6.4 Newton-Cote求积公式
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|I − C | = 6.630 × 10
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.
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) F/úª
T =
|I − T | = 2.535 × 10 Simpsonúª
S=
1−0 1 [f (0) + f (1)] = (1 + 0.841471) = 0.920735, 2 2
−2
.
|I − S| = 6.281 × 10 Cotesúª
[a, b]
§Newton-Cotesú §Newton-Cotesú
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[a, b]
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3 ξ ∈ (a, b)§¦&¨ n ´Ûê§ f ∈ C
b a
n+1
[a, b]
§k
(6.4.5)
¡F/úª§Ù¥ § § 5 F/úªäk1gê°Ý" ¨ n = 2: Simpsonúª
b a
(6.4.6)
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Ù¥ a < ξ < b§
h5 (4) b−a f (x)dx = [f (x0) + 4f (x1) + f (x2)] − f (ξ ), 6 90
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6.4.3
A«~^Newton-Cotes¦Èúª
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¨ n = 1: F/úª
b
Ù¥ a < ξ < b§
a
b−a h3 [f (x0) + f (x1)] − f (ξ ), f (x)dx = 2 12 b−a [f (x0) + f (x1)] 2 xi = a + ih i = 0, 1 h = b − a. T =
§
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¨n = 4: Cotesúª
b
f (x)dx =
a
Ù¥ a < ξ < b§
b−a [7f (x0) + 32f (x1) + 12f (x2) + 32f (x3) + 7f (x4)] 90 8h7 (6) − f (ξ ), (6.4.9) 945
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6.4 Newton-Cotes
6.4.1
¦Èúª
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ÄVg
6.4.2 Newton-Cotes 6.4.3
¦Èúªê°Ý A«~^Newton-Cotes¦Èúª
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6.4.1
½Â6.4.1 e¦È!X x , x , · · · , x 3«m[a, b]S1å©Ù§= b−a x = x + ih, i = 0, 1, · · · , n, Ù¥ x = a, h = , n ¼ê ρ ≡ 1§Kd(6.3.10)-(6.3.11)½Â'¢.¦Èúª¡(n+1) XNewton-Cotesúª§äNúª
f (n+1)(ξ ) · hn+2 f (x)dx − In = (n + 1)!
n+2
n
t(t − 1)(t − 2) · · · (t − n)dt;
0
¨ n ´óê§ f ∈ C
b
[a, b]
§k
n+3 0 n
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f (x)dx − In =
a
f
(n+2)
(ξ ) · h (n + 2)!
(6.4.7)
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¡Simpsonúª§Ù¥ § 5 Simpsonúªäk3gê°Ý"
S=
b−a [f (x0) + 4f (x1) + f (x2)] (6.4.8) 6 xi = a + ih i = 0, 1, 2 h = (b − a)/2.
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t2(t − 1)(t − 2) · · · (t − n)dt.
Title Page
5
In
(n + 1)
XNewton-Cotesúª = (b − a) C f (x ) 'ê°Ýd
n (n) i i i=0
e n ´Ûê, ≥ n + 1, e n ´óê.
≥ n,
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~6.4.1 ©O^F/úª!SimpsonúªÚCotesúª¦e È©'Cq
1
I=
0
O(I = 0.946083070367183 · · · .
sin x dx. x
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for i = 0, 1, · · · , n, (6.4.2)
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¡CotesXê"
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6.4.2 Newton-Cotes
¦Èúªê°Ý
n+1
½n6.4.1 (1) ¨ n ´ Û ê § f ∈ C ª(6.4.1)'ê°Ýd ≥ n¶ (2) ¨ n ´ ó ê § f ∈ C ª(6.4.1)'ê°Ýd ≥ n + 1.
0 1 n i 0 0 b n
ÄVg
பைடு நூலகம்
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f (x)dx ≈ (b − a)
(n) Ci f (xi), i=0
Title Page
(6.4.1)
Ù¥
(n) Ci
a
Page 2 of 9
(−1)n−i = i! · (n − i)! · n
n 0
n
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(t − j )dt,
j =0,j =i
Title Page
b−a C= [7f (x0) + 32f (x1) + 12f (x2) + 32f (x3) + 7f (x4)] 90
i
(6.4.10)
Page 7 of 9
¡Cotesúª§Ù¥ x = a + ih§i = 0, 1, 2, 3, 4§h = (b − a)/4. 5 Adúªäk5gê°Ý"
1−0 [f (0) + 4f (1/2) + f (1)] 6 1 = (1 + 4 × 0.955851 + 0.841471) = 0.946146, 6
−5
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.
1−0 C= [7f (0) + 32f (1/4) + 12f (1/2) + 32f (3/4) + 7f (1)] 90 1 = (7 × 1 + 32 × 0.989616 + 12 × 0.955851 + 32 × 0.908852 + 7 × 0.841471) 90 = 0.946083,
Page 9 of 9
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|I − C | = 6.630 × 10
Quit
.
Full Screen
Close
Quit
) F/úª
T =
|I − T | = 2.535 × 10 Simpsonúª
S=
1−0 1 [f (0) + f (1)] = (1 + 0.841471) = 0.920735, 2 2
−2
.
|I − S| = 6.281 × 10 Cotesúª
[a, b]
§Newton-Cotesú §Newton-Cotesú
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Title Page
n+2
[a, b]
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3 ξ ∈ (a, b)§¦&¨ n ´Ûê§ f ∈ C
b a
n+1
[a, b]
§k
(6.4.5)
¡F/úª§Ù¥ § § 5 F/úªäk1gê°Ý" ¨ n = 2: Simpsonúª
b a
(6.4.6)
Home Page
Title Page
Ù¥ a < ξ < b§
h5 (4) b−a f (x)dx = [f (x0) + 4f (x1) + f (x2)] − f (ξ ), 6 90
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Quit
6.4.3
A«~^Newton-Cotes¦Èúª
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¨ n = 1: F/úª
b
Ù¥ a < ξ < b§
a
b−a h3 [f (x0) + f (x1)] − f (ξ ), f (x)dx = 2 12 b−a [f (x0) + f (x1)] 2 xi = a + ih i = 0, 1 h = b − a. T =
§
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¨n = 4: Cotesúª
b
f (x)dx =
a
Ù¥ a < ξ < b§
b−a [7f (x0) + 32f (x1) + 12f (x2) + 32f (x3) + 7f (x4)] 90 8h7 (6) − f (ξ ), (6.4.9) 945
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6.4 Newton-Cotes
6.4.1
¦Èúª
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ÄVg
6.4.2 Newton-Cotes 6.4.3
¦Èúªê°Ý A«~^Newton-Cotes¦Èúª
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6.4.1
½Â6.4.1 e¦È!X x , x , · · · , x 3«m[a, b]S1å©Ù§= b−a x = x + ih, i = 0, 1, · · · , n, Ù¥ x = a, h = , n ¼ê ρ ≡ 1§Kd(6.3.10)-(6.3.11)½Â'¢.¦Èúª¡(n+1) XNewton-Cotesúª§äNúª
f (n+1)(ξ ) · hn+2 f (x)dx − In = (n + 1)!
n+2
n
t(t − 1)(t − 2) · · · (t − n)dt;
0
¨ n ´óê§ f ∈ C
b
[a, b]
§k
n+3 0 n
Home Page
f (x)dx − In =
a
f
(n+2)
(ξ ) · h (n + 2)!
(6.4.7)
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¡Simpsonúª§Ù¥ § 5 Simpsonúªäk3gê°Ý"
S=
b−a [f (x0) + 4f (x1) + f (x2)] (6.4.8) 6 xi = a + ih i = 0, 1, 2 h = (b − a)/2.
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t2(t − 1)(t − 2) · · · (t − n)dt.
Title Page
5
In
(n + 1)
XNewton-Cotesúª = (b − a) C f (x ) 'ê°Ýd
n (n) i i i=0
e n ´Ûê, ≥ n + 1, e n ´óê.
≥ n,
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~6.4.1 ©O^F/úª!SimpsonúªÚCotesúª¦e È©'Cq
1
I=
0
O(I = 0.946083070367183 · · · .
sin x dx. x
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Title Page
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for i = 0, 1, · · · , n, (6.4.2)
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¡CotesXê"
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6.4.2 Newton-Cotes
¦Èúªê°Ý
n+1
½n6.4.1 (1) ¨ n ´ Û ê § f ∈ C ª(6.4.1)'ê°Ýd ≥ n¶ (2) ¨ n ´ ó ê § f ∈ C ª(6.4.1)'ê°Ýd ≥ n + 1.
0 1 n i 0 0 b n
ÄVg
பைடு நூலகம்
Home Page
f (x)dx ≈ (b − a)
(n) Ci f (xi), i=0
Title Page
(6.4.1)
Ù¥
(n) Ci
a
Page 2 of 9
(−1)n−i = i! · (n − i)! · n
n 0
n
Go Back
(t − j )dt,
j =0,j =i
Title Page
b−a C= [7f (x0) + 32f (x1) + 12f (x2) + 32f (x3) + 7f (x4)] 90
i
(6.4.10)
Page 7 of 9
¡Cotesúª§Ù¥ x = a + ih§i = 0, 1, 2, 3, 4§h = (b − a)/4. 5 Adúªäk5gê°Ý"
1−0 [f (0) + 4f (1/2) + f (1)] 6 1 = (1 + 4 × 0.955851 + 0.841471) = 0.946146, 6
−5
Home Page
Title Page
.
1−0 C= [7f (0) + 32f (1/4) + 12f (1/2) + 32f (3/4) + 7f (1)] 90 1 = (7 × 1 + 32 × 0.989616 + 12 × 0.955851 + 32 × 0.908852 + 7 × 0.841471) 90 = 0.946083,