Newton-Cotes-Formeln

Abgeschlossene Newton-Cotes-Formeln

Die angegebenen Stützstellen ${\displaystyle t_{i}}$ gelten für das Integrationsintervall ${\displaystyle [0,1]}$: ${\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}$. Für ein allgemeines Intervall ${\displaystyle [a,b]}$ sind die Stützstellen ${\displaystyle x_{i}=a+t_{i}\cdot (b-a)}$.

Weitere Informationen , Stützstellen ...

${\displaystyle n}$	Name	Stützstellen ${\displaystyle t_{i}}$	Gewichte ${\displaystyle w_{i}}$	${\displaystyle E(f)}$
1	Trapezregel Sehnentrapezregel	${\displaystyle 0\quad 1}$	${\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}$	${\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}$
2	Simpson-Regel Keplersche Fassregel	${\displaystyle 0\quad {\frac {1}{2}}\quad 1}$	${\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}$	${\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}$
3	3/8-Regel Pulcherrima	${\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}$	${\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}$	${\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}$
4	Milne-Regel Boole-Regel	${\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}$	${\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}$	${\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}$
5	6-Punkt-Regel	${\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}$	${\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}$	${\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}$
6	Weddle-Regel (nach Thomas Weddle, 1817–1853)[1]	${\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}$	${\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}$	${\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}$

Schließen

Die gekürzten Werte aller Gewichte bis ${\displaystyle n=10}$ betragen:^[2]

Weitere Informationen , ...

n	Gewichte
1	${\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}$
2	${\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}$
3	${\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}$
4	${\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}$
5	${\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}$
6	${\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}$
7	${\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}$
8	${\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}$
9	${\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}$
10	${\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}$

Schließen

Für ${\displaystyle n=8}$ gilt ${\displaystyle w_{i}<0}$ für ${\displaystyle i=2,4,6}$ und ${\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }$ Für ${\displaystyle n=10}$ gilt ${\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }$

Beispiel: ${\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }$

Näherung mit Simpson-Regel (${\displaystyle n=2}$). Es gilt ${\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}$ und ${\displaystyle x_{0}=a=1}$.

${\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}$

Verfahrensfehler: Mit ${\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}$ erhält man ${\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}$ mit ${\displaystyle \xi \in [1,3]}$

Fehlerabschätzung: ${\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}$

Exakter Fehler: ${\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}$

Offene Newton-Cotes-Formeln

Die Stützstellen ${\displaystyle t_{i}}$ gelten für das Integrationsintervall ${\displaystyle [0,1]}$: ${\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}$. Für ein allgemeines Intervall ${\displaystyle [a,b]}$ sind die Stützstellen ${\displaystyle x_{i}=a+t_{i}\cdot (b-a)}$.

Weitere Informationen , Stützstellen ...

${\displaystyle n}$	Name	Stützstellen ${\displaystyle t_{i}}$	Gewichte ${\displaystyle w_{i}}$	${\displaystyle E(f)}$
0	Rechteckregel Mittelpunktsregel Tangententrapezregel	${\displaystyle {\frac {1}{2}}}$	${\displaystyle 1\quad }$	${\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}$
1		${\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}$	${\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}$	${\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}$
2		${\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}$	${\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}$	${\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}$
3		${\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}$	${\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}$	${\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}$
4		${\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}$	${\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}$	${\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}$
5		${\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}$	${\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}$	${\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}$
6		${\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}$	${\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}$	${\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}$

Schließen

Für ${\displaystyle n=5}$ gilt ${\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }$ Für ${\displaystyle n=6}$ gilt ${\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }$

Von diesen Formeln ist nur die Rechteckregel empfehlenswert. Die Formel für ${\displaystyle n=1}$ hat bei höherem Aufwand die gleiche Ordnung wie die Rechteckregel, die höheren Formeln haben negative Gewichte.

Beispiel: ${\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }$

Näherung mit der Formel für ${\displaystyle n=2}$. Es gilt ${\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}$ und ${\displaystyle x_{0}=a+h={\frac {3}{2}}}$.

${\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}$.

Verfahrensfehler: Mit ${\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}$ erhält man ${\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}$ mit ${\displaystyle \xi \in [1,3]}$.

Fehlerabschätzung: ${\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}$

Exakter Fehler: ${\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}$

Maclaurin-Formeln

Diese Formeln sind nach Colin Maclaurin benannt. Die Stützstellen ${\displaystyle t_{i}}$ gelten für das Integrationsintervall ${\displaystyle [0,1]}$: ${\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}$. Für ein allgemeines Intervall ${\displaystyle [a,b]}$ sind die Stützstellen ${\displaystyle x_{i}=a+t_{i}\cdot (b-a)}$.

Weitere Informationen , Stützstellen ...

${\displaystyle n}$	Name	Stützstellen ${\displaystyle t_{i}}$	Gewichte ${\displaystyle w_{i}}$	${\displaystyle E(f)}$
0	Rechteckregel Mittelpunktsregel Tangententrapezregel	${\displaystyle {\frac {1}{2}}}$	${\displaystyle 1\quad }$	${\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}$
1		${\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}$	${\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}$	${\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}$
2		${\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}$	${\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}$	${\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}$
3		${\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}$	${\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}$	${\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}$
4		${\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}$	${\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}$	${\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}$

Schließen

Für ${\displaystyle n=6}$ gilt ${\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }$ Für ${\displaystyle n=8}$ gilt ${\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }$

Beispiel: ${\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }$

Näherung mit der Formel für ${\displaystyle n=2}$. Es gilt ${\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}$ und ${\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}$.

${\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}$

Verfahrensfehler: Mit ${\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}$ erhält man ${\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}$ mit ${\displaystyle \xi \in [1,3]}$.

Fehlerabschätzung: ${\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}$

Exakter Fehler: ${\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}$