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t k = cos¨ ¸, k = 0,1,..., n − 1
© 2n ¹
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2 n −1
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18. (VHPSLR (VHPSLR
f ( x) = ln( x), [a, b] = [0.4,0.8] ( x − x0 )( x − x1 )( x − x2 )( x − x3 ) § 6 ·
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6 6
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), m-
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Q P LQFRJQLWH
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i =0 ° xi − xi +1 f ''( i )
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°
altrove f ∈ C ([a, b])
2
2
¯ x, i ∈ [ xi , xi +1 ]
­ x − x1 ­ x − xm
° , x ∈ [ x0 , x1 ] ° , x ∈ [ xm , xm +1 ] M
l0 ( x) = ® x0 − x1 lm +1 ( x) = ® xm +1 − xm f ( x) − si ( x) ≤ max ( x − xi )( x − xi +1 )
°0, °0, 2 x∈[ xi , xi+1 ]
¯ altrove ¯ altrove f ' ' ( x) ≤ M
M ( xi +1 − xi ) 2
=
2 4
M 2
h = max ( xi +1 − xi ) max f ( x) − s ( x) ≤ h
i = 0 ,..., m x∈[ a ,b ] 8
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25. RPH VL ULFDYD OD VSOLQH FXELFD GL RPH VL ULFDYD OD VSOLQH FXELFD GL
LQWHUSROD]LRQH LQWHUSROD]LRQH
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si ( x) = i + i ( x − xi ) + i ( x − xi ) + i ( x − xi )
2 3
° °
®si ' ( xi +1 ) = si +1 ' ( xi +1 ) ®si ' ( xi +1 ) = zi +1
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si ' ( x) = i + 2 i ( x − xi ) + 3 i ( x − xi ) 2 3RQLDPR KL
P [L [L
­ i + hi i + hi2 i + hi3 i = yi +1 ­ ª § y − y ·º 1
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® ¨ h ¸ h
° ¬ © ¹¼ i
° i + 2hi i + 3hi2 i = zi +1
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i
si ( xi ) = = yi ° ª § yi +1 − yi · º 1
i
+ 3 i hi = i +1 ° i = «3¨ h
¨ ¸ − ( zi +1 + 2 zi )»
¸
i
¯ ¬ © i ¹ ¼ hi
si ' ( xi ) = = zi §y −y · § y − yi +1 ·
i hi +1 zi + 2(hi + hi +1 ) zi +1 + hi zi + 2 = 3hi +1 ¨ i +1 i ¸ + 3hi ¨ i + 2
¨ h ¸ ¨ h ¸
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§y −y · § y − yi +1 · §y −y · § y − yi +1 ·
hi +1 zi + 2(hi + hi +1 ) zi +1 + hi zi + 2 = 3hi +1 ¨ i +1 i ¸ + 3hi ¨ i + 2
¨ h ¸ ¨ h ¸
¸ hi +1 zi + 2(hi + hi +1 ) zi +1 + hi zi + 2 = 3hi +1 ¨ i +1 i ¸ + 3hi ¨ i + 2
¨ h ¸ ¨ h ¸
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§ y1 − y0 ·
·§ z0 · ¨ ¸
3
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¨ ¸¨ ¸=¨ ¸ ¨ ¸¨ ¸ ¨ ¸
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¨ hm − 2 ¸¨ z m −1 ¸ ¨ bm −1 ¸ ¨ ¸¨ z ¸ ¨ ym +1 − ym ¸
¨ ¸¨ z ¸ ¨ − h z + b ¸
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hi +1 zi + 2(hi + hi +1 ) zi +1 + hi zi + 2 = 3hi +1 ¨ i +1 i ¸ + 3hi ¨ i + 2
¨ h ¸ ¨ h ¸
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y0 = ym +1 , s ' ( x0 ) = s' ( xm +1 ) = z0 = zm +1
s' ' ( x0 ) = s ' ' ( xm +1 ) → 0 = m + 3 m hm
§ §y −y · § y − y ··
§ 2(h0 + hm ) hm h0 ·§ z0 · ¨ 3h0 ¨ m +1 m ¸ + 3hm ¨ 1 0 ¸ ¸
¨ ¸¨ ¸ ¨ ¨ ¸ ¨ h ¸¸
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2 2 2
a a a a
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b
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