Численные методы. Ионкин (2009) (формат хуже), страница 7

Íàçîâåì èíòåðïîëÿöèîííûì ïîëèíîìîì Íüþòîíà ôóíê-f (x)öèèïî óçëàì{Xi }n0ïîëèíîì ñòåïåíèn:n−1Nn (x) = f (x0 ) + (x − x0 )f (x0 , x1 ) + . . . + Πi=0f (x0 , x1 , . . . , xn )Ïîêàæåì, ÷òîNn (x)(2)èíòåðïîëÿöèîííûé ïîëèíîì:i−1Nn (xi ) = f (x0 ) + (xi − x0 )f (x0 , x1 ) + . . . + Πi=0f (x0 , x1 , . . . , xi ) + 0Ýòà ñóììà ïðåäñòàâëÿåò ñîáîé ðàçäåëåííóþ ðàçíîñòü ïîðÿäêà i, ðàâíóþêàê ðàçf (xi ).Çàìå÷àíèå. Ïîëó÷åííûé ïîëèíîì òîëò æå ïîëèíîì Ëàãðàíæà, òîëüêî çàïèñàííûé â äðóãîé ôîðìå.Ñîîòâåòñòâåííî, åãî ïîãðåøíîñòü òà æå, ÷òî è ó ïîëèíîìà Ëàãðàíæà.Îòëè÷èå ïîëèíîìà Íüþòîíà îò Ëàãðàíæà â òîì, ÷òî äëÿ óâåëè÷åíèÿòî÷íîñòèNn (x)íàäî òîëüêî äîáàâèòü èíôîðìàöèþ î íîâûõ óçëàõ è íåíàäî ïåðåñ÷èòûâàòü çíà÷åíèÿ äëÿ ñòàðûõ.Èíòåðïîëèðîâàíèå ñ êðàòíûìè óçëàìè.

Èíòåðïîëÿöèîííàÿ ôîðìóëàÝðìèòàŸ460Èíòåðïîëèðîâàíèå ñ êðàòíûìè óçëàìè. Èíòåðïîëÿöèîííàÿ ôîðìóëà Ýðìèòàx0 , x1 , . . . , xm , ïðè ýòîì ak ∈ N, k = 1, m a0 + . . . am = n + 1, ãäå n ñòåïåíü èíòåðïîëè-Ïóñòü èìååòñÿ m óçëîâ:êðàòíîñòü êàæäîãî óçëà (ðóþùåãî ïîëèíîìà ).Îïðåäåëåíèå. Íàçîâåì èíòåðïîëÿöèîííûì ïîëèíîìîì Ýðìèòà ïîëèíîì:Hn (x) =m aXk −1Xck,i (x)f (i) (xk ),(1)k=0 i=0ãäåck,i (x)- ïîëèíîìn-éñòåïåíè, êîýôôèöèåíòû êîòîðîãî íàõîäÿòñÿèç óñëîâèÿ:Hn(i) (xk ) = F (i) (xk ), k = 0, n, i = 0, ak − 1Ñóùåñòâîâàíèå è åäèíñòâåííîñòü äàííîãî ïîëèíîìà î÷åâèäíû, ïåðåéäåì ñðàçó ê ïîñòðîåíèþHn (x). îáùåì ñëó÷àå âûðàæåíèå äëÿ ïîëèíî-ìà Ýðìèòà äîñòàòî÷íî ãðîìîçäêî, ïîýòîìó îãðàíè÷èìñÿ ðàññìîòðåíèåìêîíêðåòíîé çàäà÷è:ÏîñòðîèòüH3 (x) = c0 (x)f (x0 ) + c1 (x)f (x1 ) + c2 (x)f (x2 ) + b(x)f 0 (x1 ).Çàïèøåì óñëîâèÿ, ïðè êîòîðûõ äàííûé ïîëèíîì áóäåò èíòåðïîëÿöèîííûì:c0 (x0 ) = 1, c1 (x0 ) = 0, c2 (x0 ) = 0, b(x0 ) = 0,c0 (x1 ) = 0, c1 (x1 ) = 1, c2 (x1 ) = 0, b(x1 ) = 0,c0 (x2 ) = 0, c1 (x2 ) = 0, c2 (x2 ) = 1, b(x2 ) = 0,c00 (x1 ) = 0, c01 (x1 ) = 0, c02 (x1 ) = 0, b0 (x1 ) = 1.Áóäåì èñêàòüc0 (x)â âèäåk(x − x1 )2 (x − x2 ), kâûáèðàåì èç óñëîâèÿc0 (x0 ) = 1:1 = k(x0 − x1 )2 (x0 − x2 ) =⇒ c0 (x) =Àíàëîãè÷íî ïîëó÷àåì âûðàæåíèå äëÿc2 (x) =(x − x1 )2 (x − x2 )(x0 − x1 )2 (x0 − x2 )c2 (x):(x − x1 )2 (x − x0 )(x2 − x1 )2 (x2 − x0 )Èíòåðïîëèðîâàíèå ñ êðàòíûìè óçëàìè.

Èíòåðïîëÿöèîííàÿ ôîðìóëàÝðìèòà61Òåïåðü âû÷èñëèìb(x):b1 (x) = k(x − x0 )(x − x1 )(x − x2 )b01 (x1 ) = k(x − x0 )(x − x2 ) = 1(x−x0 )(x−x2 )b1 (x) = (x−x(x − x1 ).22 ) (x1 −x2 )c1 (x) = (x − x0 )(x − x2 )(ax + b):Îòêóäà ïîëó÷àåìÄàëåå ïóñòüc1 (x1 ) = 1 = (x1 − x0 )(x1 − x2 )(ax1 + b)c01 (x1 ) = 0 = a(x1 − x0 )(x1 − x2 ) + (ax1 + b)(2x1 − x0 − x2 )Èç ýòèõ óðàâíåíèé ïîëó÷àåì:a=−b=(2x1 − x0 − x2 )(x1 − x0 )2 (x1 − x2 )2x1 (2x1 − x0 − x2 )1[1 +](x1 − x0 )(x1 − x2 )(x1 − x0 )(x1 − x2 )Ïîäñòàâëÿÿ íàéäåííûå âûðàæåíèÿ, èìååì:H3 (x) = f (x0 ) ·(1 +(x − x0 )(x − x2 )(x − x1 )2 (x − x2 )+ f (x1 ) ··2(x0 − x1 ) (x0 − x2 )(x1 − x0 )(x1 − x2 )(x1 − x)(2x1 − x0 − x2 )(x − x1 )2 (x − x0 )) + f (x2 ) ·+(x1 − x0 )(x1 − x2 )(x2 − x1 )2 (x2 − x0 )f 0 (x1 ) ·(x − x0 )(x − x2 )(x − x1 )(x − x2 )2 (x1 − x2 )Ïîãðåøíîñòü ïîëèíîìà ÝðìèòàψH3 (x) = f (x) − H3 (x)Ââåäåì ôóíêöèþ g(s) = f (s) + H3 (s) − kω(s), ãäå k êîíñòàíòà, ω(s)(x − x0 )(x − x1 )2 (x − x2 ).

Êîíñòàíòó k ïîëó÷àåì èç óñëîâèÿ g(x) = 0 :k=f (x) − H3 (x)ω(x)=Èñïîëüçîâàíèå ïîëèíîìà Ýðìèòà òðåòüåé ñòåïåíè äëÿ ïîëó÷åíèÿòî÷íîé îöåíêè ïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû ÑèìïñîíàÄàëåå äëÿ ïðèìåíåíèÿ òåîðåìû Ðîëëÿ òðåáóåì62∃f (4) (x) ∈ [a, b]. Ïðèìåíèâíåñêîëüêî ðàç òåîðåìó Ðîëëÿ ê g(s), ïîëó÷èì:∃ξ ∈ (a, b) : g (4) (ξ) = 0,îòêóäà è ïîëó÷åì îêîí÷àòåëüíóþ îöåíêóf (4)f (x) − H3 (x) =ω(x)4!Ïîãðåøíîñòü ïîëèíîìà Ýðìèòà n-îé ñòåïåíè ðàâíà:f (x) − Hn (x) =f (n+1) (ξ)(x − x0 )a0 (x − x1 )a1 . . . (x − xn )an(n + 1)!a0 + a1 + · · · + an = n + 1Çàäà÷à.

Ïóñòü çàäàíû óçëû x0 , x1 , x2 , x3 , ïðè÷åì x3 6= xi , i = 0, 1, 2, èçíà÷åíèÿ ôóíêöèè f â ýòèõ óçëàõ. Äîêàçàòü, ÷òîlim L3 (x) = H3 (x).x3 →x1Äîêàçàòåëüñòâî. Ðàññìîòðèì ïîëèíîì Ëàãðàíæà äëÿ ôóíêöèè f:ÏðèL3 (x) =(x − x1 )(x − x2 )(x − x3 )f (x0 ) + . . .(x0 − x1 )(x0 − x2 )(x0 − x3 )L3 (x) →(x − x1 )2 (x − x2 )f (x0 ) + · · · = H3 (x)(x0 − x1 )2 (x0 − x2 )x3 → x1 :Ÿ5Èñïîëüçîâàíèå ïîëèíîìà Ýðìèòà òðåòüåéñòåïåíè äëÿ ïîëó÷åíèÿ òî÷íîé îöåíêè ïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû ÑèìïñîíàÐàññìîòðèì êâàäðàòóðíóþ ôîðìóëó Ñèìïñîíà äëÿðåçêå[a, b]Rbañ ðàçáèåíèåì íà ÷àñòè÷íûå îòðåçêè [xi−1 , xi ],f (x)dxíà îò-xi − xi−1 = h,Èñïîëüçîâàíèå ïîëèíîìà Ýðìèòà òðåòüåé ñòåïåíè äëÿ ïîëó÷åíèÿòî÷íîé îöåíêè ïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû Ñèìïñîíàîáúåäèíåíèå êîòîðûõ äàåò[a, b].Zxif (x)dx =63Íà i-îì ÷àñòè÷íîì îòðåçêå èìååì:h(fi−1 + 4fi− 1 + fi )26xi−1fi = f (xi ), fi− 12Åñëèf (x) = a0 + a1 x + a2 x2 ,h= f xi −2òî êâàäðàòóðíàÿ ôîðìóëà Ñèìïñîíà òî÷-íà (ïî ïîñòðîåíèþ).

Ôîðìóëà Ñèìïñîíà áóäåò òî÷íà è äëÿ êóáè÷åñêèõ23ìíîãî÷ëåíîâ (f (x) = a0 + a1 x + a2 x + a3 x ). ×òîáû ïîêàçàòü ýòî, íàéäåìRxi 3x dx:xi−1Zxi1h1x3 dx = (x4i − x4i−1 ) = (x2i − x2i−1 )(x2i + x2i−1 ) = (xi + xi−1 )(x2i + x2i−1 )444xi−1RxiÒåïåðü çàïèøåì ôîðìóëó Ñèìïñîíà äëÿx3 dxè ïðåîáðàçóåì åå:xi−1 hh 3xi−1 + 4x3i− 1 + x3i =266(xi + xi−1 )333xi + xi−1 + 4=2x2 + 2xi xi−1 + x2i−1h(xi + xi−1 )(x2i − xi xi−1 + x2i−1 + i)62 23xi + 3x2i−1hh= (xi + xi−1 )= (xi + xi−1 )(x2i + x2i−1 )624=Òàêèì îáðàçîì, ìû ïîêàçàëè, ÷òî ôîðìóëà Ñèìïñîíà òî÷íà è äëÿ ìíîãî÷ëåíîâ òðåòüåé ñòåïåíè.Ïðèáëèçèì ïîäûíòåãðàëüíóþ ôóíêöèþf (x) ïîëèíîìîì Ýðìèòà H3 (x):H3 (xi−1 ) = fi−1H3 (xi− 1 ) = fi− 122H3 (xi ) = fi0H30 (xi− 1 ) = fi−122Èñïîëüçîâàíèå ïîëèíîìà Ýðìèòà òðåòüåé ñòåïåíè äëÿ ïîëó÷åíèÿòî÷íîé îöåíêè ïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû Ñèìïñîíàf (x) = H3 (x) + ψH3 (x)ZxiZxiZxif (x)dx =H3 (x)dx +ψH3 (x)dx =xi−1xi−1xi−1Zxih= (H3 (xi−1 ) + 4H3 (xi− 1 ) + H3 (xi )) +26ψH3 (x)dx =xi−1Zxih= (fi−1 + 4fi− 1 + fi ) +26ψH3 (x)dxxi−1Íàéäåì ïîãðåøíîñòü íà i-îì ÷àñòè÷íîì îòðåçêå:Zxiψi =hf (x)dx − (fi−1 + 4fi− 1 + fi ) =26xi−1ZxiψH3 (x)dxxi−1Ïîãðåøíîñòü äëÿ ïîëèíîìà Ýðìèòà èìååò âèä:ψH3 (x) =ÏóñòüM4 =supf (4) (ξ)(x − xi−1 )(x − xi− 1 )2 (x − xi )24!|f (4) (ξ)|,òîãäà ñïðàâåäëèâà îöåíêà:ξ∈[xi−1 ,xi ]M4|ψH3 (x)| ≤4!Zxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx = O(h5 )2xi−1Çàäà÷à.

Äîêàçàòü, ÷òîRxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx =2xi−1h5.120Äîêàçàòåëüñòâî. Ïðîâåäåì çàìåíó â ïîäûíòåãðàëüíîì âûðàæåíèè:x = xi−1 + th,0≤t≤1Òîãäà:dx = hdtx − xi−1 = th64Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèè65xi − x = h(1 − t)1(x − xi− 1 )2 = h2 (t − )222Òàêèì îáðàçîì, òðåáóåìûé èíòåãðàë ëåãêî âû÷èñëèòü:Zxi(x − xi−1 )(x − xi− 1 )2 (xi − x)dx =2xi−1= h5Z11t (1 − t) t −22Z1 th55 2453dt = hdt =2t − t − t +4412000Òåïåðü ìû ìîæåì îöåíèòü ïîãðåøíîñòü íà âñåì îòðåçêåΨ=nX[a, b]:ψi (h)i=1|Ψ| ≤Ó÷òåì, ÷òîM4 h5 n4! 120hn = b − a:M4 h4 (b − a)|Ψ| ≤=4!120 4h M4 (b − a)2180Çàìå÷àíèå. Åñëè ïîäûíòåãðàëüíóþ ôóíêöèþ çàìåíèòü ñîîòâåòñòâóþùèì ïîëèíîìîì Ëàãðàíæà, òî ïîãðåøíîñòü êâàäðàòóðíîé ôîðìóëûÑèìïñîíà óâåëè÷èòñÿ.Ÿ6Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèÎïðåäåëåíèå.

Ôóíêöèÿ f(x) íàçûâàåòñÿ èíòåãðèðóåìîé ñ êâàäðàòîìíà îòðåçêå [a, b], åñëèRbaf 2 (x)dx < ∞.Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèÐàññìîòðèì ëèíåéíîå ïðîñòðàíñòâîñ êâàäðàòîì íà îòðåçêåf(x) è g(x) (f, g[a, b].66H = L2 ôóíêöèé, èíòåãðèðóåìûõÂâåäåì ñêàëÿðíîå ïðîèçâåäåíèå ôóíêöèé∈ L2 ):Zb(f, g) =f (x)g(x)dxaÒåïåðü îïðåäåëèì íîðìó:Zb1||f || = (f, f ) = ( f 2 (x)dx) 212aÐàññìîòðèì ñîâîêóïíîñòü ôóíêöèé:φ0 (x), φ1 (x), . .

. , φn (x) ËÍÇ è èíòåãðèðóåìûå ñ êâàäðàòîì, φi∈ L2(1)Ðàññìîòðèì îáîáùåííûé ìíîãî÷ëåí:φ(x) =nXCk φk (x),Ck ÷èñëà(2)k=0Ñðåäè âñåõ îáîáùåííûõ ìíîãî÷ëåíîâ íàì íåîáõîäèìî íàéòè îáîáùåííûéìíîãî÷ëåíφ(x),òàêîé ÷òî:||f (x) − φ(x)|| = min ||f (x) − φ(x)|| =φ∈L2Zbf (x) −= min φ∈L2aÎáîáùåííûé ìíîãî÷ëåíφ(x)nX!2Ck φk (x) 12dxk=0íàçûâàåòñÿ íàèëó÷øèì ñðåäíåêâàäðàòè÷-íûì ïðèáëèæåíèåì ôóíêöèè f(x). Ïîêàæåì, ÷òî îíî ñóùåñòâóåò è åäèíñòâåííî.Óòâåðæäåíèå. Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ñóùåñòâóåò è åäèíñòâåííî.Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèèÄîêàçàòåëüñòâî.

Ðàññìîòðèì ñíà÷àëà ñëó÷àé67n = 0:φ0 (x) ∈ L2 , φ(x) = C0 φ0 (x)Ââåäåì ôóíêöèþF (C0 ):Zb2F (C0 ) = ||f − φ(x)|| =(f (x) − φ(x))2 dx =aZb=f 2 (x)dx − 2C0aZbf (x)φ0 (x)dx + C02aZbφ20 (x)dx =a= (f, f ) − 2C0 (f, φ0 ) + C02 (φ0 , φ0 )Íåîáõîäèìîå óñëîâèå ìèíèìóìà äëÿ ôóíêöèè F:dF=0dC0Ìèíèìóì ýòîé ôóíêöèè ïî ïåðåìåííîéëû:C0 =C0íàõîäèòñÿ â âåðøèíå ïàðàáî-(f, φ0 )(φ0 , φ0 )Òàêèì îáðàçîì:φ(x) = C0 φ0 (x)Ðàññìîòðèì ïðèìåð. ÏóñòüRbC0 =φ0 (x) = 1,f (x)dxaRbdxòîãäà:1=b−aZbf (x)dxaa1φ(x) =b−aZbf (x)dxaÏîëó÷èëè ñðåäíåå çíà÷åíèå ôóíêöèèf (x)íà[a, b].Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèè68Ðàññìîòðèì òåïåðü îáùèé ñëó÷àé. Ïóñòü çàäàíà ñèñòåìà ôóíêöèé (1).Ââåäåì ôóíêöèþF (C0 , C1 , .

. . , Cn ):F (C0 , C1 , . . . , Cn ) = ||f − φ(x)||2 =Zbf (x) −2f (x)dx − 2=nXZbCkk=0a= (f, f ) − 2f (x)φ0 (x)dx +nXCk (f, φk ) +nXk=0k=0Ck φk (x)nXCkk=0anX!2dx =k=0aZbnXCkZbCll=0nXφ20 (x)dx =aCl (φk , φl )l=0Çàïèøåì íåîáõîäèìîå óñëîâèå ìèíèìóìà äëÿ ôóíêöèè F:∂F (C0 , C1 , . . . , Cn )= 0,∂Ckk = 0, 1, . . . , nÒîãäà ïîëó÷èì ñèñòåìó óðàâíåíèé äëÿ íàõîæäåíèÿ êîýôôèöèåíòîânXCl (φk , φl ) = (f, φk ),k = 0, 1, . .

. , nCi(3)l=0Ìàòðèöåé ýòîé ñèñòåìû ÿâëÿåòñÿ ìàòðèöà Ãðàìà:(φ0 , φ0 ) (φ0 , φ1 ) · · · (φ0 , φn ) (φ1 , φ0 ) (φ1 , φ1 ) · · · (φ1 , φn ) G =  ........ ....(φ0 , φn ) (φ1 , φn ) · · · (φn , φn )Òàê êàê ñèñòåìà ôóíêöèé (1) ëèíåéíî íåçàâèñèìà, òî îïðåäåëèòåëü ÃðàìàdetG 6= 0,à çíà÷èò íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ñó-ùåñòâóåò è åäèíñòâåííî (ìîæíî îäíîçíà÷íî íàéòè êîýôôèöèåíòûÇàìå÷àíèå.

Åñëè ñèñòåìà ôóíêöèé(φk , φl ) = δkl ,òîCk = (f, φk )Ci ).(1) - îðòîíîðìèðîâàííàÿ, òî åñòü- êîýôôèöèåíòû Ôóðüå.Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèè69Çàìå÷àíèå. Ïóñòü çàäàíà ñèñòåìà ôóíêöèé:1, x, x2 . . . , xnÏóñòüρ(x) > 0- âåñîâàÿ ôóíêöèÿ.Zβρ(x)φk (x)φl (x)dx = 0αÂûáèðàÿÏóñòüρ(x), α, β{φi }n0 -ìîæíî ïîëó÷èòü îðòîãîíàëüíûå ìíîãî÷ëåíû.îðòîíîðìèðîâàííàÿ ñèñòåìà. Òîãäà íàèìåíüøåå îòêëî-íåíèå:bZf (x) −0≤aZ!2Ck φk (x)dx =k=0bf 2 (x)dx − 2anXnXCk (φk , f ) +Ck2 =k=0k=0(f, f ) −nXnXCk2 ≥ 0k=0Òàêèì îáðàçîì, ìû ïîëó÷èëè íåðàâåíñâòî Áåññåëÿ:nXCk2 ≤ ||f ||2k=0Åñëè ñèñòåìà{φk }n0 îðòîíîðìèðîâàííûé áàçèñ, è(φk , φl ) = δkl ,ïîëó÷åííîå íåðàâåíñòâî Áåññåëÿ ñòàíåò ðàâåíñòâîì Ïàðñåâàëÿ:2||f || =∞Xk=0Ck2òîÃëàâà III×èñëåííîå ðåøåíèå íåëèíåéíûõóðàâíåíèé è ñèñòåì íåëèíåéíûõóðàâíåíèéŸ1ÂâåäåíèåÏóñòü çàäàíà ôóíêöèÿf (x), x ∈ R, ïðè÷åì[a, b]ôóíêöèÿ f íåïðåûâíà.Áóäåì ðåøàòü óðàâíåíèå íà îòðåçêåf (x) = 0, x ∈ [a, b]Ïðîöåññ ðåøåíèÿ ðàçáèâàþò íà 2 ýòàïà:1.