Me tod Shtu rma vikoristovuyetsya dlya viokremlennya dijsnih koreniv mnogochleniv tobto znahodzhennya intervaliv yaki mistyat rivno po odnomu korenyu Nadali otrimanu informaciyu pro rozmishennya koreniv mozhna vikoristati dlya yih znahodzhennya chiselnimi metodami Metod ryad i teorema nazvani na chest francuzkogo matematika Zhaka Sharlya Fransua Shturma Zmist 1 Osnovni viznachennya 2 Teorema Shturma 3 Pobudova ryadu Shturma dlya mnogochlena 4 DzherelaOsnovni viznachennya red Poslidovnist neperervnih funkcij f 0 f 1 f n displaystyle f 0 f 1 ldots f n nbsp nazivayetsya ryadom Shturma na vidrizku a b displaystyle a b nbsp yaksho vikonuyutsya taki umovi f n displaystyle f n nbsp ne maye koreniv na a b displaystyle a b nbsp Yaksho pri deyakih c a b 0 lt i lt n displaystyle c in a b 0 lt i lt n nbsp vikonuyetsya rivnist f i c 0 displaystyle f i c 0 nbsp to f i 1 c f i 1 c lt 0 displaystyle f i 1 c f i 1 c lt 0 nbsp Yaksho f 0 c 0 displaystyle f 0 c 0 nbsp pri deyakomu c a b displaystyle c in a b nbsp to pri perehodi cherez c displaystyle c nbsp funkciya f 0 x f 1 x displaystyle f 0 x f 1 x nbsp zminyuye znak z na Yaksho v tochci a displaystyle alpha nbsp vsi f i a i 0 n displaystyle f i alpha i 0 ldots n nbsp ne rivni nulyu to mozhna viznachiti kilkist znakozmin w a displaystyle omega alpha nbsp yak kilkist indeksiv 0 i lt n displaystyle 0 leq i lt n nbsp takih sho f i a f i 1 a lt 0 displaystyle f i alpha f i 1 alpha lt 0 nbsp Teorema Shturma red Nehaj na vidrizku a b displaystyle a b nbsp zadano ryad Shturma f i i 0 n displaystyle left f i right i 0 n nbsp prichomu f i a 0 f i b 0 displaystyle f i a neq 0 f i b neq 0 nbsp pri vsih i 0 n displaystyle i 0 ldots n nbsp Todi na a b displaystyle a b nbsp isnuye rivno w a w b displaystyle omega a omega b nbsp koreniv rivnyannya f 0 x 0 displaystyle f 0 x 0 nbsp Pobudova ryadu Shturma dlya mnogochlena red Nehaj mnogochlen P x displaystyle P x nbsp ne maye kratnih koreniv Todi dlya nogo mozhna pobuduvati ryad Shturma za takim algoritmom f 0 x P x displaystyle f 0 x P x nbsp f 1 x P x displaystyle f 1 x P x nbsp f k 1 f k x g k x f k 1 x deg f k 1 lt deg f k displaystyle f k 1 f k x g k x f k 1 x quad deg f k 1 lt deg f k nbsp tobto f k 1 displaystyle f k 1 nbsp ce vzyata z protilezhnim znakom ostacha vid dilennya f k 1 displaystyle f k 1 nbsp na f k displaystyle f k nbsp Ostannij krok povtoryuyetsya do otrimannya nulovogo mnogochlena Ryad Shturma utvoryuyut vsi nenulovi mnogochleni f k displaystyle f k nbsp Opisanij proces duzhe nagaduye algoritm Evklida znahodzhennya najbilshogo spilnogo dilnika mnogochleniv a ostannij v ryadi mnogochlen z tochnistyu do chislovogo mnozhnika zbigayetsya z najbilshim spilnim dilnikom P x displaystyle P x nbsp ta P x displaystyle P x nbsp Oskilki P x displaystyle P x nbsp ne maye kratnih koreniv to P x displaystyle P x nbsp i P x displaystyle P x nbsp ye vzayemno prostimi a tomu f n x c o n s t 0 displaystyle f n x equiv const neq 0 nbsp Dzherela red Shafarevich I R O reshenii uravnenij vysshih stepenej metod Shturma M Gostehizdat 1954 ros nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Metod Shturma amp oldid 38743689
