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£®¬®¦® § ¯¨á âì ¢ ¡®«¥¥ ª®¬¯ ªâ®© ä®à¬¥, ¢¢¥¤ï ¢á¯®¬®£ ⥫ìë© ¬®£®ç«¥ !n(x) á⥯¥¨ n + 1:!n(x) = (x x0 )(x x1 ) : : :(x xi 1 )(x xi )(x xi+1 ) : : :(x xn ):ந§¢®¤ ï í⮣® ¬®£®ç«¥ ¢ â®çª¥ xi ¨¬¥¥â ¢¨¤!n0 (xi ) = (xi x0)(xi x1 ) : : :(xi xi 1)(xi xi+1 ) : : :(xi xn ):®£¤ 'i (x) = (x !xn)(x)i !n0 (xi)¨, ®ª®ç ⥫ì®,Ln (x) =nXfi (x !xn)(x):i !n0 (xi)i=0®£®ç«¥ £à ¦ ¨¬¥¥â ®á®¡¥® ¯à®á⮩ ¢¨¤, ª®£¤ ã§«ë ¨â¥à¯®«¨à®¢ ¨ï ïîâáï à ¢®®âáâ®ï騬¨, â.
¥.xi xi 1 = h const. ਠí⮬, ¥á«¨ ¢¢¥á⨠®¡®§ 票¥x x0 = t, â® ¯®«ã稬h17YYj 6=ij 6=i=Yj 6=ij 6=i'i (x) = Y(x xj )(xi xj )(th jh)(ih jh)=n) ( 1)n i = ( 1)n i Cni t(t 1) : : :(t n) := t(t 1)t : : :(tii!(n i)!t in!®£¤ niXLn (x) = Ln (x0 + th) = ( 1)n t(t 1) :n!: :(t n) ( 1)i Ct n fii :i=02.1.2. â¥à¯®«ïæ¨®ë© ¬®£®ç«¥ ¬®¦® à áᬠâਢ âì ¨ª ª à §®áâë© «®£ àï¤ ¥©«®à , ¯à¨ í⮬ «®£®¬ ¯®ïâ¨ï ¯à®¨§¢®¤®© ï¥âáï ¯®ï⨥ à §¤¥«¥®© à §®áâ¨(á â®ç®áâìî ¤® ç¨á«®¢®£® ª®íä䍿¨¥â ). áᬮâਬ ¬®¦¥á⢮ â®ç¥ª x0; x1; : : :; xn (¢á¥ â®çª¨ à §«¨çë), ¢ ª®â®àëå äãªæ¨ï f(x) ᮮ⢥âá⢥® ¯à¨¨¬ ¥â § 票ï f0 ; f1; : : :; fn.
§¤¥«¥ë¬¨ à §®áâﬨ ¯¥à¢®£®¯®à浪 §ë¢ îâáï ®â®è¥¨ïf(xi ; xi+1) = xfi+1 fxi ; i = 0; 1; : : :; n 1;i+1ià §¤¥«¥ë¬¨ à §®áâﬨ ¢â®à®£® ¯®à浪 | ®â®è¥¨ïf(xi ; xi+1; xi+2) = f(xi+1 ; xxi+2) xf(xi ; xi+1) ;i+2ii = 0; 1; : : :; n 2:¨, ¢®®¡é¥, à §¤¥«¥ë¬¨ à §®áâﬨ k-£® ¯®à浪 | ®â®è¥¨ïi ; : : :; xi+k 1) ;f(xi ; : : :; xi+k ) = f(xi+1 ; : : :; xi+xk ) f(xxi+k18ii = 0; 1; : : :; n k:¥¬¬ .¬¥¥â ¬¥áâ® à ¢¥á⢮f(xi ; : : :; xi+k) =i kX+j =iiY+kl6=jl=if(xj )(xj xl ):®ª § ⥫ìá⢮ ¯à®¢¥¤¥¬ ¯® ¨¤ãªæ¨¨.
ਠk = 1ã⢥ত¥¨¥ áà §ã á«¥¤ã¥â ¨§ ®¯à¥¤¥«¥¨ï à §¤¥«¥ëå à §®á⥩ ¯¥à¢®£® ¯®à浪 . ãáâì ã⢥ত¥¨¥ ¤®ª § ® ¤«ïk m. ®£¤ i ; : : :; xi+m ) =f(xi ; : : :; xi+m+1 ) = f(xi+1 ; : : :; xxi+m+1) f(xxi+m+1i=x1i+m+1xii+Xm+1j =i+1f(xj )i+Ym+1l6=jl=i+1(xj xl )iX+mj =if(xj )iY+ml6=jl=i(xj xl )!: ¯®«ã祮¬ ¢ëà ¦¥¨¨ f(xi ) ¨ f(xi+m+1 ) ¢áâà¥ç îâáï ¯®®¤®¬ã à §ã, ¯à¨â®¬ ᮮ⢥âáâ¢ãî騥 ª®íä䍿¨¥âë ¨¬¥îââà¥¡ã¥¬ë© ¢¨¤.
áâ «ìë¥ f(xj ) ¢å®¤ïâ ¢ ¢ëà ¦¥¨¥ ¤¢ ¦¤ë, ª®íä䍿¨¥â ¯à¨ f(xj ) à ¢¥11i+Ym+1iY+ml6=jl=i+1l6=jl=ixi(xj xl )(xj xl )=xi+m+1= (xj xi ) (xi+j m+1xi+m+1 ) = i+m+1 1;YY(xi+m+1 xi)(xj xl )(xj xl )l6=jl=içâ® ¨ âॡ®¢ «®áì ãáâ ®¢¨âì.l6=jl=i«¥¤á⢨¥ 1. §¤¥«¥ ï à §®áâì ï¥âáï «¨¥©ë¬®¯¥à â®à®¬ äãªæ¨¨ .f19«¥¤á⢨¥ 2. §¤¥«¥ ï à §®áâì ¥áâì ᨬ¬¥âà¨ç¥áª ï äãªæ¨ï ᢮¨å à£ã¬¥â®¢.¥à¥©¤¥¬ ª ¢ë¢®¤ã ä®à¬ã«ë ¨â¥à¯®«ï樮®£® ¬®£®ç«¥ ìîâ® . «ï í⮣® § ¯¨è¥¬ ¬®£®ç«¥ £à ¦ ¢¢¨¤¥nXLn (x) = L0(x) + (Lj (x) Lj 1(x));j =1£¤¥ Lj (x) | ¬®£®ç«¥ £à ¦ , ¯®áâà®¥ë© ¯® ã§« ¬x0; x1; : : :; xj , ¯à¨ç¥¬ L0 (x) = f(x0 ).
¬¥â¨¬, çâ® Lj (x)Lj 1(x) ¥áâì ¬®£®ç«¥ á⥯¥¨ j, ®¡à é î騩áï ¢ ã«ì¢ â®çª å x0 ; x1; : : :; xj 1, â® ¥áâìLj (x) Lj 1 (x) = Aj (x x0 )(x x1 ) : : :(x xj 1): ª ª ª Lj (xj ) = f(xj ), â®j ) Lj 1 (xj )Aj = (x x f(xx1) : : :(xj xj 1) :j0 )(xjâáî¤ , ¯®«ì§ãïáì ®¯à¥¤¥«¥¨¥¬ ¬®£®ç«¥ £à ¦ , ¯®«ã稬Aj = jY1Y01(xj xk )jXBCC =6 iBf(x)f(xi ) kYj@(xi xk ) Ai11(xj xk )=0k=0jX= Y f(xi )i=0 ª¨¬ ®¡à §®¬,l6=i(xi xl )=k6=i= f(x0 ; : : :; xj ):Ln (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 ; x1; : : :; xn):â¥à¯®«ïæ¨®ë© ¬®£®ç«¥, § ¯¨á ë© ¢ â ª®¬ ¢¨¤¥, §ë¢ ¥âáï ¨â¥à¯®«ïæ¨®ë¬ ¬®£®ç«¥®¬ ìîâ® .20⬥⨬, çâ® ¬®£®ç«¥ë £à ¦ ¨ ìîâ® ¯à¥¤áâ ¢«ïîâ ᮡ®© ¢á¥£® «¨èì à §«¨çë¥ ä®à¬ë § ¯¨á¨ ®¤®£® ¨â®£® ¦¥ ¨â¥à¯®«ï樮®£® ¬®£®ç«¥ , ¯à¨ í⮬ ¬®£®ç«¥ £à ¦ 楫¥á®®¡à §® ¨á¯®«ì§®¢ âì ¢ â¥å á«ãç ïå, ª®£¤ ¨â¥à¯®«¨àã¥âáï ¥áª®«ìª® äãªæ¨©, § ç¥¨ï ª®â®àëå § ¤ ë ¢ ®¤¨å ¨ â¥å ¦¥ ã§« å, ¬®£®ç«¥ ìîâ® | ¯à¨¨â¥à¯®«¨à®¢ ¨¨ ®¤®© ¨ ⮩ ¦¥ äãªæ¨¨ á ¯®á⥯¥® 㢥«¨ç¨¢ î騬áï ç¨á«®¬ ã§«®¢.2.1.3.
¬¥ äãªæ¨¨ f(x) ¨â¥à¯®«ïæ¨®ë¬ ¬®£®ç«¥®¬Ln (x) ¯à¨¢®¤¨â ª ¯®ï¢«¥¨î ¯®£à¥è®áâ¨rn (x) = f(x) Ln (x); §ë¢ ¥¬®© â ª¦¥ ®áâ â®çë¬ ç«¥®¬ ¨â¥à¯®«ï樮®©ä®à¬ã«ë. 祢¨¤®, ¢ ã§« å ¯®£à¥è®áâì à ¢ ã«î. «ï®æ¥ª¨ ®áâ â®ç®£® ç«¥ ¢ ¨ëå â®çª å à áᬮâਬ ¢á¯®¬®£ ⥫ìãî äãªæ¨î'(s) = f(s) Ln(s) K!n (s);¯à¨ç¥¬ ¯®áâ®ïãî K ¢ë¡¥à¥¬ ¨§ ãá«®¢¨ï '(x) = 0, £¤¥x | â®çª , ¢ ª®â®à®© ®æ¥¨¢ ¥âáï ¯®£à¥è®áâì. «ï í⮣®§ 䨪á¨à㥬 â®çªã x ¨ ¯®«®¦¨¬Ln (x) :K = f(x)! (x)n ª¨¬ ®¡à §®¬, äãªæ¨ï '(s) ®¡à é ¥âáï ¢ ã«ì ¯® ªà ©¥©¬¥à¥ ¢ n+2 â®çª å: x0; x1; : : :; xn; x.
ãáâì äãªæ¨ï f(x) ¨¬¥¥â n+1 ¥¯à¥àë¢ãî ¯à®¨§¢®¤ãî. ®£« ᮠ⥮६¥ ®««ï¯à®¨§¢®¤ ï '0 (s) ®¡à é ¥âáï ¢ ã«ì ¯® ªà ©¥© ¬¥à¥ ¢ n+ 1â®çª¥, '00(s) | ¢ n â®çª å ¨ â. ¤., ª®¥æ, '(n+1) (s) ®¡à é ¥âáï ¢ ã«ì ¯® ªà ©¥© ¬¥à¥ ¢ 1 â®çª¥ , ¯à¨ç¥¬ íâ â®çª ¯à¨ ¤«¥¦¨â ®â१ªã [y1 ; y2 ], £¤¥ y1 = min (x0; x1; : : :; xn; x),y2 = max(x0 ; x1; : : :; xn; x).
ª ª ª'(n+1) (s) = f (n+1) (s) K(n + 1)!;â® ¨§ ãá«®¢¨ï '(n+1) () = 0 ¯®«ã稬21(n+1)() ;K = f(n + 1)!®âªã¤ nn (x)f(x) Ln (x) = f (n ()!+ 1)! ; 2 [y1 ; y2 ]:ª®ç â¥«ì® ¨¬¥¥¬jf(x) Ln (x)j M(n(n++1)1)!() j!n(x)j;( +1)(2:2)£¤¥ Mn+1 = x2maxjf (n+1) (x)j. âáî¤ ¢¨¤®, çâ® ¯®£à¥è[y ; y ]®áâì ¢ â®çª¥ x ®â®á¨â¥«ì ¥¢¥«¨ª , ¥á«¨12min(x0 ; x1; : : :; xn) < x < max(x0 ; x1; : : :; xn): ¯à®â¨¢®¬ á«ãç ¥, §ë¢ ¥¬®¬ íªáâà ¯®«¨à®¢ ¨¥¬, १ª®¢®§à á⠥⠧ 票¥ j!n(x)j.ਠxi xi 1 = h const ¬®¦® £®¢®à¨âì ® ¯®à浪¥ á室¨¬®á⨠¯®£à¥è®á⨠ª ã«î ¯à¨ h, áâ६ï饬áï ª ã«î.â®â ¯®à冷ª à ¢¥ á⥯¥¨ ¯®«¨®¬ !n (x) ¢ ®áâ â®ç®¬ç«¥¥, â® ¥áâì n + 1, ¯®áª®«ìªã ¢ ¥£® ¡ã¤¥â ¢å®¤¨âì ¬®¦¨â¥«¥¬ hn+1. ¯®¬®éìî à §¤¥«¥ëå à §®á⥩ ¬®¦® ¯®«ãç¨âì ¤àã£ãî ®æ¥ªã ¯®£à¥è®á⨠¨â¥à¯®«¨à®¢ ¨ï, ¥ ¨á¯®«ì§ãîéãî ¯à¥¤¯®«®¦¥¨ï ® áãé¥á⢮¢ ¨¨ ¯à®¨§¢®¤ëå ¢ë᮪®£®¯®à浪 äãªæ¨¨ f(x):nXY (x xj )=fi (xij 6 i i xj )!nXf(x)iY+=rn (x) = f(x) Ln(x) = f(x)=0=Yn(x xi )nYf(x)=(x xi ) j =0 (xi x) j 6=i(xi xj )i=0= !n(x)f(x; x0 ; x1; : : :; xn):(2:3)à ¢¨¢ ï ®æ¥ª¨ (2.2) ¨ (2.3), ¢¨¤¨¬, çâ® ¯à¨ f 2 C n+1áãé¥áâ¢ã¥â â®çª 2 [y1; y2 ], ¤«ï ª®â®à®©i=022n+1f(x; x0; x1; : : :; xn) = f(n + ()1)! : ᫨ ¦¥ f 2= C n+1, â® ®æ¥ª (2.2) ¥ ¢ë¯®«ï¥âáï, ¨ ¯®à冷ªá室¨¬®á⨠¯®£à¥è®á⨠¢ ®¡é¥¬ á«ãç ¥ ¡ã¤¥â ¬¥ìè¥, 祬n + 1.2.1.4.
áᬮâਬ ®¡éãî ¯®áâ ®¢ªã § ¤ ç¨ ¨â¥à¯®«¨à®¢ ¨ï. ã§« å xk , k = 0; 1; : : :; m, á।¨ ª®â®àëå ¥â ᮢ¯ ¤ îé¨å, § ¤ ë § 票ï äãªæ¨¨ f(xk ) ¨ ¥¥ ¯à®¨§¢®¤ëåf (i) (xk ), i = 1; : : :; Nk 1, ¯à¨ í⮬ ç¨á«® Nk §ë¢ ¥âáïªà â®áâìî ã§« xk . 묨 á«®¢ ¬¨, ¢ ª ¦¤®¬ ã§«¥ ¨â¥à¯®«¨à®¢ ¨ï ¨§¢¥áâë § 票ïf(xk ); f 0 (xk ); : : :; f (Nk1)(xk );â® ¥áâì ¢á¥£® ¨§¢¥áâ® N0 +N1 +: : :+Nm ¢¥«¨ç¨. ॡã¥âáﯮáâநâì ¬®£®ç«¥ Hn(x) á⥯¥¨ n = N0 +N1 +: : :+Nm 1,㤮¢«¥â¢®àïî騩 ãá«®¢¨ï¬Hn(i)(xk ) = f (i) (xk ); k = 0; 1; : : :; m; i = 0; 1; : : :; Nk 1:(2:4)®£®ç«¥ Hn(x) §ë¢ ¥âáï ¨â¥à¯®«ïæ¨®ë¬ ¬®£®ç«¥®¬ ନâ .®ª ¦¥¬, çâ® ¨â¥à¯®«ïæ¨®ë© ¬®£®ç«¥ ନâ áãé¥áâ¢ã¥â ¨ ¥¤¨á⢥.
ãáâìHn (x) =nXi=0ai xi:®¤áâ ¢«ïï íâ® ¢ëà ¦¥¨¥ ¢ ä®à¬ã«ë (2.4), ¯®«ã稬 á¨á⥬㫨¥©ëå ãà ¢¥¨© ®â®á¨â¥«ì® ai , ¯à¨ í⮬ ç¨á«® ãà ¢¥¨© à ¢® ç¨á«ã ¥¨§¢¥áâëå ¨ á®áâ ¢«ï¥â N0 +N1 +: : :+Nm . áᬠâਢ ï ãá«®¢¨ï (2.4) á ã«¥¢®© ¯à ¢®© ç áâìî, ¯®«ã稬Hn(i) (xk ) = 0; k = 0; 1; : : :; m; i = 0; 1; : : :; Nk 1; (2:5)23®âªã¤ á«¥¤ã¥â, çâ® xk ï¥âáï ª®à¥¬ ªà â®á⨠Nk ¬®£®ç«¥ Hn(x). ª¨¬ ®¡à §®¬, ¬®£®ç«¥ Hn(x) á⥯¥¨ n ¨¬¥¥â á ãç¥â®¬ ªà â®á⨠¥ ¬¥¥¥ N0 +N1 +: : :+Nm = n+1 ª®àï.
â® ®§ ç ¥â, çâ® Hn(x) 0, â® ¥áâì ¢á¥ ª®íä䍿¨¥âëai à ¢ë ã«î. â ª, ®¤®à®¤ ï á¨á⥬ (2.5) ¨¬¥¥â ⮫쪮㫥¢®¥ à¥è¥¨¥, ¢á«¥¤á⢨¥ 祣® ¥®¤®à®¤ ï á¨á⥬ (2.4)®¤®§ ç® à §à¥è¨¬ ¯à¨ «î¡ëå ¯à ¢ëå ç áâïå, ¯à¨ç¥¬ ª®íä䍿¨¥âë ai «¨¥©® ¢ëà ¦ îâáï ç¥à¥§ § 票ï f (i) (xk ).¥¬ á ¬ë¬ ¬®£®ç«¥ ନ⠬®¦® ¯à¥¤áâ ¢¨âì ¢ ¢¨¤¥Hn (x) =m NXkX1k=0 i=0cki(x)f (i) (xk );£¤¥ cki(x) | ¬®£®ç«¥ë á⥯¥¨ n. ¢ë© ¢¨¤ íâ¨å ¬®£®ç«¥®¢ ¬ë ¥ ¯à¨¢®¤¨¬ ¢¢¨¤ã ¨å £à®¬®§¤ª®áâ¨. ¤ ç 2.2.
®ª § âì, çâ® ¬®£®ç«¥ ନâ , ¯®áâà®¥ë© ¯® § ç¥¨ï¬ äãªæ¨¨ i ¨ ¥¥ ¯à®¨§¢®¤®© i0 , ª®â®à륧 ¤ ë ¢ ã§« å i ,, ¨¬¥¥â ¢¨¤fx i = 0; 1; : : :; mH(x) =m Xi=0ffi + ( 2'0i (xi)fi + fi0 )(x xi) '2i (x);'i (x) | ¡ §¨áë¥ ¬®£®ç«¥ë £à ¦ .楪 ®áâ â®ç®£® ç«¥ ¨â¥à¯®«¨à®¢ ¨ï ¯à®¢®¤¨âáïâ ª ¦¥, ª ª ¤«ï ¬®£®ç«¥ £à ¦ . ¢¥¤¥¬ ¢á¯®¬®£ ⥫ìãî äãªæ¨î'(s) = f(s) Hn(s) K!(s);£¤¥mY£¤¥ !(s) = (s xi)Ni : ®áâ®ïãî K ¢ë¡¥à¥¬ ¨§ ãá«®¢¨ïi=0'(x) = 0, £¤¥ x | â®çª , ¢ ª®â®à®© ®æ¥¨¢ ¥âáï ¯®£à¥è®áâì:Hn(x) :K = f(x)!(x)ãªæ¨ï '(s) ¨¬¥¥â á ãç¥â®¬ ªà â®á⨠¥ ¬¥¥¥ N0 + N1 ++ : : : + Nm + 1 = n + 1 ª®àï. ®á«¥¤®¢ â¥«ì® ¯à¨¬¥ïï⥮६㠮««ï ª ¯à®¨§¢®¤ë¬ äãªæ¨¨ ' ¨ ãç¨âë¢ ï ªà â®áâì ª®à¥©, ¯®«ã稬, çâ® áãé¥áâ¢ã¥â â®çª â ª ï, çâ®'(n+1)() = 0, ®âªã¤ 24m(n+1)() Yf(x) Hn(x) = f(n + 1)!(x xi)Ni :(2:6)i=0x2.2.
2.2.1. ¨á«¥ë¬ ¤¨ää¥à¥æ¨à®¢ ¨¥¬ §ë¢ ¥âáï ¯à®æ¥áá ¢ëç¨á«¥¨ï ¯à¨¡«¨¦¥®£® § ç¥¨ï ¯à®¨§¢®¤®© f (k) (x) ¥ª®â®à®© äãªæ¨¨ f(x) ¯® ¥¥ § ç¥¨ï¬ f0 ; f1; : : :; fn, § ¤ 묢 â®çª å x0 ; x1; : : :; xn.à®á⥩訥 ä®à¬ã«ë ç¨á«¥®£® ¤¨ää¥à¥æ¨à®¢ ¨ï ¯®«ãç îâáï ¯ã⥬ ¤¨ää¥à¥æ¨à®¢ ¨ï ¨â¥à¯®«ï樮ëå ä®à¬ã«. ãáâì § ¤ á¥âª (â.¥. ¡®à ã§«®¢) x0 < x1 < : : : < xná è £ ¬¨ hi = xi xi 1, i = 1; 2; : : :; n. áᬮâਬ ¬®£®ç«¥ £à ¦ L1;i (x), ¯®áâà®¥ë© ¤«ï äãªæ¨¨ f(x) ¯® â®çª ¬xi 1 ¨ xi. 祢¨¤®,L01;i (x) = fi hfi 1 :iâ® ¢ëà ¦¥¨¥ ¬®¦® ¯à¨ïâì § ¯à¨¡«¨¦¥®¥ § 票¥f 0 (x) ¢ «î¡®© â®çª¥ x 2 [xi 1; xi ]. ¯à ªâ¨ç¥áª¨å § ¤ ç å ç áâ® ¢áâà¥ç ¥âáï à ¢®¬¥à ïá¥âª , hi = h; i = 1; 2; : : :; n.