Диссертация (1149808), страница 6
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Íà ðèñóíêå 26 ïðèáëèæåíèå òîé æå ôóíêöèè è ïîãðåøíîñòü ïðèáëèæåíèÿ ïðèh = 0.1.Íà ðèñóíêå 27 ïîêàçàí ñðåç ïðèáëèæåíèÿ ôóíêöèè1+ 0.08 sin(20x)(1 + 15x2 )(1 + 25x2 )ïðèx ∈ [0, 1], y = 0.43, h = 0.2(ñëåâà) è ïîãðåøíîñòü ýòîãî ïðèáëèæåíèÿ(ñïðàâà). Íà ðèñóíêå 28 ñðåç ïðèáëèæåíèÿ è ïîãðåøíîñòü ïðèáëèæåíèÿ ïðèh = 0.1.Ðèñ. 25: Ãðàôèêè ïðèáëèæåíèÿ ôóíêöèè[0, 1], y ∈ [0, 1], h = 0.21(1+15x2 )(1+25x2 )+ 0.08 sin(20x)ïðè(ñëåâà) è ïîãðåøíîñòü ýòîãî ïðèáëèæåíèÿ (ñïðàâà).53x∈Ðèñ.
26: Ãðàôèêè ïðèáëèæåíèÿ ôóíêöèè[0, 1], y ∈ [0, 1], h = 0.11(1+15x2 )(1+25x2 )+ 0.08 sin(20x)ïðèx∈(ñëåâà) è ïîãðåøíîñòü ýòîãî ïðèáëèæåíèÿ (ñïðàâà).Ðèñ. 27: Ãðàôèêè ñðåçà ïðèáëèæåíèÿ ôóíêöèèx ∈ [0, 1], y = 0.43, h = 0.21(1+15x2 )(1+25x2 ) +0.08 sin(20x) ïðè(ñëåâà) è ïîãðåøíîñòè ýòîãî ïðèáëèæåíèÿ (ñïðàâà).54Ðèñ.
28: Ãðàôèêè ñðåçà ïðèáëèæåíèÿ ôóíêöèèx ∈ [0, 1], y = 0.43, h = 0.11(1+15x2 )(1+25x2 ) +0.08 sin(20x) ïðè(ñëåâà) è ïîãðåøíîñòè ýòîãî ïðèáëèæåíèÿ (ñïðàâà).554.2Íåïðåðûâíîå ïðèáëèæåíèåÐàññìîòðèì ñïîñîá ìîäåëèðîâàíèÿ íåïðåðûâíîãî ïðèáëèæåíèÿ ñ ïîìîùüþèíòåãðî-äèôôåðåíöèàëüíûõ ñïëàéíîâ íà ëèíèè ïàðàëëåëüíîé îñèx ñ èñïîëüh1 = h/4 èh íà 4 ðàâíûå ÷àñòè, ïîëîæèìââåä¼ì äîïîëíèòåëüíûå óçëû Xj−1 , Xj , Xj+1 , Xj+2 . Áóäåì ìîäåëèðîâàòü˜ ∈ C[xj , xj+2 ] â âèäå:ðûâíîå ïðèáëèæåíèå ũçîâàíèåì èíòåðïîëÿöèè.
Ïîäåëèìíåïðå-j+2X ũ˜ =ũ(Xk )wk (x), x ∈[Xj , Xj+1 ] ⊂ [xj+1 − h1 , xx+1 + h1 ],k=j−1˜ũ = ũ,x ∈[xj , xj+1 − h1 ] ∪ [xj+1 + h1 , xj+2 ],wk (x) =Y x − Xk,Xj − Xkk = j − 1 . . . j + 2.k6=jÄîïîëíèòåëüíûå óçëû âûáèðàþòñÿ ïðè ñîáëþäåíèè ñëåäóþùèõ óñëîâèé:Xj−1 , Xj ∈ [xj + h1 , xj+1 − h1 ], Xj+1 , Xj+1 ∈ [xj+1 + h1 , xj+2 − h1 ].Xj−1 = xj + h1 , Xj = xj+1 − h1 , Xj+1 = xj+1 + h1 , Xj+2 = xj+2 − h1 .Âîçüì¼ìÒàáëèöà 11 ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèé ôóíêöèé áåç äîïîëíèòåëüíîé èíòåðïîëÿöèè è ïîãðåøíîñòü ïðèáëèæåíèé ñ äîïîëíèòåëüíîé èíòåðïîëíÿöèåé,h = 0.1, x ∈ [0, 1], y = 0.05.Òàáëèöà 12 ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèé ôóíêöèé áåç äîïîëíèòåëüíîé èíòåðïîëÿöèè è ïîãðåøíîñòü ïðèáëèæåíèé ñ äîïîëíèòåëüíîé èíòåðïîëíÿöèåé,h = 0.1, x ∈ [0, 1], y = 0.01.Òàáëèöà 11u(x, y)max |ũ − u|max |ũ˜ − u|x2 y 2sin2 (x) cos2 (y)x4 y 4sin(3x + 3y)sin(5x + 5y)ex eyex + ey0.2399 · 10−30.2472 · 10−30.6556 · 10−50.2195 · 10−20.9963 · 10−20.2235 · 10−30.1047 · 10−30.1697 · 10−30.2782 · 10−20.5997 · 10−50.7603 · 10−20.1282 · 10−10.6852 · 10−20.6502 · 10−256Òàáëèöà 12u(x, y)max |ũ − u|max |ũ˜ − u|x2 y 2sin2 (x) cos2 (y)x4 y 4sin(3x + 3y)sin(5x + 5y)ex eyex + ey0.7899 · 10−40.1073 · 10−30.7286 · 10−50.1435 · 10−20.6588 · 10−20.1462 · 10−30.6674 · 10−40.7864 · 10−40.2715 · 10−20.7282 · 10−50.8257 · 10−20.1639 · 10−10.6507 · 10−20.6553 · 10−2Èíòåðïîëèðóåì ôóíêöèþu(x, y),ũ(x, y) =çàäàííóþ íàj+1 Xk+1XΩjkâ ñëåäóþùåì âèäå:ui,l wi,l (x, y).i=j−1 l=k−1Èç ñîîòíîøåíèÿũ(x, y) = u(x, y),u(x, y) = 1, x, y, xy, x2 , y 2 ,ïîëó÷àåì ñèñòåìó óðàâíåíèé äëÿ îïðåäåëåíèÿ áàçèñíûõ ôóíêöèé, ãäå äåòåðìèíàíò ñèñòåìûD = −4h4j h4k .Ïîëîæèìhj = hk = h, x = xj + th, y = yk + t1 hèïîëó÷èì èòîãîâûå ôîðìóëû:wj,k (x, y) = 1 + tt1 − t2 − t21 ,wj,k (x, y) = −tt1 + (t2 + t21 + t − t1 )/2,wj,k (x, y) = −t1 (t − t1 ),wj,k (x, y) = t1 (2t − t1 + 1)/2,wj,k (x, y) = (t2 − t21 + t1 − t)/2,wj,k (x, y) = t1 (t1 − 1)/2. òàáëèöåmaxj,k |ũ(sj,k ) − u(sj,k )|,ñïëàéíîâ.
Çäåñü h = 0.1, sj,k =13 ïîêàçàíû ïîãðåøíîñòè ïðèáëèæåíèÿïîñòðîåííîãî ñ ïîìîùüþ èíòåðïîëÿöèîííûõ(xj + 0.05, yk + 0.05).57Òàáëèöà 13u(x, y)maxj,k |ũ(sj,k ) − u(sj,k )|x2 y 2sin2 (x)cos2 (y)x4 y 4sin(3x + 3y)sin(5x + 5y)0.731250 · 10−30.541150 · 10−30.605585 · 10−10.131990 · 10−10.591601 · 10−1Òåïåðü áóäåì ìîäåëèðîâàòü íåïðåðûâíîå ïðèáëèæåíèå ñ èíòåðïîëèðóþùèìè ñïëàéíàìè íà ëèíèè ïàðàëëåëüíî îñèxñ èñïîëüçîâàíèåì äîïîëíèòåëüíîéèíòåðïîëÿöèè.Ðåçóëüòàò ïðèáëèæåíèÿ ñ ïîìîùüþ èíòåðïîëèðóþùèõ ñïëàéíîâ, íåïðåðûâíîãî ïðèáëèæåíèÿ è ïîãðåøíîñòåé ïðèáëèæåíèé äëÿ ôóíêöèè1(1+25x2 )(1+25y 2 ) ïðèN = 10, x ∈ [0 .
. . 1], h = 0.1, y = 0.08ïîêàçàí íà ãðàôèêàõ29 30.Àíàëîãè÷íûå ãðàôèêè äëÿ1(1+(x−0.5)2 )(1+(y−0.5)2 )20, x ∈ [−2 . . . 2], h = 0.2, y = 0.948Ðèñ. 29: Ïðèáëèæåíèÿ äëÿ2)− sin(17xy+ cos(30x20502)ïðèN=èçîáðàæåíû íà ãðàôèêàõ 31 32.1(1+25x2 )(1+25y 2 ) . Èñõîäíîå ðàçðûâíîå ïðèáëèæåíèå(ñëåâà) è íåïðåðûâíîå ïðèáëèæåíèå ñ äîïîëíèòåëüíîé èíòåðïîëÿöèåé (ñïðàâà).Ó íåïðåðûâíîãî ïðèáëèæåíèÿ íåïðåðûâíîé ëèíèåé îáîçíà÷åíû ÷àñòè èñõîäíîãî ïðèáëèæåíèÿ, òî÷êàìè - ÷àñòè, äîñòðîåííûå ñ ïîìîùüþ èíòåðïîëÿöèè.58Ðèñ. 30: Ïîãðåøíîñòè ïðèáëèæåíèÿ äëÿ1(1+25x2 )(1+25y 2 ) . Ïîãðåøíîñòü èñõîäíî-ãî ðàçðûâíîãî ïðèáëèæåíèÿ (ñëåâà), ïîãðåøíîñòü íåïðåðûâíîãî ïðèáëèæåíèÿ(ñïðàâà).Ðèñ. 31: Ïðèáëèæåíèÿ äëÿ1(1+(x−0.5)2 )(1+(y−0.5)2 )−sin(17xy 2 )20+cos(30x2 ). Èñõîäíîå50ðàçðûâíîå ïðèáëèæåíèå (ñëåâà) è íåïðåðûâíîå ïðèáëèæåíèå ñ äîïîëíèòåëüíîé èíòåðïîëÿöèåé (ñïðàâà).
Ó íåïðåðûâíîãî ïðèáëèæåíèÿ íåïðåðûâíîé ëèíèåé îáîçíà÷åíû ÷àñòè èñõîäíîãî ïðèáëèæåíèÿ, òî÷êàìè - ÷àñòè, äîñòðîåííûå ñïîìîùüþ èíòåðïîëÿöèè.59Ðèñ. 32: Ïîãðåøíîñòè ïðèáëèæåíèÿ äëÿ1(1+(x−0.5)2 )(1+(y−0.5)2 )2)− sin(17xy+ cos(30x20502).Ïîãðåøíîñòü èñõîäíîãî ðàçðûâíîãî ïðèáëèæåíèÿ (ñëåâà), ïîãðåøíîñòü íåïðåðûâíîãî ïðèáëèæåíèÿ (ñïðàâà).605Ïðèáëèæåíèÿ ïîëèíîìèàëüíûìè è òðèãîíîìåòðè÷åñêèìè èíòåãðî-äèôôåðåíöèàëüíûìèñïëàéíàìè äâóõ ïåðåìåííûõn, m íàòóðàëüíûå ÷èñëà, à a, b, c, d äåéñòâèòåëüíûå ÷èñëà, h =(b − a)/n, h1 = (d − c)/m. Ðàññìîòðèì ñåòêó óçëîâ èíòåðïîëÿöèè xj = a + jh,j = 0, 1, . .
. , n, yk = c + kh1 , k = 0, 1, . . . , m.Ðàññìîòðèì ïðÿìîóãîëüíóþ îáëàñòü Ω, ãäåÏóñòüΩ = {(x, y) | a ≤ x ≤ b, c ≤ y ≤ d}.Ââåä¼ì ñåòü ëèíèé íàΩ,êîòîðûå äåëÿò îáëàñòüΩíà ïðÿìîóãîëüíèêèΩj,k :Ωj,k = {(x, y)|x ∈ [xj , xj+1 ], y ∈ [yk , yk+1 ]}.5.1Ïîëèíîìèàëüíûå ñïëàéíû îäíîé ïåðåìåííîéÂîçüì¼ì ôóíêöèþu(x)u ∈ C 5 ([a, b]).u(xj ), u0 (xj ), j = 0, 1, . . . , n,òàêóþ ÷òîÏðåäïîëîæèì, ÷òî èçâåñòíûèR xj+1xju(t)dt, j =0, . .
. , n − 1.Îáîçíà÷èìue(x)ïðèáëèæåíèå ôóíêöèèu(x)íà ïðîìåæóòêå[xj , xj+1 ] ⊂[a, b]:ue(x) = u(xj )ωj,0 (x) + u(xj+1 )ωj+1,0 (x)+u0 (xj )ωj,1 (x) + u0 (xj+1 )ωj+1,1 (x)+xj+1Z+ u(t)dt ωj<0> (x).(70)xjÁàçèñíûå ñïëàéíûωj,0 (x), ωj+1,0 (x), ωj,1 (x), ωj+1,1 (x), ωj<0> (x),ïîëó÷àåì èçñèñòåìû óðàâíåíèé:ue(x) ≡ u(x), u(x) = xi−1 , i = 1, 2, 3, 4, 5.Ïîëîæèì, ÷òîsuppωk,α = [xk−1 , xk+1 ], α = 0, 1, suppωk<0> = [xk , xk+1 ].61(71)Äëÿx = xj + th, t ∈ [0, 1]ïîëó÷àåì ñëåäóþùèå ôîðìóëû:ωj,0 (xj + th)= − 18 t2 + 32 t3 − 15 t4 + 1,(72)ωj+1,0 (xj + th)= − 12 t2 + 28 t3 − 15 t4 ,(73)ωj,1 (xj + th)= − (9/2) h t2 + 6 h t3 − (5/2) h t4 + t h,(74)ωj+1,1 (xj + th)=(3/2) h t2 − 4 h t3 + (5/2) h t4 ,(75)ωj<0> (xj + th)=(30 t2 − 60 t3 + 30 t4 )/h.(76)Íà ðèñóíêàõ 33, 34, 35 ïîêàçàíû ãðàôèêè áàçèñíûõ ôóíêöèé ïðèÐèñóíîê 35 (ñïðàâà) ïîêàçûâàåò îøèáêó ïðèáëèæåíèÿ ôóíêöèè Ðóíãå1/(1 + 25x2 )ñ ïîìîùüþ ïîëèíîìèàëüíûõ ñïëàéíîâ,Ðèñ.
33: Ãðàôèêè áàçèñíûõ ôóíêöèé62h = 0.1, x ∈ [−1, 1].ωj,0 (ñëåâà), ωj+1,0 (ñïðàâà)h = 1.u(x) =Ðèñ. 34: Ãðàôèêè áàçèñíûõ ôóíêöèé(ñïðàâà)Ðèñ. 35: Ãðàôèêè áàçèñíûõ ôóíêöèéωj,1ωj,1ïðèïðèh = 1 (ñëåâà), ωj+1,1h = 1 (ñëåâà),áëèæåíèÿ ôóíêöèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè ïðè(ñïðàâà)63ïðèh=1ïîãðåøíîñòü ïðè-h = 0.1, x ∈ [−1, 1]Èç 70, 71, 72-76 ïîëó÷àåì ñëåäóþùèå ôîðìóëû:(5t + 1)(1 − 3t)(t − 1)2 , t ∈ [0, 1],ωj,0 (xj + th) =(3t + 1)(1 − 5t)(1 + t)2 , t ∈ [−1, 0],0, t ∈/ [0, 1].− 1 th(5t − 2)(t − 1)2 , t ∈ [0, 1], 2ωj,1 (xj + th) =− 12 th(2 + 5t)(1 + t)2 , t ∈ [−1, 0],0, t ∈/ [0, 1]. 30t2 (t − 1)2 , t ∈ [0, 1],hωj<0> (xj + th) =0, t ∈/ [0, 1].ωk,0 , ωk,1 , ωk<0> ∈ C 1 (R1 ).Ðàññìîòðèì Ũ (x), x ∈ [a, b], òàêóþÏðè ýòîìÒåîðåìà 3.÷òîŨ (x) = u(x), x ∈ [xj , xj+1 ].Ïóñòü ôóíêöèÿ u(x) òàêàÿ, ÷òî u ∈ C 5 ([a, b]). Äëÿ ïðèáëèæåíèÿu(x) ∈ C 1 (R1 ), x ∈ [xj , xj+1 ], ïîñòðîåííîãî èç ôîðìóë (70),÷òî|ũ(x) − u(x)|[xj ,xj+1 ] ≤ K1 h5 ku(5) k[xj ,xj+1 ] ,(72)(76),âåðíî,(77)K1 = 0.0138,|ũ0 (x) − u0 (x)|[xj ,xj+1 ] ≤ K2 h4 ku(5) k[xj ,xj+1 ] ,(78)|Ũ (x) − u(x)|[a,b] ≤ K1 h5 ku(5) k[a,b] , K1 = 0.0138.(79)K2 = 0.0125,Äîêàçàòåëüñòâî.Íåðàâåíñòâî (77) ñëåäóåò èç òåîðåìû Òåéëîðà è íåðàâåíñòâ:|ωj,0 (x)| ≤ 1,|ωj+1,0 (x)| ≤ 1,|ωj,1 (x)| ≤ 0.06779h,|ωj+1,1 (x)| ≤ 0.06779h,|ωj<0> (x)| ≤ 1.875/h.Íåðàâåíñòâî (79) ñëåäóåò èç (77).
Íåðàâåíñòâî (78) ñëåäóåò èç òåîðåìû Òåéëîðà64è íåðàâåíñòâ:ω 0 j,0 (xj + th) = −12t(5t − 3)(t − 1)/h,ω 0 j+1,0 (xj + th) = −12t(5t2 − 7t + 2)/h,<0>ω0j(xj + th) = 60t(2t − 1)(t − 1)/h2 ,ω 0 j,1 (xj + th) = −(t − 1)(10t2 − 8t + 1),ω 0 j+1,1 (xj + th) = t(3 − 12t + 10t2 ).Ðàññìîòðèì íåðàâíîìåðíóþ ñåòü óçëîâ. Åñëèxj + thj ,ãäåhj = xj+1 − xj , t ∈ [0, 1].x ∈ [xj , xj+1 ],òîãäàxj+1 =Òåïåðü ìîæåì èñïîëüçîâàòü áàçèñíûåñïëàéíû â âèäå:ω 0 j,0 (xj + th) = −(t + 5t)(−1 + 3t)(t − 1)2 ,ω 0 j+1,0 (xj + th) = −t2 (−2 + 3t)(−6 + 5t),ω 0 j,1 (xj + th) = −(1/2)thj (5t − 2)(t − 1)2 ,ω 0 j+1,1 (xj + th) = (1/2)t2 hj (t − 1)(5t − 3),<0>ω0j(xj + th) = 30t2 (t − 1)2 /hj .Ðèñóíîê 35 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåx ∈ [−1, 1], xj+1 = xj + h + j, hj = 0.1, j = 0, 1, ..., n − 1, n = 20, çäåñü ||u −Ũ ||[−1,1] = ε, ε = 0.207 · 10−3 .
Íàøà öåëü - ñíèçèòü n è óìåíüøèòü ïîãðåøíîñòüïðèàïïðîêñèìàöèè.Ïðèáëèæåíèå ìîäåëèðóåòñÿ íà êàæäîì èíòåðâàëå[xj , xj+1 ]ïî îòäåëüíîñòè.Ñõåìà ñïëàéíîâîãî ïðèáëèæåíèÿ ïîçâîëÿåò êîíòðîëèðîâàòü ýôôåêò îò ðàñïîëîæåíèÿ óçëîâ íà ïîãðåøíîñòü ñïëàéíîâîé àïïðîêñèìàöèè. Òàêèì îáðàçîì, áóäåì ïðèìåíÿòü èçâåñòíóþ ôîðìóëó äëÿ ïîñòðîåíèÿ àäàïòèâíîé ñåòêè ñ øàãîìhj = xj+1 − xj ,ãäåhjïîëó÷àåì èç óñëîâèÿ:Zxj +hjIj =p1 + (u0 (x))2 dx = I0 ,(80)xjRx pI0 = x01 1 + (u0 (x))2 dx.Ïðèìåð 1à.
Ðèñóíîê 36 (ñëåâà) ïîêàçûâàåò îøèáêó ïðèáëèæåíèÿ ôóíêöèèÐóíãå, x ∈ [−1, 1] ïðè ïîìîùè ïîëèíîìèàëüíûõ ñïëàéíîâ, êîãäà hj ïîëó÷àåìèç (80), n = 15, h0 = 0.2, ãäå ||u − Ũ ||[−1,1] = ε, ε = 0.000018.ãäå65Ðèñóíîê 36 (ñïðàâà) ïîêàçûâàåò îøèáêó ïðèáëèæåíèÿ ôóíêöèè Ðóíãå, ãäåhjïîëó÷àåì èç (80),n = 15, h0 = 0.205,çäåñüε = 0.0000077.Ðèñ.
36: Ãðàôèêè ïîãðåøíîñòè ïðèáëèæåíèÿ ôóíêöèè Ðóíãå,(80),n = 15, h0 = 0.2 (ñëåâà), h0 = 0.205 (ñïðàâà)Ïðèìåð 1á.hjïîëó÷àåì èçÐèñóíîê 37 ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèèsin(exp(3x)), x ∈ [−1, 1], ãäå h0 = h1 = 1/50(80), j = 0, 1, ...48, h0 = 0.28 (ñïðàâà).(ñëåâà), è ïðèhj ,ïîëó÷àåìîìó èçsin(exp(3x)), h0 = hj =n = 48 (ñïðàâà)Ðèñ. 37: Ãðàôèêè ïîãðåøíîñòè ïðèáëèæåíèÿ ôóíêöèè1/50, n = 100 (ñëåâà), h0 = 0.28, hjïîëó÷àåì èç (80),Âàæíî àïïðîêñèìèðîâàòü ôóíöèþ ñ ïîìîùüþ ñïëàéíîâ õîðîøî, èñïîëüçóÿêàê ìîæíî ìåíüøåå êîëè÷åñòâî óçëîâ. Ðèñóíîê 35 (ñïðàâà) ïîêàçûâàåò ïîãðåø-u(x) = 1/(1 + 25x2 )h = 0.1, x ∈ [−1, 1].íîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåàëüíûõ ñïëàéíîâ,ñ ïîìîùüþ ïîëèíîìè-Ïðèìåð 2. Ðèñóíîê 38 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíê-h0 = 0.37, n = 10.
Çäåñü hj ïîëó= xj + hj , hj = h2 /2, j = 3, 4, 5, 6, ||u −öèè Ðóíãå ïîëèíîìèàëüíûìè ñïëàéíàìè ïðèj = 0, 1, 2, 7, 8, 9Ũ ||[−1,1] || = ε, ε = 0.000206.÷àåì èç (80),èxj+166Ðèñóíîê 38 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåïîëèíîìèàëüíûìè ñïëàéíàìè, ãäåxj+1 = xj + hj , hj = 0.2, n = 10.Çäåñüε=0.00248.Ðèñóíîê 39 (ñëåâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåh0 = 0.344, n = 9. Çäåñü hj ïîëó÷àåì èç (80),= xj + hj , hj = 0.73h2 , j = 3, 4, 5, ||u − Ũ ||[−1,1] || = ε, ε =ïîëèíîìèàëüíûìè ñïëàéíàìè ïðèj = 0, 1, 2, 6, 7, 80.0000139.èxj+1Ðèñóíîê 39 (ñïðàâà) ïîêàçûâàåò ïîãðåøíîñòü ïðèáëèæåíèÿ ôóíêöèè Ðóíãåïîëèíîìèàëüíûìè ñïëàéíàìè, ãäåxj+1 = xj + hj , hj = 2/9, n = 9.Çäåñüε=0.00222.Ðèñ.
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