=1Simpson’s rule (a weighted average of therectangle and trapezoid rules):𝐼𝑖=23𝑖𝑓 �𝑥𝑖+𝑥𝑖+12+13𝑖𝑓(𝑥𝑖) +𝑓(𝑥𝑖+1)2Composite Simpson’s rule:𝑆(𝑓) =16𝑖𝑛𝑖=1�𝑓(𝑥𝑖) + 4𝑓 �𝑥𝑖+𝑥𝑖+12+𝑓(𝑥𝑖+1)Function evaluations:2𝑛+ 1Error𝐼(𝑓)− 𝑆(𝑓) =12880� ℎ𝑖5𝑓𝑖𝑣(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 of𝑖4)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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