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រូបមន្តញូតុន-កូត្ស ឬ ក្បួនញូតុន-កូត្ស គឺជាក្រុមរូបមន្ត សំរាប់គណនាអាំងតេក្រាល លេខ។ រូបមន្តយកឈ្មោះតាមលោក អ៊ីសាក់ ញូតុន និង រ៉ូចឺ កូត្ស (Roger Cotes) ។
អនុគមន៍ f ស្គាល់ត្រង់ចំនុចមានចន្លោះស្មើៗគ្នា
x
i
{\displaystyle \ x_{i}}
i
=
0
;
1
;
2
;
⋯
;
n
{\displaystyle \ i=0;1;2;\cdots ;n}
∫
a
b
f
(
x
)
d
x
≈
∑
i
=
0
n
w
i
f
(
x
i
)
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=0}^{n}w_{i}\,f(x_{i})}
ដែល
x
i
=
i
h
+
x
0
{\displaystyle \ x_{i}=ih+x_{0}}
h
=
x
n
−
x
0
n
{\displaystyle \ h={\frac {x_{n}-x_{0}}{n}}}
w
i
{\displaystyle \ w_{i}}
w
i
{\displaystyle \ w_{i}}
ពហុធាឡាហ្ក្រង់គ្រឹះ (Lagrange basis polynomials) ។ វាអាស្រ័យតែនឹង
x
i
{\displaystyle \ x_{i}}
f
{\displaystyle \ f}
L
(
x
)
{\displaystyle \ L(x)}
(
x
0
,
f
(
x
0
)
)
,
⋯
(
x
n
,
f
(
x
n
)
)
{\displaystyle \ (x_{0},f(x_{0})),\cdots \ (x_{n},f(x_{n}))}
∫
a
b
f
(
x
)
d
x
≈
∫
a
b
L
(
x
)
d
x
=
∫
a
b
∑
i
=
0
n
f
(
x
i
)
l
i
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \int _{a}^{b}L(x)\,dx=\int _{a}^{b}\sum _{i=0}^{n}f(x_{i})\,l_{i}(x)\,dx}
=
∑
i
=
0
n
∫
a
b
f
(
x
i
)
l
i
(
x
)
d
x
=
∑
i
=
0
n
f
(
x
i
)
∫
a
b
l
i
(
x
)
d
x
⏟
w
i
{\displaystyle =\sum _{i=0}^{n}\int _{a}^{b}f(x_{i})l_{i}(x)dx=\sum _{i=0}^{n}f(x_{i})\underbrace {\int _{a}^{b}l_{i}(x)dx} _{w_{i}}}
រូបមន្តញូតុន-កូត្សប្រភេទបិទដែលមានដឺក្រេ n ត្រូវបានគេពោលថា
∫
a
b
f
(
x
)
d
x
≈
∑
i
=
1
n
−
1
w
i
f
(
x
i
)
{\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{i=1}^{n-1}w_{i}\,f(x_{i})}
រូបមន្តញូតុន-កូត្សប្រភេទបិទ
ដឺក្រេ
ឈ្មោះ
រូបមន្ត
តួអ៊ែររឺ
1
ក្បួនចតុកោណព្នាយ
h
2
(
f
0
+
f
1
)
{\displaystyle {\frac {h}{2}}(f_{0}+f_{1})}
−
h
3
12
f
(
2
)
(
ξ
)
{\displaystyle -{\frac {h^{3}}{12}}\,f^{(2)}(\xi )}
2
ក្បួនស៊ីម្ពសុន (Simpson's rule)
h
3
(
f
0
+
4
f
1
+
f
2
)
{\displaystyle {\frac {h}{3}}(f_{0}+4f_{1}+f_{2})}
−
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {h^{5}}{90}}\,f^{(4)}(\xi )}
3
ក្បួនស៊ីម្ពសុន 3/8
3
h
8
(
f
0
+
3
f
1
+
3
f
2
+
f
3
)
{\displaystyle {\frac {3h}{8}}(f_{0}+3f_{1}+3f_{2}+f_{3})}
−
3
h
5
80
f
(
4
)
(
ξ
)
{\displaystyle -{\frac {3h^{5}}{80}}\,f^{(4)}(\xi )}
4
ក្បួនបូដ
2
h
45
(
7
f
0
+
32
f
1
+
12
f
2
+
32
f
3
+
7
f
4
)
{\displaystyle {\frac {2h}{45}}(7f_{0}+32f_{1}+12f_{2}+32f_{3}+7f_{4})}
−
8
h
7
945
f
(
6
)
(
ξ
)
{\displaystyle -{\frac {8h^{7}}{945}}\,f^{(6)}(\xi )}
សំគាល់៖
f
i
{\displaystyle \ f_{i}}
f
(
x
i
)
{\displaystyle \ f(x_{i})}
រូបមន្តញូតុន-កូត្សប្រភេទបើក
ដឺក្រេ
ឈ្មោះ
រូបមន្ត
តួអ៊ែររឺ
0
ក្បួនចតុកោណកែង
2
h
f
1
{\displaystyle 2hf_{1}\,}
h
3
24
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{24}}\,f^{(2)}(\xi )}
1
គ្មានឈ្មោះ
3
h
2
(
f
1
+
f
2
)
{\displaystyle {\frac {3h}{2}}(f_{1}+f_{2})}
h
3
4
f
(
2
)
(
ξ
)
{\displaystyle {\frac {h^{3}}{4}}\,f^{(2)}(\xi )}
2
គ្មានឈ្មោះ
4
h
3
(
2
f
1
−
f
2
+
2
f
3
)
{\displaystyle {\frac {4h}{3}}(2f_{1}-f_{2}+2f_{3})}
28
h
5
90
f
(
4
)
(
ξ
)
{\displaystyle {\frac {28h^{5}}{90}}f^{(4)}(\xi )}
3
គ្មានឈ្មោះ
5
h
24
(
11
f
1
+
f
2
+
f
3
+
11
f
4
)
{\displaystyle {\frac {5h}{24}}(11f_{1}+f_{2}+f_{3}+11f_{4})}
95
h
5
144
f
(
4
)
(
ξ
)
{\displaystyle {\frac {95h^{5}}{144}}f^{(4)}(\xi )}
ពហុធាឡាហ្ក្រង់
L
(
x
)
{\displaystyle L(x)\,}
f
{\displaystyle f\,}
L
(
x
)
=
v
n
(
x
)
∑
i
=
0
n
y
i
(
x
−
x
i
)
v
n
′
(
x
i
)
{\displaystyle L(x)=v_{n}(x)\sum _{i=0}^{n}{\frac {y_{i}}{(x-x_{i})v'_{n}(x_{i})}}}
ដែល
v
n
(
x
)
=
∏
j
=
0
n
(
x
−
x
j
)
{\displaystyle v_{n}(x)=\prod _{j=0}^{n}(x-x_{j})}
ហេតុនេះ
w
i
=
∫
a
b
v
n
(
x
)
(
x
−
x
i
)
v
n
′
(
x
i
)
d
x
{\displaystyle w_{i}=\int _{a}^{b}{\frac {v_{n}(x)}{(x-x_{i})v'_{n}(x_{i})}}dx}
ដោយប្តូរអថេរ
y
=
x
−
a
h
{\displaystyle y={\frac {x-a}{h}}}
w
i
=
b
−
a
n
(
−
1
)
n
−
i
i
!
(
n
−
i
)
!
∫
0
n
∏
k
=
0
,
k
≠
i
n
(
y
−
k
)
d
y
{\displaystyle w_{i}={\frac {b-a}{n}}{\frac {(-1)^{n-i}}{i!(n-i)!}}\int _{0}^{n}\prod _{k=0,k\neq i}^{n}(y-k)dy}
n
=
1
{\displaystyle n=1\,}
w
0
=
h
(
−
1
)
1
−
0
0
!
(
1
−
0
)
!
∫
0
1
∏
k
=
0
,
k
≠
0
1
(
y
−
k
)
d
y
=
−
h
∫
0
1
(
y
−
1
)
d
y
=
−
h
[
(
y
−
1
)
2
2
]
0
1
=
h
2
{\displaystyle {\begin{matrix}w_{0}&=&h{\frac {(-1)^{1-0}}{0!(1-0)!}}\int _{0}^{1}\prod _{k=0,k\neq 0}^{1}(y-k)dy\\&=&-h\int _{0}^{1}(y-1)dy\\&=&-h\left[{\frac {(y-1)^{2}}{2}}\right]_{0}^{1}\\&=&{\frac {h}{2}}\end{matrix}}}
w
1
=
h
2
{\displaystyle w_{1}={\frac {h}{2}}}
![{\displaystyle {\begin{matrix}w_{0}&=&h{\frac {(-1)^{1-0}}{0!(1-0)!}}\int _{0}^{1}\prod _{k=0,k\neq 0}^{1}(y-k)dy\\&=&-h\int _{0}^{1}(y-1)dy\\&=&-h\left[{\frac {(y-1)^{2}}{2}}\right]_{0}^{1}\\&=&{\frac {h}{2}}\end{matrix}}}](/media/api/rest_v1/media/math/render/svg/87dc917d9beaa1751120211322982567c0217329)
