EGM6321 - Principles of Engineering Analysis 1, Fall 2009
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Mtg 35:
Gauss Legendre quadrature (numerical integration}
=
∑
{\displaystyle =\sum }
=
∑
{\displaystyle =\sum }
I
(
f
)
:=
∫
−
1
1
f
(
x
)
d
x
{\displaystyle I(f):=\int _{-1}^{1}f(x)\,dx\ }
I
n
(
f
)
:=
∑
j
=
1
n
w
j
f
(
x
j
)
d
x
{\displaystyle I_{n}(f):=\sum _{j=1}^{n}w_{j}f(x_{j})\,dx\ }
{
x
j
}
{\displaystyle \left\{x_{j}\right\}\ }
P
n
(
x
)
=
0
{\displaystyle P_{n}(x)=0\ }
P
n
(
x
)
{\displaystyle P_{n}(x)\ }
w
j
{\displaystyle w_{j}\ }
−
1
<
x
1
<
x
2
<
.
.
.
<
x
n
<
1
{\displaystyle -1<x_{1}<x_{2}<...<x_{n}<1\ }
I
(
f
)
=
I
n
(
f
)
+
E
n
(
f
)
{\displaystyle \displaystyle {\begin{aligned}I(f)=I_{n}(f)+E_{n}(f)\end{aligned}}}
(1)
w
j
=
−
2
(
n
+
1
)
P
n
′
(
x
j
)
P
n
+
1
(
x
j
)
{\displaystyle \displaystyle {\begin{aligned}w_{j}={\frac {-2}{(n+1)P_{n}'(x_{j})P_{n+1}(x_{j})}}\end{aligned}}}
(2)
where j=1,2,...,n
E
n
(
f
)
=
2
2
n
+
1
(
n
!
)
4
(
2
n
+
1
)
[
(
2
n
)
!
]
2
f
(
2
n
)
(
η
)
(
2
n
)
!
{\displaystyle \displaystyle {\begin{aligned}E_{n}(f)={\frac {2^{2n+1}(n!)^{4}}{(2n+1)\left[(2n)!\right]^{2}}}{\frac {f^{(2n)}(\eta \ )}{(2n)!}}\end{aligned}}}
(3)
for
η
∈
[
−
1
,
+
1
]
{\displaystyle \eta \ \in \left[-1,+1\right]\ }
n
=
2
{\displaystyle n=2\ }
Eq.(3) P.31-3
P
2
(
x
)
=
1
2
(
3
x
2
−
1
)
{\displaystyle P_{2}(x)={\frac {1}{2}}(3x^{2}-1)\ }
⇒
x
1
,
2
=
±
1
3
{\displaystyle \Rightarrow \ x_{1,2}=\pm \ {\frac {1}{\sqrt {3}}}\ }
Eq.(4) P.31-3
P
2
′
(
x
)
=
3
x
,
P
3
(
x
)
=
1
2
(
5
x
3
−
3
x
)
{\displaystyle P_{2}'(x)=3x,P_{3}(x)={\frac {1}{2}}(5x^{3}-3x)\ }
W
1
=
2
(
2
+
1
)
(
3
)
(
−
1
3
)
1
2
[
5
(
−
1
3
)
3
−
3
(
−
1
3
)
]
=
1
{\displaystyle W_{1}={\frac {2}{(2+1)(3)({\frac {-1}{\sqrt {3}}}){\frac {1}{2}}\left[5({\frac {-1}{\sqrt {3}}})^{3}-3({\frac {-1}{\sqrt {3}}})\right]}}=1\ }
W
2
=
1
{\displaystyle W_{2}=1\ }
{
x
j
}
{\displaystyle \left\{x_{j}\right\}\ }
{
w
j
}
,
j
=
1
,
.
.
.
,
n
{\displaystyle \left\{w_{j}\right\},j=1,...,n\ }
n
=
1
,
.
.
.
,
5
{\displaystyle n=1,...,5\ }
P
n
(
x
)
{\displaystyle P_{n}(x)\ }
n
=
1
,
.
.
.
,
6
{\displaystyle n=1,...,6\ }
HW p31-3 )
{
x
j
}
{\displaystyle \left\{x_{j}\right\}\ }
{
w
j
}
{\displaystyle \left\{w_{j}\right\}\ }
E
n
(
f
)
{\displaystyle E_{n}(f)\ }
Eq.(3) P.35-2 . Consider
f
∈
P
2
n
−
1
{\displaystyle f\in \mathrm {P} \ _{2n-1}\ }
References
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![{\displaystyle \displaystyle {\begin{aligned}E_{n}(f)={\frac {2^{2n+1}(n!)^{4}}{(2n+1)\left[(2n)!\right]^{2}}}{\frac {f^{(2n)}(\eta \ )}{(2n)!}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/767634aaa629905574bd11cbb7a39edbbac678e4)
![{\displaystyle \eta \ \in \left[-1,+1\right]\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b17d2a5f6968347a2e6e1e6edf23745c1be85b)
![{\displaystyle W_{1}={\frac {2}{(2+1)(3)({\frac {-1}{\sqrt {3}}}){\frac {1}{2}}\left[5({\frac {-1}{\sqrt {3}}})^{3}-3({\frac {-1}{\sqrt {3}}})\right]}}=1\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b479ee0b21573d59e4f33c538eb95ae21bad984)
