Н.И. Ионкин - Электронные лекции (2008) (1135232), страница 6
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. . , xj+k .kïî óçëàìÒîãäà ìû ìîæåì îïðåäåëèòü ðàçäåë¼ííóþ ðàçíîñòük+1ïî-ðÿäêà:f (xj , . . . , xj+k+1 ) =Ââåäåì ïîëèíîìf (xj+1 , . . . , xj+k+1 ) − f (xj , . . . , xj+k ).xj+k+1 − xJβQωα,β (x) =(xi − xj ).Åãî ïðîèçâîäíàÿ â óçëàõxk :j=α0ωα,β(xk )=βY(xk − xj ).(2.6)k6=j,j=αÓòâåðæäåíèå 6. Ðàçäåë¼ííûå ðàçíîñòè ïîðÿäêà k ïðåäñòàâëÿþòñÿ â âè-äå:f (x0 , x1 , . . . , xk ) =Äîêàçàòåëüñòâî.kXf (xi )0 (x ) .ωi=0 0,k iÄîêàæåì ïî ìåòîäó ìàòåìàòè÷åñêîé èíäóêöèè.1. Áàçèñ èíäóêöèè.f (x0 , x1 ) =f (x1 ) − f (x0 )f (x0 )f (x1 ), f (x1 , x2 ) =+=x1 − x0x0 − x1x1 − x0=f (x0 )f (x1 )+ 0.0w0,1 (x0 ) w0,1 (x1 )(2.7)Ãëàâà 2. Èíòåðïîëèðîâàíèå è ïðèáëèæåíèå ôóíêöèé442. Ïðåäïîëîæåíèå èíäóêöèè. Ïðåäïîëîæèì, ÷òî:f (x0 , .
. . , xl ) =ll+1XXf (xk )f (xk ),f(x,...,x)=.0l+10 (x )0ω0,lω0,l+1(xk )kk=0k=13. Èíäóêöèîííûé ïåðåõîä.f (x1 , . . . , xl+1 ) − f (x0 , . . . , xl )=xl+1 − x0!l+1lXXf (xi )f (xi )−=0 (x )ω0(x ) i=0 ω0,lii=1 1,l+1 if (x0 , . . . , xl+1 ) ==1xl+1 − x01=xl+1 − x0lXf (xl+1 )f (x0 )− 0+0ω1,l+1 (xl+1 ) ω0,l (x0 ) i=1f (xi )f (xi )− 00ω1,l+1 (xi ) ω0,l (xi )!!.(2.8)Ðàññìîòðèì:f (xl+1 )f (xl+1 )= 0,0(xl+1 − x0 )ω1,l+1 (xl+1 )ω0,l+1 (xl+1 )f (x0 )−f (x0 )−f (x0 )= 0.0 (x ) = (x − x0 (x(xl+1 − x0 )ω0,l)ω)ω00l+10,l l+10,l+1 (x0 )Äàëåå:lXf (xi )i=1=lX11− 00ω1,l+1(xi ) ω0,l(xi )f (xi )i=1=lXi=1=f (xi )=xi − x0xi − xl+1−00 (x )(xi − x0 )ω1,l+1 (xi ) (xi − xl+1 )ω0,li!xi − x0xi − xl+1− 0=0ω0,l+1(xi ) ω0,l+1(xi )lX(xl+1 − x0 )f (xi )i=1!0ω0,l+1(xi )!=.Èòàê, ïîñëå âñåõ ïðåîáðàçîâàíèé ïîëó÷àåì èñêîìîå:f (x0 , .
. . , xl+1 ) =l+1Xk=0f (xk ).0ω0,l+1(xk )Èç îïðåäåëåíèÿ ðàçäåë¼ííîé ðàçíîñòè íåïîñðåäñòâåííî ñëåäóåò:f (x1 ) = (x1 − x0 )f (x0 , x1 ) + f (x0 )(2.9)2.4. Èíòåðïîëÿöèîííàÿ ôîðìóëà Íüþòîíà45è:f (x0 , x1 , x2 ) =f (x0 )f (x1 )f (x2 )++.(x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x2 − x0 )(x2 − x1 )Äàëåå:(x2 − x0 )(x2 − x1 )f (x0 , x1 , x2 ) =f (x0 )(x2 − x1 ) f (x1 )(x2 − x0 )++ f (x2 ).x1 − x0x0 − x1Ïðåîáðàçóåì:f (x1 )(x2 − x0 )f (x0 )(x2 − x0 ) + (x2 − x0 )(x1 − x0 )f (x0 , x1 )==x0 − x1x0 − x1f (x0 )(x2 − x0 )=− (x2 − x0 )f (x0 , x1 ).x0 − x1Ân-ìóçëå:f (xn ) = f (x0 ) + (xn − x0 )f (x0 , x1 ) + (xn − x0 )(xn − x1 )f (x0 , x1 , x2 ) + .
. . ++ (xn − x0 )(xn − x1 ) . . . (xn − xn−1 )f (x0 , . . . , xn ).(2.10)2.4Èíòåðïîëÿöèîííàÿ ôîðìóëà ÍüþòîíàÏóñòü íà îòðåçêå[a, b]çàäàíû íåêîòîðûå çíà÷åíèÿ:a ≤ x0 < x1 < x2 < . . . < xN ≤ b,ãäå{xi }N0- óçëû,(2.11)f (xi ) = fi , i = 0, N .Òîãäà, èíòåðïîëÿöèîííîé ôîðìóëîé Íüþòîíà áóäåì íàçûâàòü ñëåäóþùèé ïîëèíîì:Nn (x) = f (x0 ) + (x − x0 )f (x0 , x1 ) + . . . + (x − x0 )(x − x1 )f (x0 , x1 , x2 )++ .
. . + (x − x0 )(x − x1 ) . . . (x − xn−1 )f (x0 , . . . , xn ).Ïîêàæåì, ÷òî ýòî äåéñòâèòåëüíî èíòåðïîëÿöèîííûé ïîëèíîì:Nn (xi ) = f (x0 ) + (xi − x0 )f (x0 , x1 ) + . . . + (xi − x0 )(xi − x1 )f (x0 , . . . , xn ) == f (xi ), i = 0 . . . n, ïî ôîðìóëå (2.10).Çàìå÷àíèå.Ïîãðåøíîñòü äàííîãî ïîëèíîìà òàêàÿ æå, êàê è ó ïîëèíîìàËàãðàíæà, òàê êàê ýòî îäèí è òîò æå ïîëèíîìn-éñòåïåíè, çàïèñàííûé âðàçíûõ ôîðìàõ:ψNn (x) = f (x) − Nn (x),|f (x) − Nn (x)| ≤Çàìå÷àíèå.ÅñëèfnYMn+1|ω(x)|, ω(x) =(x − xi ).(n + 1)!i=0- îäíà è òà æå ôóíêöèÿ, à ìû òîëüêî äîáàâëÿåì íîâûåóçëû, òî óäîáíåå ïîëüçîâàòüñÿ ôîðìóëîé Íüþòîíà.
Åñëè óçëû ôèêñèðîâàííûå, à ôóíêöèÿ èçìåíÿåòñÿ, òî ñëåäóåò èñïîëüçîâàòü ïîëèíîì Ëàãðàíæà.Ãëàâà 2. Èíòåðïîëèðîâàíèå è ïðèáëèæåíèå ôóíêöèé462.5Èíòåðïîëèðîâàíèå ñ êðàòíûìè óçëàìè. Ïîëèíîìû ÝðìèòàÏóñòü çàäàím+1óçåë:{xi }m0(àíàëîãè÷íî ïðåäûäóùèì ïàðàãðàôàì) èïóñòü íàì èçâåñòíû çíà÷åíèÿ ôóíêöèè è åå ïðîèçâîäíûõ â óçëàõ:• x0 : f (x0 ), f 0 (x0 ), . . . , f (a0 −1)•.........• xm : f (xm ), f 0 (xm ), . . . , f (am −1)ak - íàòóðàëüíûå ÷èñëà ðàâíûå êðàòíîñòè óçëà xk , a0 + a1 + . . . +am = n + 1. Òîãäà èíòåðïîëÿöèîííûì ïîëèíîìîì Ýðìèòà áóäåò ÿâëÿòüñÿ(i)ïîëèíîì Hn , òàêîé ÷òî:Ïðè ýòîìHn(i) (xn ) = f (i) (xk ), k = 0 . .
. m, i = 0 . . . ak−1 ,Hn (x) =m aXk−1X(2.12)Ck,i (x)f (i) (xk ),(2.13)k=0 i=0Ck,i (x)ãäå- ìíîãî÷ëåí n-é ñòåïåíè. Ïîñòðîèì ïîëèíîì ÝðìèòàH3 (x)äëÿòðåõ óçëîâ :• x0 , f (x0 )• x1 , f (x1 ), f 0 (x1 )• x2 , f (x2 )Áóäåì èñêàòü ïîëèíîì 3-é ñòåïåíè â âèäåH3 (x) = C0 (x)f (x0 ) + C1 (x)f (x1 ) + C2 (x)f (x2 ) + b1 (x)f10 (x),C0 (x), C1 (x), C2 (x)ãäåèb1 (x) - ìíîãî÷ëåíû 3-é ñòåïåíè.H3 (x) áûë èíòåðïîëÿöèîííûì,Äëÿ òîãî, ÷òîáû ïîëèíîì(2.14)ïîòðåáóåì âû-ïîëíåíèÿ óñëîâèé:• H3 (x0 ) = f (x0 ), C0 (x0 ) = 1, C1 (x0 ) = 0, C2 (x0 ) = 0, b1 (x0 ) = 0• H3 (x1 ) = f (x1 ), C0 (x1 ) = 0, C1 (x1 ) = 1, C2 (x1 ) = 0, b1 (x1 ) = 0• H3 (x2 ) = f (x2 ), C0 (x2 ) = 0, C1 (x2 ) = 0, C2 (x2 ) = 1, b1 (x2 ) = 0• H30 (x1 ) = f 0 (x1 ), C00 (x1 ) = 0, C10 (x1 ) = 0, C20 (x1 ) = 0, b01 (x1 ) = 1C0 (x): C0 (x) = k(x − x1 )2 (x − x2 ), òàê êàê x1 - êîðåíü êðàòíîñòèx2 - êîðåíü êðàòíîñòè 1.
Òàê êàê C0 (x0 ) = 1 = k(x0 − x1 )2 (x0 − x2 ). ÂÍàéä¼ì2,àèòîãå ïîëó÷àåì:C0 (x) =(x − x1 )2 (x − x2 ).(x0 − x1 )2 (x0 − x2 )Àíàëîãè÷íî ïîëó÷àåì âûðàæåíèå äëÿC2 (x) =(2.15)C2 :(x − x1 )2 (x − x0 ).(x2 − x1 )2 (x2 − x0 )(2.16)2.5. Èíòåðïîëèðîâàíèå ñ êðàòíûìè óçëàìè. Ïîëèíîìû ÝðìèòàÒåïåðü âû÷èñëèì47b1 (x):b1 (x) = k(x − x0 )(x − x1 )(x − x2 ),b01 (x1 ) = k(x1 − x0 )(x1 − x2 ) = 1.⇓b1 (x) =(x − x0 )(x − x2 )(x − x1 ).(x1 − x0 )(x1 − x2 )×óòü áîëåå ñëîæíî âû÷èñëÿåòñÿ êîýôôèöèåíòC1 (x):C1 (x) = (x − x0 )(x − x2 )(ax + b),C1 (x1 ) = (x1 − x0 )(x1 − x2 )(ax1 + b)⇓ax1 + b =1,(x1 − x0 )(x1 − x2 )C10 (x) = a(x − x0 )(x − x2 ) + (ax + b)(2x − x0 − x2 ),C10 (x1 ) = a(x1 − x0 )(x1 − x2 ) + (ax1 + b)(2x1 − x0 − x2 ).⇓ax1 + b = −a(x1 − x0 )(x1 − x2 ).2x1 − x0 − x2⇓2x1 − x0 − x2a=−,(x1 − x0 )2 (x1 − x2 )2x1 (2x1 − x0 − x2 )11+.b=(x1 − x0 )(x1 − x2 )(x1 − x0 )(x1 − x2 )⇓(x − x0 )(x − x2 )(x1 − x)(2x1 − x0 − x2 )C1 (x) =1+.(x1 − x0 )(x1 − x2 )(x1 − x0 )(x1 − x2 )Òåïåðü, ïîäñòàâëÿÿ âû÷èñëåííûå êîýôôèöèåíòû â (2.14) ïîëó÷èì èñêîìûéïîëèíîì.Îöåíèì ïîãðåøíîñòü ïîëó÷åííîãî ïîëèíîìà:ψH3 (x) = f (x) − H3 (x).Ââåäåì ìíîãî÷ëåíû:ω(s) = (s − x0 )(s − x1 )2 (s − x2 ),g(s) = f (s) − H3 (s) − kω(s).Îòìåòèì, ÷òîgèìååò òðè íóëÿ â óçëàõ(x0 , x1 , x2 ).
Íàéäåì kèç ñëåäóþùèõñîîáðàæåíèé:g(x) = 0, x ∈ [x0 , x2 ] ⇒ f (x) − H3 (x) = kω(x) ⇒ k =ψH3 (x).ω(x)g(x0 ) = g(x1 ) = g(x2 ) = g(x) = 0 ⇒ g 0 (x) èìååò íå ìåíåå ÷åòûðåõ íóëåé:00òðè ïî òåîðåìå Ðîëëÿ è åùå îäèí èç-çà êðàòíîñòè êîðíÿ x1 ; g (x) - íå ìåíååÒ.ê.Ãëàâà 2. Èíòåðïîëèðîâàíèå è ïðèáëèæåíèå ôóíêöèé48g 000 (x)òðåõ íóëåé,- íå ìåíåå äâóõ,∃ξ ∈ [x0 , x1 ] : g (4) (ξ) = 0(ïî òåîðåìåÐîëëÿ). Òîãäà:g (4) (ξ) = f (4) (ξ) − 4!k = 0,⇓f (x) − H3 (x) =f (4) (ξ)ω(x),4!⇓|f (x) − H3 (x)| ≤ãäåM4 = sup |f (4) (x)|.M4|ω(x)|,4!Îáîáùèì ïîëó÷åííóþ îöåíêó äëÿ ïîëèíîìàn-éñòå-xïåíè:Mn+1|ω(x)|,(n + 1)!ω(x) = (x − x0 )a0 (x − x1 )a1 .
. . (x − xm )am ,|ψHn (x)| ≤a1 + a2 + . . . + am = n + 1.2.6Èñïîëüçîâàíèå H3 (x) äëÿ îöåíêè ïîãðåøíîñòè êâàäðàòóðíîé ôîðìóëû ÑèìïñîíàÂûïèøåì êâàäðàòóðíóþ ôîðìóëó Ñèìïñîíà äëÿ èíòåãðàëàRbf (x)dx è ðàç-aáèåíèÿ[a, b]íà ÷àñòè÷íûå îòðåçêèêîòîðûõ äàåòh = xi − xi−1 , i = 1, N ,îáúåäèíåíèå[a, b]:Zxif (x)dx =h(fi−1 + 4fi− 12 + fi ),6(2.17)xi−1h2 ). Ýòà ôîðìóëà òî÷íà íà ïàðàáîëàõ (ïîïîñòðîåíèþ, òàê êàê ïîãðåøíîñòü áóäåò ïðîñòî ðàâíà íóëþ). Äîêàæåì, ÷òîãäåfi = f (xi ), fi− 12 = f (xi −íà ñàìîì äåëå ôîðìóëà Ñèìïñîíà òî÷íà è äëÿ êóáè÷åñêèõ ìíîãî÷ëåíîâ:Zxix3 dx =111 4 4(xi −xi−1 ) = (x2i −x2i−1 )(x2i +xi−1 )2 = h(xi +xi−1 )(x2i +xi−1 )2 .444xi−1Òåïåðü ïðåîáðàçóåì ôîðìóëó Ñèìïñîíà:h 3h 3(xi + xi−1 )3333(x+ 4xi− 1 + xi ) = (xi + xi−1 + 4=26 i−162x2 + 2xi xi−1 + x2i−1h= (xi + xi−1 ) x2i − xi xi−1 + x2i−1 + i=62 23xi + 3x2i−11h= h(xi + xi−1 )(x2i + xi−1 )2 ,= (xi + xi−1 )624à ýòî â òî÷íîñòè ñîâïàäàåò ñ ïðåäûäóùèì ïîëó÷åííûì âûðàæåíèåì(2.18)⇒ êâàä-ðàòóðíàÿ ôîðìóëà Ñèìïñîíà âåðíà íà ìíîãî÷ëåíàõ òðåòüåé ñòåïåíè.
Ïðèáëèçèì ïîäûíòåãðàëüíóþ ôóíêöèþ ïîëèíîìîì Ýðìèòà. Ïóñòü çàäàíû òðèóçëàxi−1 , xi− 21 , xièH3 (x):2.7. Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèé49• H3 (xi−1 ) = fi−1• H3 (xi− 12 ) = fi− 21• H3 (xi ) = fi0• H30 (xi− 12 ) = fi−12Äàëåå:Zxihh1(H3 (xi−1 )+4H3 (xi− 21 )+H3 (xi )) = = (fi−1 +4f i − +fi ),662H3 (x)dx =xi−1Zxif (x)dx −Ψi (f ) =h(fi−1 + 4fi− 21 + fi ), i = 1, N ,6(2.19)xi−1 ïîãðåøíîñòü íà ÷àñòè÷íîì îòðåçêå. Òîãäà åñëèZxiZxif (x)dx =xi−1f (x) = H3 (x) + ψH3 (x), òîZxiH3 (x)dx +xi−1ψH3 (x)dx =(2.20)xi−1h= (fi−1 + 4fi− 12 + fi ) + Ψi (f ).6Ò.ê.ψH3 (x) =f (4) (ξ)4! (x− xi−1 )(x − xi− 21 )2 (x − xi ),Zxi|ψi (f )| ≤òî:M4(x − xi−1 )(x − xi− 21 )2 (x − xi )dx =4!xi−1= {x − xi−1 = th, 0 ≤ t ≤ 1, dx = dt} ==M44!Z1(2.21)1M4 h5h5 t(t − )2 (1 − t)dt =,24! 1200 îáîñíîâàíèå ïîñëåäíåãî ïåðåõîäà îñòàâèì â êà÷åñòâå óïðàæíåíèÿ ÷èòàòåëþ. Òàêèì îáðàçîì, ìû îöåíèëè ïîãðåøíîñòü íà îäíîì èç ðàçáèåíèé îòðåçêà.
Îöåíèì òåïåðü ïîãðåøíîñòü íà âñåì îòðåçêå|Ψn (f )| = |nXi=1ψi (f )| ≤M4 h5n=4! 120M4 h4= {nh = b − a} =(b − a) ≤4! 1202.7Ïóñòü 4hM4 (b − a).2180(2.22)Íàèëó÷øåå ñðåäíåêâàäðàòè÷íîå ïðèáëèæåíèå ôóíêöèéx ∈ [a, b].Ðàññìàòðèâàåì ïðîñòðàícòâî ôóíêöèé (Hèíòåãðèðóåìû ñ êâàäðàòîì, òî åñòü:∀f (x) ∈ L2 :Rba2= L2 ),êîòîðûåf (x)dx < ∞. ÑêàëÿðíîåÃëàâà 2.
Èíòåðïîëèðîâàíèå è ïðèáëèæåíèå ôóíêöèé50ïðîèçâåäåíèå çàäàäèì òàê:Zb(f (x), g(x)) =f (x)g(x)dx.aÒîãäà íîðìó ìîæíî îïðåäåëèòü ñëåäóþùèì îáðàçîì:||f ||L2Zb1= ||f || = (f, f ) = ( f 2 (x)dx) 2 .12aÏîñòàíîâêà çàäà÷è: ïóñòü åñòü ñîâîêóïíîñòü ôóíêöèéφ0 (x), φ1 (x), . . . , φn (x);Rbφ2i (x)dx < ∞, i = 0, n, òàêèå, ÷òî îíè ëèíåéíî íåçàâèñèìû (íîaíå îáÿçàòåëüíî îáðàçóþò áàçèñ). Ïî èõ ñîâîêóïíîñòè ïîñòðîèì îáîáùåííûénPCi φi (x), Ci - ÷èñëà. Ñðåäè âñåõ îáîáùåííûõ ìíîãî÷ëåìíîãî÷ëåí: φ(x) =i=0φi (x) ∈ H :φ(x)(îííîâ íóæíî íàéòè îáîáîùåííûé ìíîãî÷ëåííàçûâàåòñÿ íàèëó÷øèìñðåäíåêâàäðàòè÷íûì ïðèáëèæåíèåì):Zb||f (x) − φ(x)|| = min ||f (x) − φ(x)|| = min (f (x) −Ckφ(x)∈Hφ(x)1/2Ck φ(x))2 dx.k=0aÏîêàæåì ÷òînXñóùåñòâóåò è åäèíñòâåíåí (êîýôôèåíòûCkîïðåäåëÿþò-ñÿ îäíîçíà÷íî).Rbφ20 (x)dx < ∞, φ(x) = C0 φ0 (x).
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