Skinchenna riznicya matematichnij viraz vidu f x b f x a sho shiroko vikoristovuyetsya v chislovih metodah v metodi skinchennih riznic dlya aproksimaciyi znachen funkciyi ta yiyi pohidnih Zmist 1 Prava liva ta centralna riznicya 2 Zv yazok z pohidnoyu 3 Riznici vishih poryadkiv 4 Div takozhPrava liva ta centralna riznicya RedaguvatiPrava riznicya viraz vidu D h f x f x h f x displaystyle Delta h f x f x h f x nbsp Liva riznicya viraz vidu h f x f x f x h displaystyle nabla h f x f x f x h nbsp Centralna riznicya viraz vidu d h f x f x 1 2 h f x 1 2 h displaystyle delta h f x f x tfrac 1 2 h f x tfrac 1 2 h nbsp Zv yazok z pohidnoyu RedaguvatiPohidna funkciyi f v tochci x viznachena yak granicya rozdilenoyi riznici f x lim h 0 f x h f x h lim h 0 D h f x h displaystyle f x lim h to 0 frac f x h f x h lim h to 0 frac Delta h f x h nbsp Otzhe prava riznicya podilena na h aproksimuye pohidnu yaksho h ye malim Pohibka aproksimaciyi otrimuyetsya z teoremi Tejlora Liva ta centralna riznici tezh aproksimuyut pohidnu D h f x h f x O h h 0 displaystyle frac Delta h f x h f x O h quad h to 0 nbsp h f x h f x O h displaystyle frac nabla h f x h f x O h nbsp d h f x h f x O h 2 displaystyle frac delta h f x h f x O h 2 nbsp Riznici vishih poryadkiv RedaguvatiAnalogichno do pohidnih vishih poryadkiv mozhna otrimati skinchenni riznici vishih poryadkiv Napriklad zastosuvavshi centralnu riznicyu v formulah f x h 2 displaystyle f x h 2 nbsp ta f x h 2 displaystyle f x h 2 nbsp dlya aproksimaciyi drugoyi pohidnoyi f displaystyle f nbsp v tochci x otrimayemo f x d h 2 f x h 2 f x h 2 f x f x h h 2 displaystyle f x approx frac delta h 2 f x h 2 frac f x h 2f x f x h h 2 nbsp V zagalnomu vipadku pravi livi ta centralni riznici nth togo poryadku virazhayutsya formulami D h n f x i 0 n 1 i n i f x n i h displaystyle Delta h n f x sum i 0 n 1 i binom n i f x n i h nbsp h n f x i 0 n 1 i n i f x i h displaystyle nabla h n f x sum i 0 n 1 i binom n i f x ih nbsp d h n f x i 0 n 1 i n i f x n 2 i h displaystyle delta h n f x sum i 0 n 1 i binom n i f left x left frac n 2 i right h right nbsp Dlya neparnih n displaystyle n nbsp koeficiyent pered h displaystyle h nbsp bude ne cilim Ce chasom ye problemoyu oskilki h displaystyle h nbsp ye intervalom diskretizaciyi Dlya virishennya problemi vikoristovuyut serednye vid d n f x h 2 displaystyle delta n f x h 2 nbsp ta d n f x h 2 displaystyle delta n f x h 2 nbsp Zv yazok skinchennih riznic vishih poryadkiv z pohidnimi vishih poryadkiv d n f d x n x displaystyle frac d n f dx n x nbsp D h n f x h n O h displaystyle frac Delta h n f x h n O h nbsp h n f x h n O h displaystyle frac nabla h n f x h n O h nbsp d h n f x h n O h 2 displaystyle frac delta h n f x h n O h 2 nbsp Skinchenni riznici vishih poryadkiv mozhut vikoristovuvatis dlya pokrashennya aproksimaciyi Napriklad D h f x 1 2 D h 2 f x h f x 2 h 4 f x h 3 f x 2 h displaystyle frac Delta h f x frac 1 2 Delta h 2 f x h frac f x 2h 4f x h 3f x 2h nbsp aproksimaye f x z tochnistyu do h2 Dovoditsya zapisom vishenavedenogo virazu cherez ryad Tejlora ta zvedennyam podibnih dodankiv Div takozh RedaguvatiKoeficiyent skinchennih riznicCya stattya ne mistit posilan na dzherela Vi mozhete dopomogti polipshiti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Material bez dzherel mozhe buti piddano sumnivu ta vilucheno gruden 2010 Otrimano z https uk wikipedia org w index php title Skinchenni riznici amp oldid 40173605
