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Âîñïîëüçóåìñÿ êâàäðàòóðíîé ôîðìóëîé äëÿ ïðèáëèæåííîãî âû÷èñëåíèÿ èíòåãðàëîâF <k> =R xk+1xku(t)dt, k =0, 1, . . . , n − 1,F<k>23 2 0h2 0h3 003= hu(xk )+ hu(xk+1 )+ h u (xk )− u (xk+1 )+ u (xk )+O(h5 )),55202060à òàêæå ôîðìóëàìè ÷èñëåííîãî äèôôåðåíöèðîâàíèÿ äëÿu00 (xk ) =k = 2, 3, . . . , n − 2,1(−2u(xk−2 ) + 32u(xk−1 ) − 60u(xk ) + 32u(xk+1 ) − 2u(xk+2 ))+24h2+ O(h6 ); (39)u0 (xk ) =äëÿ(38)1(u(xk−2 ) − 8u(xk−1 ) + 8u(xk+1 ) − u(xk+2 )) + O(h5 ),12h(40)k = 0, 1u00 (xk ) =1(70u(xk ) − 208u(xk+1 ) + 228u(xk+2 ) − 112u(xk+3 ) + 22u(xk+4 ))+24h2+ O(h5 ), (41)26u0 (xk ) =äëÿ1(−25u(xk ) + 48u(xk+1 ) − 36u(xk+2 ) + 16u(xk+3 ) − 3u(xk+4 ))+12h+ O(h5 ) (42)k = n − 1, nu00 (xk ) =1(70u(xk ) − 208u(xk−1 ) + 228u(xk−2 ) − 112u(xk−3 ) + 22u(xk−4 ))+24h2+ O(h5 ), (43)u0 (xk ) =1(25u(xk ) − 48u(xk−1 ) + 36u(xk−2 ) − 16u(xk−3 ) + 3u(xk−4 ))+12h+ O(h5 ). [101] îòìå÷åíî, ÷òîèñïîëüçîâàíèå(44)íà ïðàêòèêå âåðîÿòíîñòíîãî ïîäõîäàê îöåíêå ïîãðåøíîñòåé ðåçóëüòàòîâ èçìåðåíèé ïðåæäå âñåãî ïðåäïîëàãàåò çíàíèå àíàëèòè÷åñêîé ìîäåëè çàêîíà ðàñïðåäåëåíèÿ ðàññìàòðèâàåìîé ïîãðåøíîñòè.

Âñòðå÷àþùèåñÿ â ìåòðîëîãèè ðàñïðåäåëåíèÿ äîñòàòî÷íî ðàçíîîáðàçíû,ïðè÷åì èõ çíà÷èòåëüíóþ ÷àñòü ñîñòàâëÿþò ðàçëè÷íûå âèäû äâóõìîäàëüíûõðàñïðåäåëåíèé.Íà ðèñ. 5à ïðåäñòàâëåíû ãèñòîãðàììà, ïëîòíîñòü ðàñïðåäåëåíèÿ è àïïðîêñèìàöèÿ ïëîòíîñòè ðàñïðåäåëåíèÿf0 = √íà ïðîìåæóòêå[−2, 2],x21e− 2σ2 ,2πσãäåσ = 0.5,à íà ðèñ. 5 ïðåäñòàâëåíû ãèñòîãðàììà, ïëîòíîñòü ðàñ-f =i = 1, 2, σ1 = 0.5, σ2 = 0.8, α1 = −0.8,íà ïðîìåæóòêå [−2, 3].ïðåäåëåíèÿ è àïïðîêñèìàöèÿ ïëîòíîñòè äâóõìîäàëüíîãî ðàñïðåäåëåíèÿ2/2σi2,1(f1 + f2 )/2, ãäå fi = √2πσe−(x−αi )iα2 = 1, ïîëèíîìèàëüíûìè ñïëàéíàìè27Ðèñ. 5: Ãèñòîãðàììà, ïëîòíîñòü ðàñïðåäåëåíèÿ è àïïðîêñèìàöèÿ ïëîòíîñòè ðàñïðåäåëåíèÿ ñëåâà f0 ,ñïðàâà f.282Ñðåäíåêâàäðàòè÷åñêîå ïðèáëèæåíèåÑðåäíåêâàäðàòè÷åñêîå ïðèáëèæåíèå íà ïðîìåæóòêåíîøåíèåì:˜ũ(x)=nX[a, b]îïðåäåëÿåì ñîîò-cj Ωj (x),j=0Ωj (x)ãäåcjîïðåäåëÿþòñÿ èç óñëîâèÿ˜ 2 dx → min .[u − ũ](45) áàçèñíûå ôóíêöèè, à êîýôôèöèåíòûìèíèìàëüíîñòèE:ZbE=aÍåîáõîäèìûì óñëîâèåì ýêñòðåìóìà ÿâëÿåòñÿ âûïîëíåíèå ñîîòíîøåíèé∂E= 0, j = 0, 1, .

. . , n,∂cjMC = F ,êîòîðûå ïðèâîäÿò ê ñèñòåìå óðàâíåíèéãäåM = (mi,j ) ïîëîæè-òåëüíî îïðåäåëåííàÿ ìàòðèöà,ZbΩi (x)Ωj (x)dx,mi,j = (Ωi , Ωj ) =aZbF = (fi ) , fi = (u, Ωi ) =u(x)Ωi (x)dx.aÒàêèì îáðàçîì, ñèñòåìà óðàâíåíèé èìååò åäèíñòâåííîå ðåøåíèå, êîòîðîå äîñòàâëÿåò ìèíèìóì ôóíêöèîíàëó (45).2.1Ôîðìèðîâàíèå è õðàíåíèå ýëåìåíòîâ ìàòðèöû ÃðàìàÁóäåì ìîäåëèðîâàòü ñðåäíåêâàäðàòè÷åñêîå ïðèáëèæåíèå íà ïðîìåæóòêå[0, 1]c ïîìîùüþ ïîñòðîåííûõ íåïðåðûâíî äèôôåðåíöèðóåìûõ èíòåãðî-äèôôåðåíöèàëüíûõ áàçèñíûõ ñïëàéíîâ. ñëó÷àå ïîëèíîìèàëüíûõ èíòåãðî-äèôôåðåíöèàëüíûõ ñïëàéíîâ ýëåìåíòû29ìàòðèöû Ãðàìà èìåþò âèä:xj+1xj+1ZZ(ωi,k , ωj,s ) =ωi,k (x)ωj,s (x)dx, (ωi,k , ωj<1> ) =ωi,k (x)ωj<1> (x)dx,xj−1xjk, s = 0, 1, i, j = 0, . .

. , n.Ìàòðèöà Ãðàìà M ñèñòåìûMC = Fóðàâíåíèéìîæåò áûòü ïðåäñòàâëåíàâ áëî÷íîì âèäåM1 M2 M5M =  M3 M4 M6  ,M7 M8 M9ïðè÷åìM3 = (M2 )T , M7 = (M5 )T , M8 = (M6 )T ,ëåíòî÷íûå ìàòðèöû. Íîñèòåëè áàçèñíûõ ôóíêöèéòî÷íûõ èíòåðâàëîâ, à íîñèòåëüMâ ìàòðèöåωj<1>M1 , M2 , M5 , M4 , M6 , M9 ωj,0 , ωj,1 ñîñòîÿò èç äâóõ ñå-à èç îäíîãî ñåòî÷íîãî èíòåðâàëà, ïîýòîìóìàëî îòëè÷íûõ îò íóëÿ ýëåìåíòîâ.

Ðåçóëüòàòû âû÷èñëåíèé ïî-êàçûâàþò, ÷òî ðàçëè÷íûõ ìåæäó ñîáîé ýëåìåíòîâ ìàòðèöû 14. Òàêèì îáðàçîì,ïðè âû÷èñëåíèÿõ íåò íåîáõîäèìîñòè õðàíèòü â ïàìÿòè êîìïüþòåðà âñþ ìàòðèöó è äîñòàòî÷íî îãðàíè÷èòüñÿ õðàíåíèåì ÷åòûðíàäöàòè ýëåìåíòîâ:a1 =16hh8h, b1 =, c1 = − ,353570a2 =h2h2h2h2, a3 = − , c2 =, c3 = −,6060210210h3h3h33a4 =, b4 =, c4 =, a5 = − ,630315126014Ïðèâåäåì âèä ìàòðèö Mi ïðè n = 3:a1 c1M1 =  00c1b1c10a5 a5M5 =  000c1b1c10a5a5000c1a1a2 , M2 =  c3 0000a5a5c20c30a6 , M6 =  b6 00300c20c30a6b60a6 = −00c2a300a6b6hh, b6 = ,8484a4 , M4 =  c4 00a9 =c4b4c400c4b4c410.7h00c4a4a9 0 0 , M9 =  0 a9 0  .0 0 a9,F ñèñòåìû óðàâíåíèé ïðåäñòàâèìà â âèäå áëîêîâTF = F 1 , F 2 , F 3 , ýëåìåíòû fiα êîòîðûõ âû÷èñëÿþòñÿ ïî ôîðìóëàì:Ïðàâàÿ ÷àñòüZxi+1fiα =ωi,α (x)u(x)dx, α = 1, 2, i = 1, 2, .

. . , n − 1,xi−1f1α =Zx1fnα =ω0,α (x)u(x)dx, α = 1, 2,x0Zxnωn,α (x)u(x)dx, α = 1, 2,xn−1Zxi+1fi3 =ωi<1> (x)u(x)dx, i = 0, 1, . . . , n − 1.xi2.2Ðåçóëüòàòû âû÷èñëåíèéÂû÷èñëåíèÿ ïðîâîäèëèñü íàòîâ âåêòîðàF α , α = 1, 2, 3,C++ ñ ïîääåðæêîé OpenMP. Âû÷èñëåíèå ýëåìåí-ïðîèçâîäèòñÿ ïàðàëëåëüíî ñ äèíàìè÷åñêèì ðàñïðå-äåëåíèåì âû÷èñëåíèé ïî ïðîöåññîðàì.Äëÿ ðåøåíèÿ ñèñòåìû óðàâíåíèé ïðèìåíÿåòñÿ ïàðàëëåëüíûé âàðèàíòâñòðå÷íîé ïðîãîíêè.

Ñðåäíåêâàäðàòè÷åñêîå ïðèáëèæåíèå ôóíêöèèu(x)˜ũ(x)= ck,0 ωk,0 (x) + ck+1,0 ωk+1,0 (x) + ck,1 ωk,1 (x) + ck+1,1 ωk+1,1 (x) + ck,2 ωk<1> (x),è ñðåäíåêâàäðàòè÷åñêîå ïðèáëèæåíèå ïðîèçâîäíîé ôóíêöèèu(x)<1>00ũ˜0 (x) = ck,0 ω 0 k,0 (x) + ck+1,0 ωk+1,0(x) + ck,1 ωk,1(x) + ck+1,1 ω 0 k+1,1 (x) + ck,2 ω 0 k (x)ñòðîèì ïàðàëëåëüíî íà äâóõ ïðîöåññîðàõ ïðè(ïðåäïîëàãàåì, ÷òîx ∈ [x0 , xN −1 ]èx ∈ [xN , xn ]n = 2N − 1).Íà ðèñ. 6, 7 ïðåäñòàâëåíû ðåçóëüòàòû ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿy = sin(5x) + (1/5) cos(50x) + (1/20) sin(150x) è å¼ ïðîèçâîäíîé ïðåäëàãàåìûìè ñïëàéíàìè íà ïðîìåæóòêå [0,1] ïðè n = 6 (÷èñëî îáóñëîâëåííîñòè6−22 22ìàòðèöû cond(M ) ≈ 7 · 10 , det(M ) ≈ 0.2 · 10h ).ôóíêöèèÍà ðèñ. 8, 9 ïðåäñòàâëåíû ðåçóëüòàòû ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿýòîé æå ôóíêöèè ïðèÄëÿ ñðàâíåíèÿ íàn = 9 (cond(M ) ≈ 4 · 107 , det(M ) = 0.5 · 10−31 h31 ).ðèñ.

10, 11 ïðèâåäåì ãðàôèêè ôóíêöèè y(x) = sin(5x) +31(1/5) cos(50x) + (1/20) sin(150x)ïðè n = 6 è ïðè n = 9.è åå èíòåðïîëÿöèè ïðåäëàãàåìûìè ñïëàéíàìèÍà ðèñ. 12 ïðèâåäåì ãðàôèê ïîãðåøíîñòè èíòåðïîëÿöèè ýòîé æå ôóíêöèèïðèn = 50ýòîì ñëó÷àå(çàìåòèì, ÷òî òåîðåòè÷åñêàÿ îöåíêà ïîãðåøíîñòè èíòåðïîëÿöèè â|R| ≤ 0.26).Ðèñ. 6: Ãðàôèêè ôóíêöèèy(x) = sin(5x) + (1/5) cos(50x) + (1/20) sin(150x)(ïðåðûâèñòàÿ ëèíèÿ) è å¼ ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿ (íåïðåðûâíàÿëèíèÿ) ïðèn = 6.Ðèñ. 7: Ãðàôèêè ïðîèçâîäíîé ôóíêöèè(1/20) sin(150x)y(x) = sin(5x) + (1/5) cos(50x) +(ïðåðûâèñòàÿ ëèíèÿ) è å¼ ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæå-íèÿ (íåïðåðûâíàÿ ëèíèÿ) ïðèn = 6.Íà ðèñ.13 ïðèâåäåì ãðàôèêè ôóíêöèè Ðóíãå è åå ñðåäíåêâàäðàòè÷åñêîãîïðèáëèæåíèÿ ïðèn=6è ãðàôèê ïîãðåøíîñòè ýòîãî ïðèáëèæåíèÿ.32Ðèñ.

8: Ãðàôèêè ôóíêöèèy(x) = sin(5x) + (1/5) cos(50x) + (1/20) sin(150x)(ïðåðûâèñòàÿ ëèíèÿ) è å¼ ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿ (íåïðåðûâíàÿëèíèÿ) ïðèn = 9.Ðèñ. 9: Ãðàôèêè ïðîèçâîäíîé ôóíêöèè(1/20) sin(150x)y(x) = sin(5x) + (1/5) cos(50x) +(ïðåðûâèñòàÿ ëèíèÿ) è å¼ ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæå-íèÿ (íåïðåðûâíàÿ ëèíèÿ) ïðèn = 9.Íà ðèñ.14 15 ïðèâåäåíû ãðàôèêè áàçèñíûõ ôóíêöèéïðèh = 1.33ω00 (t), ω01 (t), ω0<1> (t)Ðèñ.

10: Ãðàôèêè èíòåðïîëÿöèè ôóíêöèè(1/20) sin(150x)ïðèn = 6.Ðèñ. 11: Ãðàôèêè èíòåðïîëÿöèè ôóíêöèè(1/20) sin(150x)ïðèy(x) = sin(5x) + (1/5) cos(50x) +n = 9.34y(x) = sin(5x) + (1/5) cos(50x) +Ðèñ.12:Ãðàôèêïîãðåøíîñòè(1/5) cos(50x) + (1/20) sin(150x)èíòåðïîëÿöèèïðèôóíêöèèy(x) = sin(5x) +n = 50.Ðèñ. 13: Ãðàôèêè ôóíêöèè Ðóíãå è å¼ ñðåäíåêâàäðàòè÷åñêîãî ïðèáëèæåíèÿ ïðèn=6(ñëåâà), ïîãðåøíîñòè ýòîãî ïðèáëèæåíèÿ (ñïðàâà).35Ðèñ. 14: Ãðàôèêèω00 (t)(ñëåâà),Ðèñ. 15: Ãðàôèê36ω01 (t)ω0<1> (t).(ñïðàâà)3Ïðèáëèæåíèÿ ëåâîñòîðîííèìè èíòåãðîäèôôåðåíöèàëüíûìè ñïëàéíàìè ïÿòîãî ïîðÿäêà àïïðîêñèìàöèè ïåðâîé âûñîòû3.1Ïîñòðîåíèå áàçèñíûõ ñïëàéíîâÐàíåå áûëè ñìîäåëèðîâàíû öåíòðàëüíûå ñïëàéíû, à çäåñü áóäåì ìîäåëèðîâàòü ëåâîñòîðîííèå.

Ðàññìîòðèì ïðîìåæóòîê÷èñëà,h=n íàòóðàëüíîå ÷èñëî,n ≥ 1.[a, b], ãäåa, b âåùåñòâåííûåÏîñòðîèì ñåòêó óçëîâ (1):{xj }ñ øàãîì(b−a)n .u(x) òàêîâà, ÷òî u ∈ C 5 [a, b]. Òàêæå èçâåñòíû çíà÷åíèÿR xku(xk ), u0 (xk ), k = 0, 1, . . . , n, xk−1u(t)dt, k = 1, . . .

, n. Áóäåì ñòðîèòü ïðèáëèæåíèå u(x), x ∈ [xk , xk+1 ], k = 1, . . . , n − 1, â âèäåÏóñòü ôóíêöèÿuek (x) = u(xk )ωk,0 (x) + u(xk+1 )ωk+1,0 (x)++ u0 (xk )ωk,1 (x) + u0 (xk+1 )ωk+1,1 (x) +Z!xku(t)dtxk−1ãäåωk,0 (x), ωk+1,0 (x), ωk,1 (x), ωk+1,1 (x), ωk<−1> (x)uek (x) = u(x)Ïîëîæèìäëÿωk<−1> (x),(46)îïðåäåëÿåì èç óñëîâèéu(x) = xi , i = 0, 1, 2, 3, 4.(47)supp ωk,α = [xk−1 , xk+1 ], α = 0, 1, supp ωk<−1> = [xk , xk+1 ].Óñëîâèÿ (47) âåäóò ê ñèñòåìå ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé îòíîñèòåëüíîωs,α (x), s = k, k + 1, α = 0, 1, ωk<−1> (x):ωk,0 (x) + ωk+1,0 (x) + (xk − xk−1 )ωk<−1> (x) = 1,(48)xk ωk,0 (x) + xk+1 ωk+1,0 (x) + ωk,1 (x) + ωk+1,1 (x)++ (x2k /2 − x2k−1 /2)ωk<−1> (x) = x,(49)x2k ωk,0 (x) + x2k+1 ωk+1,0 (x) + 2xk ωk,1 (x) + 2xk+1 ωk+1,1 (x)++ (x3k+1 /3 − x3k /3)ωk<−1> (x) = x2 ,37(50)x3k ωk,0 (x) + x3k+1 ωk+1,0 (x) + 3x2k ωk,1 (x) + 3x2k+1 ωk+1,1 (x)++ (x4k /4 − x4k−1 /4)ωk<−1> (x) = x4 ,(51)x4k ωk,0 (x) + x4k+1 ωk+1,0 (x) + 4x3k ωk,1 (x) + 4x3k+1 ωk+1,1 (x)++ (x5k /5 − x5k−1 /5)ωk<−1> (x) = x5 .Î÷åâèäíî, ÷òîËåììà 3.ωk,0 , ωk,1 , ωk<−1> ∈ C 1 (R1 ).Ïóñòü(52)kf k[a,b] = max |f (x)|.[a,b]Ïóñòü ôóíêöèÿ u ∈ C (5) [xk , xk+1 ], uk îïðåäåëÿþòñÿ èç(46).Ñïðàâåäëèâû ñëåäóþùèå óòâåðæäåíèÿ:|u(x) − uek (x)| ≤ h5 K0 ku(5) k[xk−1 ,xk+1 ] , x ∈ [xk , xk+1 ], K0 = const > 0,(53)|u0 (x) − ue0k (x)| ≤ h4 K1 ku(5) k[xk−1 ,xk+1 ] , x ∈ [xk , xk+1 ], K1 = const > 0.(54)Äîêàçàòåëüñòâî.Ïðèx ∈ [xk , xk+1 ]ïðåäñòàâëÿÿu(x), u(xk+1 )èu0 (xk+1 )ñ ïî-ìîùüþ ôîðìóëû Òåéëîðà, ó÷èòûâàÿ (48)(52), ïîëó÷èìuek (x) − u(x) = R,ãäåR=1 (5)1u (τ2 )(h5 ωk+1,0 (x) + u(5) (τ3 )(h4 ωk+1,1 (x)+5!4!Z xk11+u(5) (τ1 )(t − xk )5 dtωk<−1> (x) − u(5) (τ4 )(x − xk )55! xk−15!(55)τi ∈ [xk , xk+1 ], i = 1, 2, 3, 4.Îïðåäåëèòåëü ñèñòåìû ëèíåéíûõ àëãåáðàè÷åñêèõ óðàâíåíèé (48)(52) ðàâåí9− 3130 h .Ðåøàÿ (48)(52), ïîëó÷àåì ñëåäóþùèå ôîðìóëû áàçèñíûõ ñïëàéíîâ äëÿ[xk , xk+1 ], x = xk + th, t ∈ [0, 1]:38x∈Äëÿωk,0 (x) = (15t2 + 62t + 31)(t − 1)2 /31,(56)0ωk,0(x) = 12t(5t + 13)(t − 1)/(31h),(57)ωk+1,0 (x) = −t2 (45t2 − 28t − 48)/31,(58)0ωk+1,0(x) = −12t(15t + 8)(t − 1)/(31h),(59)ωk<−1> (x) = 30t2 (t − 1)2 /(31h),(60)ωk0<−1> (x) = 60t(2t − 1)(t − 1)/(31h2 ),(61)ωk,1 (x) = th(62 + 85t)(t − 1)2 /62,(62)0ωk,1(x) = (t − 1)(170t2 + 8t − 31)/31,(63)ωk+1,1 (x) = t2 h(35t + 27)(t − 1)/62,(64)0ωk+1,1(x) = t(−12t + 70t2 − 27)/31.(65)x ∈ [xk , xk+1 ]âåðíû íåðàâåíñòâà:|ωk,0 (x)| ≤ 1,|ωk+1,0 (x)| ≤ 1,|ωk,1 (x)| ≤ C1 h, C1 ≈ 0.223,|ωk+1,1 (x)| ≤ C2 h, C2 ≈ 0.122,|ωk<−1> (x)| ≤ 15/(248h).Òåïåðü, èñïîëüçóÿ òåîðåìó î ñðåäíåì, äëÿ(5)|R| = |euk (x) − u(x)| ≤ ku k[xk−1 ,xk+1 ]x ∈ [xk , xk+1 ]h5max |ωk+1,0 (x)|+5! x∈[xk ,xk+1 ]4+îòêóäàh4!ïîëó÷èì:6maxx∈[xk ,xk+1 ]|ωk+1,1 (x)| +|euk (x) − u(x)| ≤ 0.022 h5 ku(5) k[xk−1 ,xk+1 ] .39h6!5maxx∈[xk ,xk+1 ]|ωk<−1> (x)| +!h,5!Àíàëîãè÷íî,|R1 | = |eu0k (x) − u0 (x)| ≤ ku(5) k[xk−1 ,xk+1 ]4+h4!maxx∈[xk ,xk+1 ]h50max |ωk+1,0(x)|+5! x∈[xk ,xk+1 ]0|ωk+1,1(x)| +6h6!maxx∈[xk ,xk+1 ]<−1>|ω 0 k5(x)| +!h.5!Ó÷èòûâàÿ íåðàâåíñòâà0|ωk,0(x)| ≤ 1.50963/h,0|ωk+1,0(x)| ≤ 1.57955/h,<−1>|ω 0 k(x)| ≤ 0.18624/h2 ,0|ωk,1(x)| ≤ 1,0|ωk+1,1(x)| ≤ 1,x ∈ [xk , xk+1 ],ïîëó÷àåì îöåíêó|eu0k (x) − u0 (x)| ≤ 0.06342h4 ku(5) k[xk−1 ,xk+1 ] .3.2Ñðàâíåíèå ïðèáëèæåíèé ëåâîñòîðîííèìè è öåíòðàëüíûìè èíòåãðî-äèôôåðåíöèàëüíûìè ñïëàéíàìè ðàáîòå [69] è â ðàçäåëå 1.1 ìîäåëèðîâàëèñü ïðèáëèæåíèÿu(x), x ∈ [xk , xk+1 ],k = 0, 1, .

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