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Micron Document

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<title>Newton-Cotes-Formeln</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Newton-Cotes-Formeln</h1>

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Eine Newton-Cotes-Formel (nach <a href="Isaac_Newton" title="Isaac Newton">Isaac Newton</a> und <a href="Roger_Cotes" title="Roger Cotes">Roger Cotes</a>) ist eine Formel für die <a href="Numerische_Integration" title="Numerische Integration">numerische Integration</a>, also zur näherungsweisen Berechnung von <a href="Integralrechnung" title="Integralrechnung">Integralen</a>. Diesen Formeln liegt die Idee zu Grunde, die zu integrierende <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> durch ein <a href="Polynom" title="Polynom">Polynom</a> zu <a href="Polynominterpolation" title="Polynominterpolation">interpolieren</a> und dieses als Näherung exakt zu integrieren. Die <a href="St%C3%BCtzstelle" title="Stützstelle">Stützstellen</a> der <a href="Interpolation_(Mathematik)" title="Interpolation (Mathematik)">Interpolation</a> werden dabei <a href="%C3%84quidistanz_(Geometrie)" title="Äquidistanz (Geometrie)">äquidistant</a> gewählt.

<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>

Für das zu integrierende <a href="Interpolationspolynom" class="mw-redirect" title="Interpolationspolynom">Interpolationspolynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a47cca302b556e703b68e2a12168117208f47ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}" loading="lazy"> vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle a\leq x_{0}<x_{1}<\dots <x_{n}\leq b}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5f908ab6a1a065209b94db1214334d34e95d398f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.759ex; height:2.509ex;" alt="{\displaystyle a\leq x_{0}<x_{1}<\dots <x_{n}\leq b}" loading="lazy"></dd></dl>

äquidistant mit dem konstanten Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=x_{i+1}-x_{i}}">

<msub>

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<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h=x_{i+1}-x_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bcbeb15d1bffb9c684d57b6e44ab13758a8e44b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.637ex; height:2.509ex;" alt="{\displaystyle h=x_{i+1}-x_{i}}" loading="lazy"> so gewählt, dass sie symmetrisch zur Intervallmitte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a+b}{2}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a+b}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdc7810f36b1298823a923ee35d6cd0abe79d122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.69ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a+b}{2}}}" loading="lazy"> des Integrationsintervalls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> liegen. Somit gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n-i}=a+b-x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{n-i}=a+b-x_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93377694217d1b581b88b05e89cbd335fb694e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.53ex; height:2.509ex;" alt="{\displaystyle x_{n-i}=a+b-x_{i}}" loading="lazy">.

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90490911c631400e3be106821aa5bcbc1b1b8133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.712ex; height:2.009ex;" alt="{\displaystyle x_{0}=a}" loading="lazy"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}=b}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba5205b5c212d0229bbbb4e87af574fa2534c9fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.644ex; height:2.509ex;" alt="{\displaystyle x_{n}=b}" loading="lazy">) erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervalle</a> der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\tfrac {b-a}{n}}}">

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcfd2767a84c86fff60f1d0e8837839a10b32bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.127ex; height:3.509ex;" alt="{\displaystyle h={\tfrac {b-a}{n}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+i\cdot h}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+i\cdot h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40d83d4809309e020854a89c31325171ff72d50a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.119ex; height:2.509ex;" alt="{\displaystyle x_{i}=a+i\cdot h}" loading="lazy">. Diese Formeln werden abgeschlossene Newton-Cotes-Formeln genannt.

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}\neq a}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42f45a37ac624950455b24d51909c8c9629669a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.712ex; height:2.676ex;" alt="{\displaystyle x_{0}\neq a}" loading="lazy"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\neq b}">

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<annotation encoding="application/x-tex">{\displaystyle x_{n}\neq b}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44a9e8b71b8a0879cf153515d69ef190f1248c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.644ex; height:2.676ex;" alt="{\displaystyle x_{n}\neq b}" loading="lazy">) erhält man offene Newton-Cotes-Formeln:

<ul><li>Wählt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+h}">

<msub>

</mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=a+h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc406addfb4fac0180b238cb7be9ee7cd41303c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.892ex; height:2.509ex;" alt="{\displaystyle x_{0}=a+h}" loading="lazy"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b-h}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{n}=b-h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/228fb970553f3dc0e35aef0e261e3312b91d6a61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.824ex; height:2.509ex;" alt="{\displaystyle x_{n}=b-h}" loading="lazy">), erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+2}">

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<annotation encoding="application/x-tex">{\displaystyle n+2}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaf4207e44c00a9312fe3e8710efbefcd6eafdc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+2}" loading="lazy"> Intervalle der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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<mfrac>

<mrow>

</mrow>

<mrow>

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</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n+2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e06de0014607f77b19375867a149212ba29cdba6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.36ex; height:3.843ex;" alt="{\displaystyle h={\tfrac {b-a}{n+2}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+(1+i)\cdot h}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+(1+i)\cdot h}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/136bb30fc06ec98887a850a6ab4236b2efb960cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.931ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+(1+i)\cdot h}" loading="lazy">. Diese Formeln werden offene Newton-Cotes-Formeln genannt.</li>

<li>Wählt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+{\tfrac {h}{2}}}">

<msub>

</mrow>

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<mfrac>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{0}=a+{\tfrac {h}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dad3eafa29e81e8a612214cbe08a58c5cea8bc78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.336ex; height:3.676ex;" alt="{\displaystyle x_{0}=a+{\tfrac {h}{2}}}" loading="lazy"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b-{\tfrac {h}{2}}}">

<msub>

</mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle x_{n}=b-{\tfrac {h}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eccf6c0f98a56959c9a561fc49c75aaf5e95a7f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.267ex; height:3.676ex;" alt="{\displaystyle x_{n}=b-{\tfrac {h}{2}}}" loading="lazy">), erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

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<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> Intervalle der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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<mfrac>

<mrow>

</mrow>

<mrow>

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</mfrac>

</mstyle>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n+1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d4a4ca2ea340141813f79bbf46b3b12ea855186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.36ex; height:3.843ex;" alt="{\displaystyle h={\tfrac {b-a}{n+1}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}">

<msub>

</mrow>

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<mrow>

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<mfrac>

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</mstyle>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4499426353c5c2607b27eecd6702d64491fbd1d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.747ex; height:3.509ex;" alt="{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}" loading="lazy">. Diese Formeln werden <a href="Colin_Maclaurin" title="Colin Maclaurin">Maclaurin</a>-Formeln genannt.</li></ul>

Zur <a href="Numerische_Integration" title="Numerische Integration">numerischen Integration</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}f(x)\,dx}">

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0aa70d1cb4a93ca369ce78aebef928db166fb62a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:9.931ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx}" loading="lazy"> wird das Interpolationspolynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}">

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a47cca302b556e703b68e2a12168117208f47ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}" loading="lazy"> der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1630b44c0b28a062f13a27aa8ae0c022336be4de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:22.306ex; height:6.843ex;" alt="{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}" loading="lazy">,</dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle l_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48ba35055d00bcaa522bca9247c8857730998759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.493ex; height:2.509ex;" alt="{\displaystyle l_{i}}" loading="lazy"> die <a href="Lagrange-Polynom#Lagrange-Basis" class="mw-redirect" title="Lagrange-Polynom">Lagrange-Basispolynome</a> sind. Daraus folgt:

</mrow>

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</munderover>

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</mrow>

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</mrow>

</msub>

<mfrac>

<mrow>

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</mfrac>

</mrow>

</mrow>

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</munderover>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}f(x_{i}){\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9d7eb7f06240bd6cd25df45f2bb272019567a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:46.799ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}f(x_{i}){\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Für die Newton-Cotes-Formel folgt dann:

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<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx\approx \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3c199bd043bacbd324a70678019044cbedb2edc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:45.856ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx\approx \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}" loading="lazy"></dd></dl>

mit den Gewichten

<msub>

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</munderover>

<msub>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca37815b42fca9a9c041c1ce2052b875bbb95b02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.999ex; height:9.176ex;" alt="{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}" loading="lazy"></dd></dl>

Die Gewichte sind symmetrisch, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n-i}=w_{i}}">

<msub>

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<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{n-i}=w_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a556319694114abf80a8f2d18c5eee7f20e61f35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.291ex; height:2.009ex;" alt="{\displaystyle w_{n-i}=w_{i}}" loading="lazy">.

<msub>

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<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

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</munder>

<mfrac>

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<mrow>

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<mfrac>

<mrow>

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<msub>

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<msub>

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<mrow>

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<msub>

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<msub>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle l_{i}(x)=\prod _{\begin{smallmatrix}0\leq j\leq n\\j\neq i\end{smallmatrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}={\frac {(x-x_{0})\cdots (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\cdots (x_{i}-x_{i-1})(x_{i}-x_{i+1})\cdots (x_{i}-x_{n})}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/859b802f531f730373ae543e03200bc0aa67c684.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:73.395ex; height:9.176ex;" alt="{\displaystyle l_{i}(x)=\prod _{\begin{smallmatrix}0\leq j\leq n\\j\neq i\end{smallmatrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}={\frac {(x-x_{0})\cdots (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\cdots (x_{i}-x_{i-1})(x_{i}-x_{i+1})\cdots (x_{i}-x_{n})}}}" loading="lazy"></dd></dl>

Wegen der speziellen Wahl der Stützstellen integrieren diese Formeln bei ungeradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> Polynome bis zum Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, bei geradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> sogar bis zum Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> exakt. Somit sind Newton-Cotes-Formeln mit geradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> (also einer ungeraden Anzahl an Stützstellen) denen mit ungeradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

Speziell gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea78f54e69b72f398cf6077e61c50a05b532d4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}" loading="lazy">, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}">

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<msub>

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</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c661e85b466e9a789817893f003d9a6c20504e74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:63.85ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}" loading="lazy"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}w_{i}=1}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}w_{i}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e14d38c1026d6848808d5f2aab78ba2d10dedfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.467ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}w_{i}=1}" loading="lazy">.

Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}">

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</mrow>

</munderover>

</mrow>

<msub>

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</msub>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1c293a69dedaf58c3a86aa2c7f4dcbf510fbfcfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.065ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}" loading="lazy">, was bei Gewichten mit verschiedenen <a href="Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> der Fall ist, besteht die Gefahr, dass sich die <a href="Rundungsfehler" title="Rundungsfehler">Rundungsfehler</a> aufschaukeln oder <a href="Ausl%C3%B6schung_(numerische_Mathematik)" title="Auslöschung (numerische Mathematik)">Auslöschung</a> eintritt. Daher sind aus numerischen Gründen Formeln mit positiven Gewichten zu bevorzugen. Da für großes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> das <a href="Interpolationspolynom" class="mw-redirect" title="Interpolationspolynom">Interpolationspolynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a47cca302b556e703b68e2a12168117208f47ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}" loading="lazy"> unbrauchbar ist, sind ebenso Formeln mit großem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</munderover>

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</mrow>

</munderover>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/530d1b0f18a814d822065106b14a0cbac3b725f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:31.774ex; height:9.176ex;" alt="{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}" loading="lazy"></dd></dl>

ist der Fehler (Verfahrensfehler), der bei der Anwendung der Newton-Cotes-Formel gemacht wird. Dieser hat bei der speziellen Wahl der Stützstellen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+1)}">

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<annotation encoding="application/x-tex">{\displaystyle (p+1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7da0746babaf86fc30da6b03fa83177f699790ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle (p+1)}" loading="lazy">-mal auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> <a href="Stetig_differenzierbar" class="mw-redirect" title="Stetig differenzierbar">stetig differenzierbar</a> reellwertige Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a56817275231d4e58ea376c6158009fbe3c2665e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.306ex; height:3.343ex;" alt="{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}" loading="lazy">,</dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> eine von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> unabhängige Konstante und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle \xi \in [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.426ex; height:2.843ex;" alt="{\displaystyle \xi \in [a,b]}" loading="lazy"> ein nur in Ausnahmefällen bekannter Zwischenwert ist. Wäre er generell bekannt, könnte man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d733654dab86d76c9ed9ba7f4473250dfea33d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}" loading="lazy"> und somit auch das <a href="Integralrechnung" title="Integralrechnung">Integral</a> exakt ausrechnen, im Widerspruch zu der Tatsache, dass es unendlich viele Integrale gibt, die man nicht exakt berechnen kann. Der Fehler ist Null für alle Funktionen, deren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+1)}">

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<annotation encoding="application/x-tex">{\displaystyle (p+1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7da0746babaf86fc30da6b03fa83177f699790ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle (p+1)}" loading="lazy">-te Ableitung Null ist, also für alle Polynome vom Grad kleiner oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. Somit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> der Genauigkeitsgrad. Der Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+1}">

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<annotation encoding="application/x-tex">{\displaystyle p+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5885ec01d3b5670fd5f88847f32da2b3dd62f60c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p+1}" loading="lazy"> wird auch als (polynomiale) Ordnung der Newton-Cotes-Formel bezeichnet.

Mit Hilfe des Verfahrensfehlers erhält man die Fehlerabschätzung:

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</munder>

<mrow>

<mrow>

<msup>

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</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b6ecd0c7545932b7467321dbddafa1b70a758df7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.435ex; height:4.843ex;" alt="{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}" loading="lazy">.</dd></dl>

Der exakte Fehler ist immer kleiner oder gleich dieser Fehlerabschätzung, wie auch die unten angegebenen Beispiele zeigen.

<div class="mw-heading mw-heading3"><h3 id="Abgeschlossene_Newton-Cotes-Formeln">Abgeschlossene Newton-Cotes-Formeln</h3></div>

Die angegebenen Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b61e3d4d909be4a19c9a554a301684232f59e5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}" loading="lazy"> gelten für das Integrationsintervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c9c6e38c825624bcaa5b36002f37b438ab1093c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.51ex; height:5.176ex;" alt="{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}" loading="lazy">. Für ein allgemeines Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe87fb95b4dd8d76d1700605f2575d06407f0865.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.493ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}" loading="lazy">.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</th>

<th class="hintergrundfarbe6">Name

</th>

<th class="hintergrundfarbe6">Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</th>

<th class="hintergrundfarbe6">Gewichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe22f0329d3ecb2e1880d44d191aba0e5475db68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}" loading="lazy">

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

</semantics>

</th></tr>

<tr>

<td><a href="Trapezregel" title="Trapezregel">Trapezregel</a> Sehnentrapezregel

</td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/252ae6fabd6a414a413dfa7790fa5b0cd163d53e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.648ex; height:2.176ex;" alt="{\displaystyle 0\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0bca2d9a583274c41fec309ea88cd6258e533adc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98523454e05921fcb510fe16505ff06d78ec6e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.873ex; height:5.843ex;" alt="{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}" loading="lazy">

</td></tr>

<tr>

<td><a href="Simpsonregel" title="Simpsonregel">Simpson-Regel</a> <a href="Keplersche_Fassregel" class="mw-redirect" title="Keplersche Fassregel">Keplersche Fassregel</a>

</td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{2}}\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bdc5c985e67e9d3c570c141e7bda091c2fa8984.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.969ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{2}}\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab87ddb4243faa01c77bff0356422e2d8a5577e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}" loading="lazy">

</td>

<mfrac>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6de11b4f9184cd0d5969bf132e33218cb50c14b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.657ex; height:8.009ex;" alt="{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<td>3/8-Regel Pulcherrima

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac3ad772df42f60fb64a80b734f02104a5eb02ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.29ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c37c218d021a2e30acf6b27d214a2b66e1402f73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eab897c075754eb1e96e592c5accd9988165be82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.82ex; height:8.009ex;" alt="{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<td><a href="Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a>-Regel <a href="George_Boole" title="George Boole">Boole</a>-Regel

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58695b3e100f5b2cc57f5812ff625486a2292065.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.611ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5661da12df91d0afce5c65eaa9db84416170b496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.096ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fdf4e5083f82d3c6fc8ba516dfc2e11978097bc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.82ex; height:8.176ex;" alt="{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<td>6-Punkt-Regel

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb147678b551cc4112939e686630e03409e201cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.932ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9c4ea3bc3f85586d88ae31bfb37c81c459bbb1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.554ex; height:5.176ex;" alt="{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb16f3f1acec30dcabe52cb5adee1be14b79cedf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.145ex; height:8.009ex;" alt="{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<td>Weddle-Regel (nach Thomas Weddle, 1817–1853)<a href="#cite_note-1">[1]</a>

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb374857687bbabee663985515169b3e96b5f3e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.253ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d79f5235ba0f676d61a6e11e5b7e66d3a3428a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.2ex; height:5.343ex;" alt="{\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a26ccc184f3263973f4994ef798dff7d4709dac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.82ex; height:8.176ex;" alt="{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}" loading="lazy">

</td></tr></tbody></table>

Die gekürzten Werte aller Gewichte bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=10}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=10}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fecadd7726f4d4c6c390c2f2e73533d7a3729ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=10}" loading="lazy"> betragen:<a href="#cite_note-2">[2]</a>

<th>n

</th>

<th>Gewichte

</th></tr>

<tr>

<th>1

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/046af4af8c4f600a2c4c17d74aeee5a66dfc27a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.675ex; height:3.509ex;" alt="{\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}" loading="lazy">

</td></tr>

<tr>

<th>2

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27a11e2339a75cbacf99e81cf30e1d5916012159.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.367ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}" loading="lazy">

</td></tr>

<tr>

<th>3

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fd609fa2647cd14195bccb0c753eed723ef444a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.059ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}" loading="lazy">

</td></tr>

<tr>

<th>4

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1711be765f4c0947caffc50b572bc6613574924f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.861ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}" loading="lazy">

</td></tr>

<tr>

<th>5

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/813f9ad8eda634091b66b9344542d5f531dc560f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.663ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}" loading="lazy">

</td></tr>

<tr>

<th>6

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d27f72fc20944236ba78d75fbcab157244c3c1aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.999ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}" loading="lazy">

</td></tr>

<tr>

<th>7

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/41b006ac19cad94bb14ab024d3b68bc471732714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.843ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}" loading="lazy">

</td></tr>

<tr>

<th>8

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4c3b8adac3ff19b7a9a7824124b672ce6addde2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:59.713ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}" loading="lazy">

</td></tr>

<tr>

<th>9

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2c651e4f9e8b851f3e118aa2a32c52dc2f4b7be3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:57.803ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}" loading="lazy">

</td></tr>

<tr>

<th>10

</th>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0585d7cfd837a9b331124fd946a39f1500412d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:79.235ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}" loading="lazy">

</td></tr></tbody></table>

Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=8}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=8}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1344e2c194f016e12bb978493013955fe2f445f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}<0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{i}<0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a00eb944de69d6e005c78a17633d5b22d4616ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.725ex; height:2.509ex;" alt="{\displaystyle w_{i}<0}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=2,4,6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i=2,4,6}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e6f81bff9d8706e5da992bb706f8047f4145847.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle i=2,4,6}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }">

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61c3b3e778610a00c5d311459f5bc7389b3b7656.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.842ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }" loading="lazy"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=10}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=10}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fecadd7726f4d4c6c390c2f2e73533d7a3729ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=10}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }">

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca5d56ba953d519c1f66a30c59a8b97ef77c4b20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.492ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }" loading="lazy">

Beispiel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }">

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6359d53d9a19a61d2c209f4cdda4c6c2c31eb65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.443ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }" loading="lazy">

Näherung mit Simpson-Regel (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy">). Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ef768ef226e1b051fb7ce3dc7d8e7c6e1dda1f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.699ex; height:5.343ex;" alt="{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}=a=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ab35f29660dae82321d61edd6aa69a2f4b656a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.973ex; height:2.509ex;" alt="{\displaystyle x_{0}=a=1}" loading="lazy">.

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84e953ca2ae4944c3bb0ee862a62587d401b11a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:88.344ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}" loading="lazy"></dd></dl>

Verfahrensfehler: Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}">

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}">

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ced64c025ebc2a0beb0e12d6bdf04d7c1f8d80a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.358ex; height:6.509ex;" alt="{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b658b27a1b1babd01aa16dc7426ceddb92d85fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle \xi \in [1,3]}" loading="lazy">

Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd242cc5d2742ce72415df0eb536281a55477744.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.915ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}" loading="lazy">

Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}">

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db921b35725ec99bbb8df503809830c8473bd328.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.729ex; height:9.176ex;" alt="{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Offene_Newton-Cotes-Formeln">Offene Newton-Cotes-Formeln</h3></div>

Die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/273615f07504a185c664b09c4b55fc5fd4af3b4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.723ex; height:3.843ex;" alt="{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}" loading="lazy">. Für ein allgemeines <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</th>

<th class="hintergrundfarbe6">Name

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation>

</semantics>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

</semantics>

</th></tr>

<tr>

<td><a href="Rechteckregel" class="mw-redirect" title="Rechteckregel">Rechteckregel</a> <a href="Mittelpunktsregel" title="Mittelpunktsregel">Mittelpunktsregel</a> <a href="Tangententrapezregel" class="mw-redirect" title="Tangententrapezregel">Tangententrapezregel</a></td>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\quad }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/524372005505bc6ed175f1672cbb2076db1229c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.485ex; height:2.176ex;" alt="{\displaystyle 1\quad }" loading="lazy"></td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e865534038ac36a35cf1d0823eaf519eaad6fdeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.064ex; height:5.843ex;" alt="{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/341b8b1c3600240e22e0d2f42e65c45a0a91b5bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation>

</semantics>

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ca578c0e5ed12947a5f3f743c754738f6e8f12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.815ex; height:8.009ex;" alt="{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b2d107d587b5fb3303734a8f3a8ea6a6be1ac46e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4bdd149a4007c48c240fa68d519d580ee2f8e6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.481ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25d873fa8c05755ade06c06380a2a7f01c296053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5935a09c12de72aed9a7f22ed57bac0cd95e3874.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/512517d87e128f5e6dc19d2d7577c3360d0a47e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.612ex; height:5.176ex;" alt="{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f99abe85a90d29ce2564f4190c36001e328369e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.174ex; height:8.009ex;" alt="{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/195c6149dbdef44e477683f0a27618f32cb8f881.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.283ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82706ff7457dd6ed7660449d92e2b66599327dd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.776ex; height:5.176ex;" alt="{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ca763bd582023b12a2b0ea712d995e520d32ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e1b009b0c5bbfa06f2edf2a6543a20f0a5c4c40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.604ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3cd46163387c2b7f4212a24bf5e73630990675e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.209ex; height:5.343ex;" alt="{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffceed2d9c1b0cce5498f5cd0aeb9bdad9949624.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.499ex; height:8.176ex;" alt="{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/54d9e9056c355a6237b8ffed2f947e8b99f21020.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.926ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5495220ab771f3c18ebc529e5b6d1c2578f31405.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.209ex; height:5.343ex;" alt="{\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}" loading="lazy">

</td>

<mfrac>

<mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0ee6fb99404d0104b283676652a292aac778f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.499ex; height:8.176ex;" alt="{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}" loading="lazy">

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdb41e9a10a8fd7179b9170149a8d70949ba5d03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }">

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb2ca7a0dead6c7565925ad4ac36298e525f98a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.714ex; height:3.676ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }" loading="lazy"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0365f0b9f2721ed3ebb488a96d7348d978acf8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }">

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7345feb1acf404f42a84bc2608abd4d7e71f9b7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.227ex; height:3.843ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }" loading="lazy">

Von diesen Formeln ist nur die Rechteckregel empfehlenswert. Die Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> hat bei höherem Aufwand die gleiche Ordnung wie die Rechteckregel, die höheren Formeln haben negative Gewichte.

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation>

</semantics>

Näherung mit der Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy">. Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}">

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc2fa180e0e1182b32ecefa492e7d38e5738f30c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.865ex; height:5.509ex;" alt="{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+h={\frac {3}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}=a+h={\frac {3}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea2773dcc7faf694225f407f81733dc3c8285098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.989ex; height:5.176ex;" alt="{\displaystyle x_{0}=a+h={\frac {3}{2}}}" loading="lazy">.

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d1297322df161a7f0efd1e91ddb4aabd4915d307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:97.687ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}" loading="lazy">.</dd></dl>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}">

<mfrac>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cdad15846309d01d75866d2c73f9cd1048de4fec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.742ex; height:6.509ex;" alt="{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation>

</semantics>

Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81dc0942e4f30e6673afcf0e681664b585818087.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.915ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}" loading="lazy">

Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}">

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdf669e24c61ffd0ea35a6a4aac413aa5ccbac38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.662ex; height:9.176ex;" alt="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}" loading="lazy">

<div class="mw-heading mw-heading3"><h3 id="Maclaurin-Formeln">Maclaurin-Formeln</h3></div>

Diese Formeln sind nach <a href="Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a> benannt. Die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c07733480eb1226e073fdf36a4264603b26892bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.189ex; height:3.843ex;" alt="{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}" loading="lazy">. Für ein allgemeines <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</th>

<th class="hintergrundfarbe6">Name

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation>

</semantics>

</th>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation>

</semantics>

</th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation>

</semantics>

</th></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\quad }</annotation>

</semantics>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}</annotation>

</semantics>

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cfcaae772d169ee58093dbf5c0a82e802dbe5532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation>

</semantics>

</td>

<mfrac>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22135bcb12c85492fcc6d960e315d6df29639a8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.653ex; height:8.009ex;" alt="{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}" loading="lazy">

</td></tr>

<tr>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8722e832370090ec115b31d34e6992f74d174677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}" loading="lazy">

</td>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6aeabaf129fbcf05d1b115da32c65bdc88184672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c238ca077e46da3c02d67bcf76ff8f3a8e2b416e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe4e7a56e6e5319fee0a484920d37a86b66711fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aff60968177fb371b94e9f494dcb0ec44e087678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.612ex; height:5.343ex;" alt="{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa1730d42153aefa82b2be89c8e5d7ba3ac84442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.336ex; height:8.176ex;" alt="{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/76fc33739ed99f14adb9ad08b2c568c52d831106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.096ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebcef7a491c54c66f025c4ba5bdd20f45de1e46a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.72ex; height:5.176ex;" alt="{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/219d846b00539703c7ceca175adebf83adf4b601.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.499ex; height:8.009ex;" alt="{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}" loading="lazy">

</td></tr></tbody></table>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=6}</annotation>

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</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b247046987691fc8f081f517371865e45b65d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.005ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }" loading="lazy"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=8}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=8}</annotation>

</semantics>

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3de33375d94e43e134f746d05d204483420d2429.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.005ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }" loading="lazy">

</mrow>

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</munderover>

<mfrac>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy">. Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}">

<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/78422fd209b67d980ccc7b77b330dce1059cde8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.768ex; height:5.509ex;" alt="{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}">

<msub>

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<mfrac>

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<mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f4213e097eecbde21ead6f785de5aae36e87e0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.825ex; height:5.343ex;" alt="{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}" loading="lazy">.

</mrow>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6844de1f9c1e4ad12d2b8230e0107e6eb0d8ef2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:102.222ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}" loading="lazy"></dd></dl>

<msup>

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</msup>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e44f9df123978f09dc0c74dbdc61721542e7f13c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.067ex; height:6.509ex;" alt="{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation>

</semantics>

Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}">

</mrow>

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</munder>

<mrow>

<mfrac>

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<mfrac>

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<mover>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08889179d18036b68d01def771b5f03274366c9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.565ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}" loading="lazy">

Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}">

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</munderover>

<mfrac>

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<mover>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27cba58bfb6e658597b4a67a34af6c2ec70faef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:83.359ex; height:9.176ex;" alt="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Summierte_Newton-Cotes-Formeln">Summierte Newton-Cotes-Formeln</h2></div>

Ab Grad 8 treten bei vielen Newton-Cotes-Formeln negative Gewichte auf, was die Gefahr der <a href="Ausl%C3%B6schung_(numerische_Mathematik)" title="Auslöschung (numerische Mathematik)">Auslöschung</a> mit sich bringt. Außerdem kann man im Allgemeinen keine Konvergenz erwarten, da die <a href="Polynominterpolation" title="Polynominterpolation">Polynominterpolation</a> schlecht konditioniert ist. Bei größeren Integrationsbereichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Hans R. Schwarz, Norbert Köckler: Numerische Mathematik. 6. Auflage. Teubner, Stuttgart 2006, ISBN 3-519-42960-8, S. 311–316.</li>

<li>Roland W. Freund, Ronald H. W. Hoppe: Stoer/Bulirsch: Numerische Mathematik 1. 10. Auflage. Springer, Berlin 2007, ISBN 978-3-540-45389-5, S. 164–169.</li>

<li>Michael R. Schäferkotter, Prem K. Kythe: Handbook of Computational Methods for Integration. Chapman & Hall, Boca Raton 2005, ISBN 1-58488-428-2, S. 54–62, 503–505.</li>

<li>Günter Bärwolf: Numerik für Ingenieure, Physiker und Informatiker. ISBN 978-3-8274-1689-6, Spektrum, München 2007, S. 128.</li>

<li><a href="Gisela_Engeln-M%C3%BCllges" title="Gisela Engeln-Müllges">Gisela Engeln-Müllges</a>, Klaus Niederdrenk, Reinhard Wodicka: Numerik-Algorithmen : Verfahren, Beispiele, Anwendungen. ISBN 978-3-642-13472-2, Springer, Berlin und Heidelberg 2011.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Thomas Weddle (<a href="Newcastle-upon-Tyne" class="mw-redirect" title="Newcastle-upon-Tyne">Newcastle-upon-Tyne</a>): <cite style="font-style:italic">A new simple and general method of solving numerical equations of all orders</cite>. Hamilton, Adams & Co. and J. Philipson, London 1842 (<a rel="nofollow" class="external text" href="https://archive.org/details/newsimplegeneral00weddrich">Internet Archive</a> – 52 S.).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Newton-Cotes-Formeln&rft.au=Thomas+Weddle+%28Newcastle-upon-Tyne%29&rft.btitle=A+new+simple+and+general+method+of+solving+numerical+equations+of+all+orders&rft.date=1842&rft.genre=book&rft.place=London&rft.pub=Hamilton%2C+Adams+%26+Co.+and+J.+Philipson" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a rel="nofollow" class="external text" href="https://www.wolframalpha.com/input/?i=Table%5BTable%5BIntegrate%5BProduct%5B(x-(j-1)/(m-1))/((i-j)/(m-1)),%7Bj,1,i-1%7D%5D*Product%5B(x-(j-1)/(m-1))/((i-j)/(m-1)),%7Bj,i%2B1,m%7D%5D,%7Bx,0,1%7D%5D+,%7Bi,1,m%7D%5D,%7Bm,2,9%7D%5D">WolframAlpha.</a> wolframalpha.com, abgerufen am 14. September 2019.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ANewton-Cotes-Formeln&rft.title=WolframAlpha&rft.description=WolframAlpha&rft.identifier=&rft.publisher=wolframalpha.com&rft.date=">

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