7.7 高斯型求积公式
时间：2025-07-13
7.77.7.1 7.7.2 7.7.3
pd.¦ÈúªHome Page Title Page Page 1 of 10
Ä Vg;pdúª½5ÚÂñ5
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7.7.1
Ä Vg1 2 n b n
½Â7.7.1 x, x,···, x´«m[a, b]þ n pÉ:§f´[a, b]þ'u¼êρ ȼê"e¢ .¦Èúªρ(x)f (x)dx≈a i=1
Aif (xi)
(7.7.1)
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äk 2n 1 g ê°Ý§K(7.7.1)¡ pd.¦Èúª§ p¦È3: x, x,···, x¡ pd£3¤:" p½n7.7.1¦È3: x, x,···, x´pd:¿©7 ^ ´µx, x,···, x´[a, b]þ, 'u¼êρ n g¢õ ª":"1 2 n 1 2 n 1 2 n
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~7.7.1¦ x, xÚ A, A§¦&e¦Èúª´pdF¦Èúª1 2 1 2 1 0
)Ï ¦Èúª(7.7.2)äk 2×2 1= 3 g ê°Ý§¤±(7.7.2)é f (x)= x (k= 0, 1, 2, 3)ÑO(¤á§=k
f (x)√ dx≈ A1f (x1)+ A2f (x2). x
(7.7.2)
1 1√ 0 x 1 1√ 0 x 1 1√ 0 x 1 1√ 0 x 2
· 1dx= A1· 1+ A2· 1,Home Page
· xdx= A1· x1+ A2· x2,· x2dx= A1· x2+ A2· x2, 1 2· x3dx= A1· x3+ A2· x3. 1 21 2
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)þã'u þ x, x, A, A ' §|§&1
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3 2 x1= 7 7
6, 5
3 2 x2=+ 7 7
6, 5
1 A1= 1+ 3
6, 5
1 A2= 1 3
6 . 5
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7.7.2
;pdúª1 dn pn(x)= n· 2· n! dxn x2 1n
½Â7.7.2 (V4õ ª)õ ª, n= 0, 1, 2,···, (7.7.3)
V¡ V4õ ª" 5õ ª
´Ä Xê 1 n gV4õ ª§n= 0, 1, 2,··· .½Â7.7.3 x, x,···, x´«m[ 1, 1]þ n gV4õ ª p (x)":§f´«m[ 1, 1]þ'u¼êρ≡ 1 ȼê§K¢ .¦Èúª1 2 n n 1 n
n! dn pn(x)=· (2n)! dxn
x2 1
n
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(7.7.4)Title Page
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f (x)dx≈ 1
Aif (xi)i=1
(7.7.5)
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¡ ;pdúª§½pdúª. p
L7.7.1ÑV4%õ ª'"XÚ¨ n= 2, 3, 4, 5 'pdúª'Xê A .i
n 2 3
"X x
L7.7.1i
Xê A
i
4
5
0.5773502692 0.5773502692 0.7745966692 0.0000000000 0.7745966692 0.8611363116 0.3399810436 0.3399810436 0.8611363116 0.9061798459 0.5384693101 0.0000000000 0.5384693101 0.9061798459
1.0000000000 1.0000000000 0.5555555556 0.8888888889 0.5555555556 0.3478548451 0.6521451549 0.6521451549 0.3478548451 0.2369268850 0.4786286705 0.5688888889 0.4786286705 0.2369268850
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5 e^C t=
K 3?Û«m[a, b]þ$^pdúªb a
2x a b b a
x=
b a a+b t+, 2 2
(7.7.6)Home Page
b a f (x)dx= 2 b a≈ 2
1
f 1 n
b a a+b t+ dt 2 2 b a a+b ti+ . 2 2 (7.7.7)
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Ai fi=1
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~7.7.2©O^2XpdúªÚ3Xpdúª¦eÈ©'Cq 1 0
)Ï a= 0,x1=
sin x dx. x
b= 1, f (x)= sin x/x
§'u2Xpdúª§k
A1= A2= 1, t1= 0.5773502692, t2= 0.5773502692, b a a+b 1 0 0+1 t1+=· ( 0.5773502692)+= 0.2113248654, 2 2 2 2Home Page
x2=
d(7.7.7)ª&1 0
a+b 1 0
0+1 b a t2+=· 0.5773502692+= 0.7886751346, 2 2 2 2
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é'O( I= 0.946083070367183···§¤¦Cq 0.9460411369 k4 k¨êi§ |I G2|= 4.193× 10 5.
sin x 1 0[1× f (0.2113248654)+ 1× f (0.7886751346)] dx≈ G2= x 2= 0.9460411369.
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Ï a= 0,A1= x1=
b= 1, f (x)= sin x/x
§'u3Xpdúª§kt3= 0.7745966692,
0.5555555556, A2= 0.8888888889, A3= 0.5555555556,
t1= 0.7745966692, t2= 0,
d(7.7.7)ª&1 0
b a a+b 1 0 0+1 t1+=· ( 0.7745966692)+= 0.1127016654, 2 2 2 2 a+b 1 0 0+1 b a t2+=·0+= 0.5, x2= 2 2 2 2 b a a+b 1 0 0+1 x3= t3+=· 0.7745966692+= 0.8872983346, 2 2 2 2 1 0 sin x dx≈ G3=[0.5555555556× f (0.1127016654) x 2+0.8888888889× f (0.5)+ 0.5555555556× f (0.8872983346)]= 0.9460831341.
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é'O( I= 0.946083070367183···§¤¦Cq 0.9460831341 k6 k¨êi§ |I G3|= 6.376× 10 8.
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7.7.3
½5ÚÂñ5b n
½n7.7.2 e¦Èúªρ(x)f (x)dx≈a i=1Home Page
Aif (xi)
(7.7.1)
´pdF¦Èúª§K A> 0§i= 1, 2,···, n¶ i b
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Ai=
2ρ(x)li (x)dx, a
(7.7.8)Page 9 of 10 Go Back
Ù¥li(x)=
n
j=1,j=i
x xj, xi xj
i= 1, 2,···, n.
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½n7.7.3 (a)¤k'pdF¦ÈúªÑ´½'" (b) e f∈ C[a, b]§Kn n→∞ b
lim
Aif (xi)=i=1 a
ρ(x)f (x)dx,
(7.7.9)Home Page
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7.7 高斯型求积公式.doc
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