Newton-Cotes-Formel für n = 2 Eine Newton-Cotes-Formel (nach Isaac Newton und Roger Cotes ) ist eine numerische Quadraturformel zur näherungsweisen Berechnung von Integralen . Diesen Formeln liegt die Idee zu Grunde, die zu integrierende Funktion durch ein Polynom zu interpolieren und dieses als Näherung exakt zu integrieren. Die Stützstellen der Interpolation werden dabei äquidistant gewählt.
Herleitung
Für das zu integrierende Interpolationspolynom
p
n
(
x
)
{\displaystyle p_{n}(x)}
n
{\displaystyle n}
a
≤
x
0
<
x
1
<
⋯
<
x
n
≤
b
{\displaystyle a\leq x_{0}<x_{1}<\dotsb <x_{n}\leq b}
äquidistant mit dem konstanten Abstand
h
=
x
i
+
1
−
x
i
{\displaystyle h=x_{i+1}-x_{i}}
a
+
b
2
{\displaystyle {\tfrac {a+b}{2}}}
[
a
,
b
]
{\displaystyle [a,b]}
x
n
−
i
=
a
+
b
−
x
i
{\displaystyle x_{n-i}=a+b-x_{i}}
Mit
x
0
=
a
{\displaystyle x_{0}=a}
x
n
=
b
{\displaystyle x_{n}=b}
n
{\displaystyle n}
h
{\displaystyle h}
h
=
b
−
a
n
{\displaystyle h={\tfrac {b-a}{n}}}
x
i
=
a
+
i
⋅
h
{\displaystyle x_{i}=a+i\cdot h}
abgeschlossene Newton-Cotes-Formeln genannt.
Mit
x
0
≠
a
{\displaystyle x_{0}\neq a}
x
n
≠
b
{\displaystyle x_{n}\neq b}
offene Quadratur-Formeln:
Wählt man
x
0
=
a
+
h
{\displaystyle x_{0}=a+h}
x
n
=
b
−
h
{\displaystyle x_{n}=b-h}
n
+
2
{\displaystyle n+2}
h
{\displaystyle h}
h
=
b
−
a
n
+
2
{\displaystyle h={\tfrac {b-a}{n+2}}}
x
i
=
a
+
(
1
+
i
)
⋅
h
{\displaystyle x_{i}=a+(1+i)\cdot h}
offene Newton-Cotes-Formeln genannt.
Wählt man
x
0
=
a
+
h
2
{\displaystyle x_{0}=a+{\tfrac {h}{2}}}
x
n
=
b
−
h
2
{\displaystyle x_{n}=b-{\tfrac {h}{2}}}
n
+
1
{\displaystyle n+1}
h
{\displaystyle h}
h
=
b
−
a
n
+
1
{\displaystyle h={\tfrac {b-a}{n+1}}}
x
i
=
a
+
(
1
2
+
i
)
⋅
h
{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}
Maclaurin -Formeln Zur numerischen Integration von
∫
a
b
f
(
x
)
d
x
{\displaystyle \int \limits _{a}^{b}f(x)\,dx}
p
n
(
x
)
{\displaystyle p_{n}(x)}
f
(
x
)
{\displaystyle f(x)}
p
n
(
x
)
=
∑
i
=
0
n
f
(
x
i
)
l
i
(
x
)
{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}
wobei
l
i
{\displaystyle l_{i}}
Lagrange-Basispolynome sind. Daraus folgt:
∫
a
b
p
n
(
x
)
d
x
=
(
b
−
a
)
∑
i
=
0
n
f
(
x
i
)
1
b
−
a
∫
a
b
l
i
(
x
)
d
x
{\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}
Definition
Für die Newton-Cotes-Formel folgt dann:
∫
a
b
f
(
x
)
d
x
≈
∫
a
b
p
n
(
x
)
d
x
=
(
b
−
a
)
∑
i
=
0
n
w
i
f
(
x
i
)
{\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})}
mit den Gewichten
w
i
=
1
b
−
a
∫
a
b
l
i
(
x
)
d
x
{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}
Die Gewichte sind symmetrisch, das heißt
w
n
−
i
=
w
i
{\displaystyle w_{n-i}=w_{i}}
l
i
(
x
)
=
∏
0
≤
j
≤
n
j
≠
i
x
−
x
j
x
i
−
x
j
=
(
x
−
x
0
)
⋯
(
x
−
x
i
−
1
)
(
x
−
x
i
+
1
)
⋯
(
x
−
x
n
)
(
x
i
−
x
0
)
⋯
(
x
i
−
x
i
−
1
)
(
x
i
−
x
i
+
1
)
⋯
(
x
i
−
x
n
)
{\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})\dotsm (x-x_{i-1})(x-x_{i+1})\dotsm (x-x_{n})}{(x_{i}-x_{0})\dotsm (x_{i}-x_{i-1})(x_{i}-x_{i+1})\dotsm (x_{i}-x_{n})}}}
Wegen der speziellen Wahl der Stützstellen integrieren die Quadraturformeln bei ungeradem
n
{\displaystyle n}
n
{\displaystyle n}
n
{\displaystyle n}
n
+
1
{\displaystyle n+1}
n
{\displaystyle n}
n
{\displaystyle n}
Speziell gilt für
f
(
x
)
=
1
{\displaystyle f(x)=1}
∫
a
b
f
(
x
)
d
x
=
∫
a
b
1
d
x
=
b
−
a
=
(
b
−
a
)
∑
i
=
0
n
w
i
⋅
1
=
(
b
−
a
)
∑
i
=
0
n
w
i
{\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}}
∑
i
=
0
n
w
i
=
1
{\displaystyle \sum _{i=0}^{n}w_{i}=1}
Falls
∑
i
=
0
n
|
w
i
|
>
∑
i
=
0
n
w
i
=
1
{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}
Vorzeichen der Fall ist, besteht die Gefahr, dass sich die Rundungsfehler aufschaukeln oder Auslöschung eintritt. Daher sind aus numerischen Gründen Quadraturformeln mit positiven Gewichten zu bevorzugen. Da für großes
n
{\displaystyle n}
p
n
(
x
)
{\displaystyle p_{n}(x)}
n
{\displaystyle n}
E
(
f
)
=
∫
a
b
f
(
x
)
d
x
−
∫
a
b
p
n
(
x
)
d
x
{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}
ist der Fehler (Verfahrensfehler), der bei der Anwendung der Quadraturformel gemacht wird. Dieser hat bei der speziellen Wahl der Stützstellen für
(
p
+
1
)
{\displaystyle (p+1)}
[
a
,
b
]
{\displaystyle [a,b]}
f
(
x
)
{\displaystyle f(x)}
E
(
f
)
=
K
⋅
f
(
p
+
1
)
(
ξ
)
{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}
wobei
K
{\displaystyle K}
f
(
x
)
{\displaystyle f(x)}
ξ
∈
[
a
,
b
]
{\displaystyle \xi \in [a,b]}
E
(
f
)
{\displaystyle E(f)}
(
p
+
1
)
{\displaystyle (p+1)}
p
{\displaystyle p}
p
{\displaystyle p}
p
+
1
{\displaystyle p+1}
(polynomiale) Ordnung der Quadraturformel bezeichnet.
Mit Hilfe des Verfahrensfehlers erhält man die Fehlerabschätzung:
|
E
(
f
)
|
≤
|
K
|
⋅
max
a
≤
ξ
≤
b
|
f
(
p
+
1
)
(
ξ
)
|
{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}
Der exakte Fehler ist immer kleiner/gleich als diese Fehlerabschätzung, wie auch die unten angegebenen Beispiele zeigen.
Die angegebenen Stützstellen
t
i
{\displaystyle t_{i}}
[
0
,
1
]
{\displaystyle [0,1]}
t
0
=
0
,
t
i
=
i
n
,
t
n
=
1
{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}
[
a
,
b
]
{\displaystyle [a,b]}
x
i
=
a
+
t
i
⋅
(
b
−
a
)
{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}
n
{\displaystyle n}
Name
Stützstellen
t
i
{\displaystyle t_{i}}
Gewichte
w
i
{\displaystyle w_{i}}
E
(
f
)
{\displaystyle E(f)}
1
Trapezregel
0
1
{\displaystyle 0\quad 1}
1
2
1
2
{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}
−
(
b
−
a
)
3
12
f
″
(
ξ
)
{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}
2
Simpson-Regel Keplersche Fassregel
0
1
2
1
{\displaystyle 0\quad {\frac {1}{2}}\quad 1}
1
6
4
6
1
6
{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}
−
(
b
−
a
2
)
5
90
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}
3
3/8-Regel
0
1
3
2
3
1
{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}
1
8
3
8
3
8
1
8
{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}
−
3
(
b
−
a
3
)
5
80
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}
4
Milne -Regel Boole -Regel
0
1
4
2
4
3
4
1
{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}
7
90
32
90
12
90
32
90
7
90
{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}
−
8
(
b
−
a
4
)
7
945
f
(
6
)
(
ξ
)
{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}
5
6-Punkt-Regel
0
1
5
2
5
3
5
4
5
1
{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}
19
288
75
288
50
288
50
288
75
288
19
288
{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}
−
275
(
b
−
a
5
)
7
12
096
f
(
6
)
(
ξ
)
{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}
6
Weddle-Regel (nach Thomas Weddle , 1817–1853)[ 1]
0
1
6
2
6
3
6
4
6
5
6
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}
41
840
216
840
27
840
272
840
27
840
216
840
41
840
{\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}}}
−
9
(
b
−
a
6
)
9
1400
f
(
8
)
(
ξ
)
{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}
Die gekürzten Werte aller Gewichte bis
n
=
10
{\displaystyle n=10}
[ 2]
n=1: {1/2, 1/2}
n=2: {1/6, 2/3, 1/6}
n=3: {1/8, 3/8, 3/8, 1/8}
n=4: {7/90, 16/45, 2/15, 16/45, 7/90}
n=5: {19/288, 25/96, 25/144, 25/144, 25/96, 19/288}
n=6: {41/840, 9/35, 9/280, 34/105, 9/280, 9/35, 41/840}
n=7: {751/17280, 3577/17280, 49/640, 2989/17280, 2989/17280, 49/640, 3577/17280, 751/17280}
n=8: {989/28350, 2944/14175, -464/14175, 5248/14175, -454/2835, 5248/14175, -464/14175, 2944/14175, 989/28350}
n=9: {2857/89600, 15741/89600, 27/2240, 1209/5600, 2889/44800, 2889/44800, 1209/5600, 27/2240, 15741/89600, 2857/89600}
n=10: {16067/598752 , 26575/149688 , -16175/199584 , 5675/12474 , -4825/11088 , 17807/24948 , -4825/11088 , 5675/12474 , -16175/199584 , 26575/149688 , 16067/598752}
Für
n
=
8
{\displaystyle n=8}
w
i
<
0
{\displaystyle w_{i}<0}
i
=
2
,
4
,
6
{\displaystyle i=2,4,6}
∑
i
=
0
n
|
w
i
|
=
1
,
45
…
{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dotso }
n
=
10
{\displaystyle n=10}
∑
i
=
0
n
|
w
i
|
=
3,064
79477312810646143979477312810
…
{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}06479477312810646143979477312810\dotso }
Beispiel:
∫
1
3
1
x
d
x
=
ln
(
3
)
−
ln
(
1
)
=
ln
(
3
)
=
1,098
61
23
…
{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}09861\,23\dotso }
Näherung mit Simpson-Regel (
n
=
2
{\displaystyle n=2}
h
=
b
−
a
n
=
2
2
=
1
{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}
x
0
=
a
=
1
{\displaystyle x_{0}=a=1}
∫
1
3
p
2
(
x
)
d
x
=
2
⋅
(
1
6
f
(
1
)
+
4
6
f
(
2
)
+
1
6
f
(
3
)
)
=
2
⋅
(
1
6
⋅
1
+
4
6
⋅
1
2
+
1
6
⋅
1
3
)
=
10
9
=
1
,
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
f
(
4
)
(
ξ
)
=
4
!
ξ
5
{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}
E
(
f
)
=
−
1
90
⋅
(
2
2
)
5
⋅
4
!
ξ
5
=
−
4
15
⋅
1
ξ
5
{\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}}}}
ξ
∈
[
1
,
3
]
{\displaystyle \xi \in [1,3]}
Fehlerabschätzung:
|
E
(
f
)
|
≤
4
15
⋅
max
1
≤
ξ
≤
3
|
1
ξ
5
|
=
4
15
⋅
1
1
=
0
,
2
6
¯
{\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:
|
E
(
f
)
|
=
|
∫
1
3
1
x
d
x
−
∫
1
3
p
2
(
x
)
d
x
|
=
|
1,098
61
23
…
−
1
,
1
¯
|
=
0,012
49
88
…
<
0
,
2
6
¯
{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}09861\,23\dotso -1{,}{\overline {1}}\right|=0{,}01249\,88\dotso <0{,}2{\overline {6}}}
Die Stützstellen
t
i
{\displaystyle t_{i}}
[
0
,
1
]
{\displaystyle [0,1]}
t
0
=
1
n
+
2
,
t
i
=
i
+
1
n
+
2
,
t
n
=
n
+
1
n
+
2
{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}
[
a
,
b
]
{\displaystyle [a,b]}
x
i
=
a
+
t
i
⋅
(
b
−
a
)
{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}
n
{\displaystyle n}
Name
Stützstellen
t
i
{\displaystyle t_{i}}
Gewichte
w
i
{\displaystyle w_{i}}
E
(
f
)
{\displaystyle E(f)}
0
Rechteckregel Mittelpunktsregel Tangententrapezregel
1
2
{\displaystyle {\frac {1}{2}}}
1
{\displaystyle 1\quad }
(
b
−
a
)
3
24
f
″
(
ξ
)
{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}
1
1
3
2
3
{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}
1
2
1
2
{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}
3
(
b
−
a
3
)
3
4
f
″
(
ξ
)
{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}
2
1
4
2
4
3
4
{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}
2
3
−
1
3
2
3
{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}
14
(
b
−
a
4
)
5
45
f
(
4
)
(
ξ
)
{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}
3
1
5
2
5
3
5
4
5
{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}
11
24
1
24
1
24
11
24
{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}
95
(
b
−
a
5
)
5
144
f
(
4
)
(
ξ
)
{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}
4
1
6
2
6
3
6
4
6
5
6
{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}
11
20
−
14
20
26
20
−
14
20
11
20
{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}
41
(
b
−
a
6
)
7
140
f
(
6
)
(
ξ
)
{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}
5
1
7
2
7
3
7
4
7
5
7
6
7
{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}
611
1440
−
453
1440
562
1440
562
1440
−
453
1440
611
1440
{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}
5257
(
b
−
a
7
)
7
8640
f
(
6
)
(
ξ
)
{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}
6
1
8
2
8
3
8
4
8
5
8
6
8
7
8
{\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}}}
460
945
−
954
945
2196
945
−
2459
945
2196
945
−
954
945
460
945
{\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}}}
3956
(
b
−
a
8
)
9
14
175
f
(
8
)
(
ξ
)
{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}
Für
n
=
5
{\displaystyle n=5}
∑
i
=
0
n
|
w
i
|
=
3252
1440
=
2,258
33
3
…
{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}25833\,3\dotso }
n
=
6
{\displaystyle n=6}
∑
i
=
0
n
|
w
i
|
=
9679
945
=
10
,
24
…
{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dotso }
Von diesen Formeln ist nur die Rechteckregel empfehlenswert. Die Formel für
n
=
1
{\displaystyle n=1}
Beispiel:
∫
1
3
1
x
d
x
=
ln
(
3
)
−
ln
(
1
)
=
ln
(
3
)
=
1,098
61
23
…
{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}09861\,23\dotso }
Näherung mit der Formel für
n
=
2
{\displaystyle n=2}
h
=
b
−
a
n
+
2
=
2
4
=
1
2
{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}
x
0
=
a
+
h
=
3
2
{\displaystyle x_{0}=a+h={\frac {3}{2}}}
∫
1
3
p
2
(
x
)
d
x
=
2
⋅
(
2
3
f
(
3
2
)
−
1
3
f
(
4
2
)
+
2
3
f
(
5
2
)
)
=
2
⋅
(
2
3
⋅
2
3
−
1
3
⋅
2
4
+
2
3
⋅
2
5
)
=
49
45
=
1
,
0
8
¯
{\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
f
(
4
)
(
ξ
)
=
4
!
ξ
5
{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}
E
(
f
)
=
14
45
⋅
(
2
4
)
5
⋅
4
!
ξ
5
=
7
30
⋅
1
ξ
5
{\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}}}}
ξ
∈
[
1
,
3
]
{\displaystyle \xi \in [1,3]}
Fehlerabschätzung:
|
E
(
f
)
|
≤
7
30
⋅
max
1
≤
ξ
≤
3
|
1
ξ
5
|
=
7
30
⋅
1
1
=
0
,
2
3
¯
{\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:
|
E
(
f
)
|
=
|
∫
1
3
1
x
d
x
−
∫
1
3
p
2
(
x
)
d
x
|
=
|
1,098
61
23
…
−
1
,
0
8
¯
|
=
0,009
72
33997
79
…
<
0
,
2
3
¯
{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}09861\,23\dotso -1{,}0{\overline {8}}\right|=0{,}00972\,33997\,79\dotso <0{,}2{\overline {3}}}
Diese Formeln sind nach Colin Maclaurin benannt. Die Stützstellen
t
i
{\displaystyle t_{i}}
[
0
,
1
]
{\displaystyle [0,1]}
t
0
=
1
2
n
+
2
,
t
i
=
2
i
+
1
2
n
+
2
,
t
n
=
2
n
+
1
2
n
+
2
{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}
[
a
,
b
]
{\displaystyle [a,b]}
x
i
=
a
+
t
i
⋅
(
b
−
a
)
{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}
n
{\displaystyle n}
Name
Stützstellen
t
i
{\displaystyle t_{i}}
Gewichte
w
i
{\displaystyle w_{i}}
E
(
f
)
{\displaystyle E(f)}
0
Rechteckregel Mittelpunktsregel Tangententrapezregel
1
2
{\displaystyle {\frac {1}{2}}}
1
{\displaystyle 1\quad }
(
b
−
a
)
3
24
f
″
(
ξ
)
{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}
1
1
4
3
4
{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}
1
2
1
2
{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}
(
b
−
a
2
)
3
12
f
″
(
ξ
)
{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}
2
1
6
1
2
5
6
{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}
3
8
2
8
3
8
{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}
21
(
b
−
a
3
)
5
640
f
(
4
)
(
ξ
)
{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}
3
1
8
3
8
5
8
7
8
{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}
13
48
11
48
11
48
13
48
{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}
103
(
b
−
a
4
)
5
1440
f
(
4
)
(
ξ
)
{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}
4
1
10
3
10
5
10
7
10
9
10
{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}
275
1152
100
1152
402
1152
100
1152
275
1152
{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}
5575
(
b
−
a
5
)
7
193
536
f
(
6
)
(
ξ
)
{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}
Für
n
=
6
{\displaystyle n=6}
∑
i
=
0
n
|
w
i
|
=
1,363
…
{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dotso }
n
=
8
{\displaystyle n=8}
∑
i
=
0
n
|
w
i
|
=
3,433
…
{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dotso }
Beispiel:
∫
1
3
1
x
d
x
=
ln
(
3
)
−
ln
(
1
)
=
ln
(
3
)
=
1,098
61
23
…
{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}09861\,23\dotso }
Näherung mit der Formel für
n
=
2
{\displaystyle n=2}
h
=
b
−
a
n
+
1
=
2
3
{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}
x
0
=
a
+
h
2
=
4
3
{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}
∫
1
3
p
2
(
x
)
d
x
=
2
⋅
(
3
8
f
(
4
3
)
+
2
8
f
(
6
3
)
+
3
8
f
(
8
3
)
)
=
2
⋅
(
3
8
⋅
3
4
+
2
8
⋅
3
6
+
3
8
⋅
3
8
)
=
105
96
=
1,093
75
{\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
f
(
4
)
(
ξ
)
=
4
!
ξ
5
{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}
E
(
f
)
=
21
640
⋅
(
2
3
)
5
⋅
4
!
ξ
5
=
14
135
⋅
1
ξ
5
{\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}}}}
ξ
∈
[
1
,
3
]
{\displaystyle \xi \in [1,3]}
Fehlerabschätzung:
|
E
(
f
)
|
≤
14
135
⋅
max
1
≤
ξ
≤
3
|
1
ξ
5
|
=
14
135
⋅
1
1
=
0
,
1
037
¯
{\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:
|
E
(
f
)
|
=
|
∫
1
3
1
x
d
x
−
∫
1
3
p
2
(
x
)
d
x
|
=
|
1,098
61
23
…
−
1,093
75
|
=
0,000
48
6229
…
<
0
,
1
037
¯
{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}09861\,23\dotso -1{,}09375|=0{,}00048\,6229\dotso <0{,}1{\overline {037}}}
Ab Grad 8 treten bei vielen Newton-Cotes-Formeln negative Gewichte auf, was die Gefahr der Auslöschung mit sich bringt. Außerdem kann man im Allgemeinen keine Konvergenz erwarten, da die Polynominterpolation schlecht konditioniert ist. Bei größeren Integrationsbereichen
[
a
,
b
]
{\displaystyle [a,b]}
Literatur
Hans R. Schwarz, Norbert Köckler: Numerische Mathematik. 6. Auflage. Teubner, Stuttgart 2006, ISBN 3-519-42960-8 , S. 311–316.
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.
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.
Günter Bärwolf: Numerik für Ingenieure, Physiker und Informatiker. ISBN 978-3-8274-1689-6 , Spektrum, München 2007, S. 128.
Gisela Engeln-Müllges , Klaus Niederdrenk, Reinhard Wodicka: Numerik-Algorithmen : Verfahren, Beispiele, Anwendungen. ISBN 978-3-642-13472-2 , Springer, Berlin und Heidelberg 2011.
Einzelnachweise
↑ Thomas Weddle (Newcastle-upon-Tyne ): A new simple and general method of solving numerical equations of all orders . Hamilton, Adams & Co. and J. Philipson, London 1842 (Internet Archive – 52 S.).
↑ WolframAlpha. abgerufen am 14. September 2019 .
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle \xi \in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23)
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
![{\displaystyle \xi \in [1,3]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b658b27a1b1babd01aa16dc7426ceddb92d85fb)
