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5[1].7__三次样条插值

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5[1].7__三次样条插值 Home Page Title Page JJ II J I Page 1 of 12 Go Back Full Screen Close Quit 5.7 nnnggg���^^^���ŁŁŁ 5.7.1 ÄÄÄflflflVVVggg 5.7.2 nnnggg���^^^���ŁŁŁffffff���EEE Home Page Title Page JJ II J I Page 2 of 12 Go Back Full Screen Close Quit 5.7.1 ÄÄÄ...

5[1].7__三次样条插值
Home Page Title Page JJ II J I Page 1 of 12 Go Back Full Screen Close Quit 5.7 nnnggg���^^^���ŁŁŁ 5.7.1 ÄÄÄflflflVVVggg 5.7.2 nnnggg���^^^���ŁŁŁffffff���EEE Home Page Title Page JJ II J I Page 2 of 12 Go Back Full Screen Close Quit 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)´˜ ‡ngõ‘ª§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 ¡���^^^(((:::" Home Page Title Page JJ II J I Page 3 of 12 Go Back Full Screen Close Quit >>>...^^^‡‡‡ 111˜˜˜aaa>>>...^^^‡‡‡: 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,,,>>>...^^^‡‡‡§ƒAffng�^¡ggg,,,���^^^" �¼ê y = f (x)´± b− a��±Ïff±Ï¼êž§Œæ^ e�>.^‡ 111nnnaaa>>>...^^^‡‡‡: S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4) dž§w,k S(a) = S(b). Home Page Title Page JJ II J I Page 4 of 12 Go Back Full Screen Close Quit 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. Home Page Title Page JJ II J I Page 5 of 12 Go Back Full Screen Close Quit ÷÷÷vvv111˜˜˜aaa>>>...^^^‡‡‡ffffffnnnggg���^^^ 111˜˜˜aaa>>>...^^^‡‡‡: 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. Home Page Title Page JJ II J I Page 6 of 12 Go Back Full Screen Close Quit ÷÷÷vvv111˜˜˜aaa>>>...^^^‡‡‡ffffffnnnggg���^^^ 111˜˜˜aaa>>>...^^^‡‡‡: 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)Œ±�¤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) Home Page Title Page JJ II J I Page 7 of 12 Go Back Full Screen Close Quit ÷÷÷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. Home Page Title Page JJ II J I Page 8 of 12 Go Back Full Screen Close Quit ÷÷÷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)Œ±�¤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) Home Page Title Page JJ II J I Page 9 of 12 Go Back Full Screen Close Quit ÷÷÷vvv111nnnaaa>>>...^^^‡‡‡ffffffnnnggg���^^^ 111nnnaaa>>>...^^^‡‡‡: S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4) dž¼ê 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. Home Page Title Page JJ II J I Page 10 of 12 Go Back Full Screen Close Quit ÷÷÷vvv111nnnaaa>>>...^^^‡‡‡ffffffnnnggg���^^^ 111nnnaaa>>>...^^^‡‡‡: S ′(a) = S ′(b), S ′′(a) = S ′′(b), (5.7.4) dž¼êy = 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)Œ±�¤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) Home Page Title Page JJ II J I Page 11 of 12 Go Back Full Screen Close Quit 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. Home Page Title Page JJ II J I Page 12 of 12 Go Back Full Screen Close Quit ½½½nnn5.7.1 � f ∈ C4[a, b]§S ´ f ff'u!: x0, x1, · · · , xN+1§¿ ÷ve�>.^‡ƒ˜ffng�^µ 111˜˜˜aaa>>>...^^^‡‡‡: 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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