Metod hord inodi metod linijnogo interpolyuvannya abo metod proporcijnih chastin iteracijnij chislovij metod znahodzhennya nablizhenih koreniv nelinijnogo algebrayichnogo rivnyannya V comu metodi nelinijna funkciya na vidilenomu intervali a b displaystyle a b zaminyuyetsya linijnoyu hordoyu pryamoyu sho z yednuye kinci nelinijnoyi funkciyi Pershi tri iteraciyi metodu hord Sinim namalovana funkciya f x chervonim hordi Zmist 1 Metod 2 Zbizhnist 3 Div takozh 4 PosilannyaMetodred Metod hord viznachayetsya nastupnim rekurentnim spivvidnoshennyam x n x n 1 f x n 1 x n 1 x n 2 f x n 1 f x n 2 displaystyle x n x n 1 f x n 1 frac x n 1 x n 2 f x n 1 f x n 2 nbsp Yak vidno z cogo vidnoshennya metod hord vimagaye dvoh pochatkovih tochok x 0 displaystyle x 0 nbsp i x 1 displaystyle x 1 nbsp yaki v ideali mayut buti vibrani v okoli rozv yazku Zbizhnistred Skazhimo x n x e n x n 1 x e n 1 displaystyle x n x e n x n 1 x e n 1 nbsp de x displaystyle x nbsp ye korenem f x 0 displaystyle f x 0 nbsp a e n e n 1 displaystyle e n e n 1 nbsp ce pohibki na n ta n 1 iteraciyah i x n x n 1 displaystyle x n x n 1 nbsp ce nablizhennya x displaystyle x nbsp na n ta n 1 iteraciyah Yaksho e n 1 K e n p displaystyle e n 1 Ke n p nbsp de K displaystyle K nbsp ce deyaka stala todi shvidkist zbizhnosti metoda yakij generuye x n displaystyle x n nbsp stanovit p displaystyle p nbsp Mi pokazhemo sho metod hord maye nadlinijnu zbizhnist Dovedennya Iteracijna shema dlya metoda hord taka x n 1 f x n x n 1 f x n 1 x n f x n f x n 1 displaystyle x n 1 frac f x n x n 1 f x n 1 x n f x n f x n 1 nbsp 1 x n 1 x n f x n x n x n 1 f x n f x n 1 displaystyle x n 1 x n frac f x n x n x n 1 f x n f x n 1 nbsp 2 Nehaj f x 0 displaystyle f x 0 nbsp i e n x x n displaystyle e n x x n nbsp todi pomilka na n iteraciyi v ocinyuvanni x displaystyle x nbsp stanovit x n 1 e n 1 x x n e n x x n 1 e n 1 x displaystyle begin matrix x n 1 e n 1 x x n e n x x n 1 e n 1 x end matrix nbsp 3 Vikoristovuyuchi 3 i 2 mi mayemo e n 1 e n 1 f x n f x n 1 e n f x n f x n 1 displaystyle e n 1 frac e n 1 f x n f x n 1 e n f x n f x n 1 nbsp 4 Po teoremi Lagranzha 3 n i n t x n x displaystyle exists xi n in int x n x nbsp take sho f 3 n f x n f x x n x displaystyle f xi n frac f x n f x x n x nbsp f x 0 x n x e n displaystyle because f x 0 x n x e n nbsp Mi mayemo f 3 n f x n e n displaystyle f xi n frac f x n e n nbsp todi f x n e n f 3 n displaystyle f x n e n f xi n nbsp 5 Analogichno f x n 1 e n 1 f 3 n 1 displaystyle f x n 1 e n 1 f xi n 1 nbsp 6 Pidstavlyayuchi 5 i 6 u 4 mi otrimuyemo e n 1 e n e n 1 f 3 n f 3 n 1 f x n f x n 1 displaystyle e n 1 e n e n 1 frac f xi n f xi n 1 f x n f x n 1 nbsp tobto e n 1 e n e n 1 displaystyle e n 1 propto e n e n 1 nbsp 7 Za viznachennyam shvidkosti zbizhnosti poryadku p displaystyle p nbsp e n e n 1 p e n 1 e n p displaystyle begin matrix e n propto e n 1 p e n 1 propto e n p end matrix nbsp 8 Z 7 i 8 viplivaye e n p e n 1 p e n 1 displaystyle e n p propto e n 1 p e n 1 nbsp e n e n 1 p 1 p displaystyle e n propto e n 1 p 1 p nbsp 9 Z 8 i 9 mayemo p p 1 p displaystyle p p 1 p nbsp todi p 2 p 1 0 displaystyle p 2 p 1 0 nbsp otzhe p 1 5 2 displaystyle p frac 1 pm sqrt 5 2 nbsp Tobto p gt 0 p 1 618 displaystyle p gt 0 p 1 618 nbsp i znachit e n 1 e n 1 618 displaystyle e n 1 propto e n 1 618 nbsp Otzhe zbizhnist nadlinijna Div takozhred Metod NyutonaPosilannyared Weisstein Eric W Metod hord angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Metod hord amp oldid 40334681