Recent Advances in Computers, Communications, Applied Social Science and Mathematics

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1 Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs Coutg Roots of Radom Polyomal Equatos mall Itevals EFRAI HERIG epatmet of Compute cece ad athematcs Ael Uvesty Cete of amaa cece Pa,Ael,4487 IRAE Abstact: - The poblem of fdg the pobablty dstbuto of the umbe of zeos some eal teval of a adom polyomal whose coeffcets have a gve cotuous ot desty fucto s cosdeed. A ew smulato algothm fo solvg ths poblem s peseted. The effectveess of the peseted algothm fo the case whee the eal teval s small s poved. Key-Wods: - Radom Polyomal, ote-calo Algoth umecal ethod Itoducto Aalyss of the behavo of adom polyomals ad the zeos has bee a subect of actve eseach fo seveal decades. otvato fo these studes ad most of the ealy esults ths feld wee collected ad summazed the moogaph by Bhaucha-Red ad ambadham [], whee oe ca fd applcatos such vaed felds as spectal aalyss, statstcs, flteg theoy ad ecoomcs. oe ecet esults appea, fo example, []-[8] amog othes. Cosde a adom polyomal F a, x + a x a0 of degee, whose coeffcets ae eal-valued adom vaables wth gve cotuous ot pobablty desty fucto p a,..., a. ome of the coeffcets may be o-adom o eve zeo. A mpotat poblem s the study of the behavo of the zeos of the adom polyomal that belog to some gve eal teval. A atual fst step the study of the behavo of such zeos s to estmate the dstbuto of the adom vaable B,, whch gves the umbe of zeos of the polyomal F a that belog to some abtay teval B[l; o the Eucldea space R. Ths poblem has bee wdely studed ad has bee solved fo some specfc cases whch the coeffcets a,..., a ae dstbuted accodg to some 0 ω gve law ad satsfy ceta specfc codtos. Howeve, a geeal method of soluto fo abtaly dstbuted adom coeffcets s ot yet avalable eve fo quadatc adom polyomals. It s wdely accepted that most cases smulato s equed to solve the poblem. The classcal method fo estmatg the pobablty that exactly zeoes of F a belog to B, that has bee actvely used dug the last decades, cludes the followg steps: geeatg a set of adom pots a,..., a, dstbuted accodg to p a,..., a, the coodates of whch ae the coeffcets of the polyomal fo each geeated pot, calculatg the zeos of the coespodg polyomal ad checg whethe exactly zeos belog to B. calculatg the popoto of geeated polyomals whch have zeos B. If a set s bg eough, the popoto of geeated polyomals whch have zeos B s a suffcetly good estmate fo the desed pobablty. The algothm eques geeato of a bg umbe of pots ode to acheve the desed ate of accuacy of the estmate ad s, theefoe, tme-expesve. O the othe had, the classcal method eques geeatg adom vectos dstbuted accodg to the gve ot p.d.f. of the coeffcets, ad such geeatos may ot be avalable some specfc cases. IB:

2 Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs The above metoed dsadvatages of the classcal method mae the developmet of moe effectve algothms hghly desable. Ths motvated us to develop a ew smulato algoth whch s moe effectve fo the case whee the teval B s small. The algothm s peseted ths atcle. escpto of the Algothm The poblem of calculatg P B, pobabltes P B, ω R, m that the polyomal F a zeos, fom whch exactly belog to B, wth ate of accuacy ε ε / m+. et A> 0 be a eal umbe whch satsfes the equalty wth a gve ate of accuacyε ca be educed to calculatg the has exactly m eal P max{ a } < ε, whch meas that the pobablty that eal ad magay pats of all complex zeos of F a ae smalle tha + A / absolute value ad the modul of all eal zeos of F a ae smalle tha + A,s geate tha ε. et Q m deote the set cosstg of all pots a R such that the coespodg polyomal Fa has exactly -m dstct complex zeos ad exactly m dstct eal zeos, fom whch exactly belog to B. It ca be easly show that Q m cossts of a fte umbe of o-tesectg sem-algebac sets whch ca be popely defed, theefoe p a d a Q All the polyomals F a, a Q have complex zeos z l l,, wth postve magay pats pl > 0 ad eal pats l, l, such that < <... <, ad m eal zeos,,..., such that < <... < m, m. We utlze the expesso 4 a,, whee the polyomals F a 0. desgates the sum of, ae elemetay symmetc fuctos of the oots C poducts of factos x wth dstct dces: x,...,, x x of the equato x+ x x, xx + xx +..., xxx + xx x4 +..., xx... x. ubsttutg p stead of x, whee s a odd tege less o equal to, + x, whee s a eve tege less o equal to /, stead of x, whee p stead of <, 4, we obta expessos fo a,, as polyomal fuctos f of vaables,...,,..., p,,..., m. I ode to calculate the tegal, we utlze the followg chage of vaables fom the tal set of vaables a a, a,..., a to a ew set of vaables,,...,, p,,...,. The tegal equals m 5 p f,..., f J a,, whee J a, desgates IB:

3 Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs det p p m p, p m a, desgates m m p p p m m p, the tegato doma s a uo of m + sets R, s 0, m, defed as 6 Itegal [ ; ] [... [ s [ l ; [ [ ; ]... [ s ; ]... [ ; ]. s ; ] [0; ]... [0; ] [ ; l [ [ tmes [... [ Itega. p f J s gves us the pobablty that F a, has exactly m eal oots, fom whch exactly belog to B ad exactly s belog to. et s defe a bouded domas [ ; ] [ ; ]... [ ; ] [0; ]... [0; ] [ ; l [ [... [ s [ ; ] [ [ l ; [ + ; ]... [ tmes [ ; ], s 0, m... [ m U s 0 s Obvously, vew of, the dffeece betwee the tegal 5 ove ad the tegal ove s smalle tha ε, whch eables oe to calculate the tegal ove stead of 5 ode to obta a appoxmato wth the desed ate of accuacy ε. IB:

4 Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs ce s a extemely coveet doma fom the pot of vew of geeatg adom tuples ufomly dstbuted t such tuples ca be geeated utlzg stadad ufom adom umbe geeato, classcal ote-calo algothm ca be used fo umecal calculato of the tegal Itegad. At, we have a adom polyomal x + a x + a x+ a 0. et s calculate the pobablty that the polyomal has a uque eal zeo belogg to the teval B [0;]. We have f, + p f, + f, + J a,, det 0 p p 4 p+ 4p + 4p 8 p p + + p + p B, I R, 4 p+ 4p + 4p 0 p +, + 8 p p + p, p Coclusve Remas et us stess the two ma advatages of the peseted method. Fst, whe calculatg the pobablty that a gve umbe of zeos of a adom polyomal belog to a small teval, the peseted method s moe effectve tha the classcal oe. et us pove ths asseto fo the case whee Itegad s a bouded fucto 0< G < Itega < G t ca be show that ths codto s satsfed may cases of teest. et { } B B, be a sequece of tevals such that B B B B, B,..., ad B, 0 0 B, 0 B le B 0. Hee le desgates the legth of the teval B. et desgate the pobablty that exactly m zeos of F a, belog to B ad let desgate the tegato doma fo calculatg the tegal accodg to the peseted method f the zeos of the polyomal belogg to : B ae couted. It ca be sdtpes easly poved that the sequece of atos teds to zeo as teds to fty, whee sdtpes s stdclass the mea squae devato of the ote-calo estmate fo, based o the peseted method, ad stdclass - the mea squae devato of the estmate based o the classcal method. I both cases the same amout of geeated tuples s used. The poof s based o the followg equaltes : V G K stdclass, whee K 0 IB:

5 Recet Advaces Computes, Commucatos, Appled ocal cece ad athematcs stdpes V G G, 0,,.... Hee V desgates the volume of. Obvously, V 0 sce le 0. B The secod advatage of the peseted method s that t eables oe to develop the softwae whch ca be mplemeted fo coutg the zeos of adom polyomals wth abtay ot pobablty desty fucto of ts coeffcets, whle softwae outes based o the classcal method ca t be uvesal sce they have to cooate a specfc adom umbe geeato fo each adom polyomal. Refeeces:. A.T. Bhaucha-Red ad.ambadha Radom Polyomals, Academc Pess: Olado/odo, A.Edelma ad E.Kostla, How may zeos of a adom polyomal ae eal?, Bull.Ame.ath. oc.., Vol,995,pp.-7.. Y. CastZ. Hadzbabc,. toc, J. albad ad. tga Quatzed Votces the Ideal Bose Gas: A Physcal Realzato of Radom Polyomals. Phys. Rev. ett. 96,04005, B.hffma ad. Zeldtch, Equlbum dstbuto of zeos of adom polyomals, Iteatoal ath.res. otes, Vol.00,pp5-49,00 5. E. hmelg ad K.J. Hochbeg, Asymptotc behavo of oots of adom polyomal equatos,poc.ame.ath.oc.vol.0,00,pp E. hmelg ad K.J. Hochbeg, tum s ethod coutgroots of Radom Polyomal Equatos. ethodology ad Computg Appled Pobablty,o.6,004,pp E.hmelg. Algothm fo efg the stbuto of Zeos of Radom Polyomals, Poc..of the -th WEA Iteatoal Cofeece of Computes, Vol. 4,007,pp od Zeos of Gaussa aalytc fuctos, ath.res.ett.,vol.7,000,pp.7-8. IB: