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5.7 nnnggg���^^^���ŁŁŁ
5.7.1 ÄÄÄflflflVVVggg
5.7.2 nnnggg���^^^���ŁŁŁffffff���EEE
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5.7.1 ÄÄÄflflflVVVggg
½½½ÂÂÂ5.7.1 �¼ê f ½Â3«m [a, b]þ§½ [a, b]ff©y
a = x0 < x1 < · · · < xN+1 = b,
¡÷ve�^ff¼êS ´'uf ffnnnggg���^^^���ŁŁŁªªª"
(a) é j = 1, 2, · · · , N + 1§3z�«m [xj−1, xj]þ§S(x)´
ngõª§P Sj(x)¶
(b) Sj(xj) = Sj+1(xj), j = 1, 2, · · · , N ;
(c) S ′j(xj) = S ′j+1(xj), j = 1, 2, · · · , N ;
(d) S ′′j (xj) = S ′′j+1(xj), j = 1, 2, · · · , N ;
(e) S(xj) = f (xj), j = 0, 1, · · · , N + 1,
Ù¥ x1, x2, · · · , xN ¡���^^^(((:::"
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>>>...^^^
111aaa>>>...^^^:
S ′(a) = y′0, S
′(b) = y′N+1. (5.7.1)
111���aaa>>>...^^^:
S ′′(a) = y′′0 , S
′′(b) = y′′N+1. (5.7.2)
e
S ′′(a) = 0, S ′′(b) = 0, (5.7.3)
K^(5.7.3)¡ggg,,,>>>...^^^§Affng�^¡ggg,,,���^^^"
�¼ê y = f (x)´± b− a��±Ïff±Ï¼ê§æ^
e�>.^
111nnnaaa>>>...^^^:
S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4)
d§w,k S(a) = S(b).
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5.7.2 nnnggg���^^^���ŁŁŁffffff���EEE
P
Mj = S
′′(xj), j = 0, 1, · · · , N + 1;
hj = xj − xj−1, j = 1, 2, · · · , N + 1;
Ú
S(x) = Sj(x), x ∈ [xj−1, xj],
Ù¥ j = 1, 2, · · · , N + 1.
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÷÷÷vvv111aaa>>>...^^^ffffffnnnggg���^^^
111aaa>>>...^^^:
S ′(a) = y′0, S
′(b) = y′N+1. (5.7.1)
^e�§|)ÑM0,M1, · · · ,MN+1:
2M0 + λ0M1 = d0,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 1, 2, · · · , N,
µN+1MN + 2MN+1 = dN+1,
(5.7.5)
Ù¥
λ0 = µN+1 = 1, µj =
hj
hj + hj+1
, λj = 1− µj, j = 1, 2, · · · , N,
d0 =
6
h1
(
y1 − y0
h1
− y′0
)
, dN+1 =
6
hN+1
(
y′N+1 −
yN+1 − yN
hN+1
)
,
dj =
6
hj + hj+1
(
yj+1 − yj
hj+1
− yj − yj−1
hj
)
, j = 1, 2, · · · , N.
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÷÷÷vvv111aaa>>>...^^^ffffffnnnggg���^^^
111aaa>>>...^^^:
S ′(a) = y′0, S
′(b) = y′N+1. (5.7.1)
^e�§|)ÑM0,M1, · · · ,MN+1:
2M0 + λ0M1 = d0,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 1, 2, · · · , N,
µN+1MN + 2MN+1 = dN+1,
(5.7.5)
§|(5.7.5)±�¤e�Ý
/ªµ
2 λ0
µ1 2 λ1
µ2 2 λ2
. . . . . . . . .
µN 2 λN
µN+1 2
M0
M1
M2
...
MN
MN+1
=
d0
d1
d2
...
dN
dN+1
. (5.7.6)
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÷÷÷vvv111���aaa>>>...^^^ffffffnnnggg���^^^
111���aaa>>>...^^^:
S ′′(a) = 0, S ′′(b) = 0. (5.7.3)
d^(5.7.3)µ M0 = MN+1 = 0§2^e�§|)Ñ
M1,M2, · · · ,MN :
2M1 + λ1M2 = d1,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 2, 3, · · · , N − 1,
µNMN−1 + 2MN = dN ,
(5.7.7)
Ù¥
µj =
hj
hj + hj+1
, λj = 1− µj, j = 1, 2, · · · , N,
dj =
6
hj + hj+1
(
yj+1 − yj
hj+1
− yj − yj−1
hj
)
, j = 1, 2, · · · , N.
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÷÷÷vvv111���aaa>>>...^^^ffffffnnnggg���^^^
111���aaa>>>...^^^:
S ′′(a) = 0, S ′′(b) = 0. (5.7.3)
d^(5.7.3)µ M0 = MN+1 = 0§2^e�§|)Ñ
M1,M2, · · · ,MN :
2M1 + λ1M2 = d1,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 1, 2, · · · , N − 1,
µNMN−1 + 2MN = dN ,
(5.7.7)
§|(5.7.7)±�¤e�Ý
/ªµ
2 λ1
µ2 2 λ2
µ3 2 λ32
. . . . . . . . .
µN−1 2 λN−1
µN 2
M1
M2
M3
...
MN−1
MN
=
d1
d2
d3
...
dN−1
dN
. (5.7.8)
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÷÷÷vvv111nnnaaa>>>...^^^ffffffnnnggg���^^^
111nnnaaa>>>...^^^:
S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4)
d¼ê y = f (x)´± b− a��±Ïff±Ï¼ê"
d^(5.7.4)µ M0 = MN+1§2^e�§|)Ñ
M1,M2, · · · ,MN+1:
µ1MN + 2M1 + λ1M2 = d1,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 2, 3, · · · , N − 1,
λN+1M1 + µN+1MN + 2MN+1 = dN ,
(5.7.9)
Ù¥
µj =
hj
hj + hj+1
, λj = 1− µj, j = 1, 2, · · · , N,
dj =
6
hj + hj+1
(
yj+1 − yj
hj+1
− yj − yj−1
hj
)
, j = 1, 2, · · · , N.
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÷÷÷vvv111nnnaaa>>>...^^^ffffffnnnggg���^^^
111nnnaaa>>>...^^^:
S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4)
d¼êy = f (x)´±b− a��±Ïff±Ï¼ê"
d^(5.7.4)µ M0 = MN+1§2^e�§|)Ñ
M1,M2, · · · ,MN+1:
µ1MN + 2M1 + λ1M2 = d1,
µjMj−1 + 2Mj + λjMj+1 = dj, j = 2, 3, · · · , N − 1,
λN+1M1 + µN+1MN + 2MN+1 = dN ,
(5.7.9)
§|(5.7.9)±�¤e�Ý
/ªµ
2 λ1 µ1
µ2 2 λ2
µ3 2 λ32
. . . . . . . . .
µN 2 λN
λN+1 µN+1 2
M1
M2
M3
...
MN
MN+1
=
d1
d2
d3
...
dN
dN+1
. (5.7.10)
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d¦ÑffM0,M1, · · · ,MN+1§·��¤¦ffng�^
Sj(x) =
Mj−1
6hj
(xj − x)3 + Mj
6hj
(x− xj−1)3
+
(
yj−1 −
Mj−1h2j
6
)
xj − x
hj
+
(
yj −
Mjh
2
j
6
)
x− xj−1
hj
,
Ù¥ x ∈ [xj−1, xj]§j = 1, 2, · · · , N + 1.
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½½½nnn5.7.1 � f ∈ C4[a, b]§S ´ f ff'u!: x0, x1, · · · , xN+1§¿
÷ve�>.^ffng�^µ
111aaa>>>...^^^:
S ′(a) = y′0, S
′(b) = y′N+1. (5.7.1)
111���aaa>>>...^^^:
S ′′(a) = y′′0 , S
′′(b) = y′′N+1. (5.7.2)
Kk
max
a≤x≤b
∣∣∣f (k)(x)− S(k)(x)∣∣∣ ≤ Ck max
a≤x≤b
∣∣∣f (4)(x)∣∣∣h4−k,
k = 0, 1, 2, (5.7.11)
Ù¥
h = max
1≤j≤N+1
(xj − xj−1), C0 = 5
384
, C1 =
1
24
, C2 =
3
8
.
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