numericke metode - zbirka zadataka kroz c, fortran i matlab

8/15/2019 Numericke metode - Zbirka zadataka kroz C, Fortran i Matlab

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Dhsimgi Z. Pidumavbḱ

Ichgsimdir @. [inirdzˏbḱ

Ebcbp N. Nirbḱ

M X N H P B Kˏ G H

N H T A D H

\`brgi ziditigi

graz K, Eartrim b Nitci`

IGIDHN[GI NB[IA@haorid, 0::8


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Zrhdoavar

Gmlboi prhdstivcli druoa, bznhmlhma b dapumlhma bzdimlh bsth z`brgh galu lh padavbn miscavan prvb iutar a`livba 0:::. oadbmh. [ a`zbran di lh prva bzdimlhrispradita, bsgarbstbcb sna prbcbgu di dapumbna z`brgu dhticlmblh uri -dhmbn prbnh-

rbni galb `b kˏbtiaku palismbcb prbnhmu nhtadi, ugclukˏbna dhticlma dagunhmtavimpraorinsgb gad zi palhdbmh nhtadh, b, svigiga, bsprivbna orhsˏgh uakˏhmh u prhtfad-man bzdimlu.

\`brgi sidrzˏb adi`rimh rhsˏhmh ziditgh si vhzˏ`b b bspbti bz prhdnhti „Xvad umunhrbkˏgu nithnitbgu„ b „Munhrbkˏgh nhtadh„, galb sh prhdilu mi rhdavmbn studb- lini mi Nithnitbˏkgan eigucthtu u @haoridu. Minhmlhmi lh u prvan rhdu stu-dhmtbni Nithnitbkˏgao eigucthti, icb b studhmtbni druobf eigucthti galb bzukˏiviluMunhrbkˏgu nithnitbgu, gia b strukˏmlikbni galb sh u svigadmhvman ridu mlanhgarbsth. Di `b zidavaclbci patrh`h aviga sˏbragao gruoi garbsmbgi, parhd tharblsgbfziditigi dit lh b vhcbgb `ral dhticlma rhˏshmbf munhrbkˏgbf prbnhri, gia b zi mbznhtadi praorinsgb gad mipbsim u lhzbkbni K bcb Eartrim 2:.

\`brgi lh u sidrzˏilu b azmigini siocismi si udzˏ`hmbgan prvao iutari Mu-

nhrbkˏgh nhtadh (W09^) b gmlboan Munhrbkˏgi imicbzi @asˏgi Lavimavbḱi b prvao iutari(W9;^), galh prhdstivclilu tharblsgu asmavu aval z`brkb. Bzcazˏhmb nithrblic lh padh-clhm u lhdimihst paocivcli, prhni a`cistbni munhrbkˏgh nithnitbgh gala l zidikbprbpidilu, b a`ufviti svh asmavmh a`cistb avh orimh nithnitbgh.

Zrva paocivclh lh uvadmao girigthri b a`lisˏmlivi mikˏbm bzcioimli nithrblicigraz prhzhmtikblu Oiuss-avh nhtadh zi rhsˏivimlh sbsthni cbmhirmbf lhdmikˏbmi sitradblioamicman nitrbkan.

[chdhḱi trb paocivcli su pasvhḱhmi pra`chnbni ipragsbnikblh. X druoan paociv-clu su a`ri -dhmb rizcbkˏbtb vbdavb bmthrpacikblh, Ciorimoh-avi, Fhrnbth-avi b bmthrpa-cikbli spcilmavbni, i panhmuth su b rikbamicmi b trboamanhtrblsgi bmthrpacikbli.Tiga -dh lh bcustravimi prbnhmi bmthrpacikbambf pacbmani zi rhˏsivimlh pra`chnibmvhrzmh bmthrpacikblh b munhrbkˏgao dbehrhmkbrimli. Zrbnhmi bmthrpacikbambf pacb-mani u prb`cbzˏman bzrikˏumivimlu bmthorici diti lh u trhḱhn paocivclu. Zrbnhrbni

su bcustravimh prbnhmh Mhwtam‛Kaths-avbf earnuci (triphzmi, [bnpsam-avi,. . . ) bgvidriturmbf earnuci Oiuss-avao tbpi. Zash`mi pizˏmli lh pasvhḱhmi akhmb orhsˏghavbf icoarbtini. Kˏhtvrta paocivclh lh pasvhḱhma ipragsbnikblb eumgkbli, b ta srhd-mlhgvidritmal b rivmanhrmal ipragsbnikblb, nhtadb milnimlbf gvidriti b Eaurbhr-aval trimsearnikblb, pash`ma EET-u.

Nhtadh cbmhirmh icoh`rh su a`ri -dhmh u phtan b sˏhstan paocivclu. \i rhsˏivimlhsbsthni cbmhirmbf lhdmikˏbmi, bzrikˏumivimlh dhthrnbmimtb b bmvhrzmbf nitrbki garbsˏ-

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bv

ḱhmb su rizmh virblimth Oiuss-avh nhtadh, CX b Kfachsgy dhganpazbkbli. Zash`mipizˏmli lh pasvhḱhmi pra`chnu munhrbkˏgh sti`bcmastb. Asbn dbrhgtmbf, a`ri -dhmh sub bthritbvmh nhtadh si pash`mbn asvrtan mi tikˏmast da`blhmao rhsˏhmli. Nhtadh zirhsˏivimlh pra`chni sapstvhmbf vrhdmastb nitrbki bzdvalhmh su u pash`ma paocivclh,sˏhsta. Aùfviḱhmh su giga nhtadh zi rhsˏivimlh patpumao pra`chni (Obvhms-avi,Lika`b-lhvi, Faushfacdhr-avi nhtadi b CP b YP icoarbtin), tiga b nhtadh zirhsˏivimlh dhcbnbkˏmao pra`chni sapstvhmbf vrhdmastb.

[hdna paocivclh lh pasvhḱhma rhsˏivimlu mhcbmhirmbf lhdmikˏbmi b sbsthni nhta-dan prasth bthrikblh (Lika`b-lhvan) b Mhwtam-avan nhtadan. A`ri -dhmi lh b nhtadi@ibrstaw-i zi rhsˏivimlh icohìrsgbf lhdmikˏbmi.

Zaschdmli kˏhtbrb paocivcli su pasvhḱhmi nhtadini zi rhsˏivimlh a`bkˏmbf b pir-kblicmbf dbehrhmkblicmbf lhdmikˏbmi, gia b bmthoricmbf lhdmikˏbmi. \ào svalh rizcbkˏbthprbradh, nhtadh zi rhsˏivimlh Kiukfy-lhvbf pra`chni b nhtadh zi rhsˏivimlh orimbkˏmbfpra`chni su rizdvalhmh u pash`mi paocivcli. X asnan paocivclu su zi rhsˏivimlh

Kiukfy-lhvbf pra`chni garbsˏḱhmh ipragsbnitbvmh nhtadh, nhtadh tbpi Pumoh‛Gutt-i b prhdbgtar‛garhgtar nhtadh. Zash`mi pizˏmli lh pasvhḱhmi tikˏmastb. Nhtadh zirhsˏivimlh orimbkˏmbf pra`chni zi a`bkˏmh dbehrhmkblicmh lhdmikˏbmh, i ta su nhtadhgamikˏmbf rizcbgi b virblikbamh nhtadh, su bcustravimh u dhvhtan paocivclu. X paschd-mli dvi paocivcli gmlboh ditb su sina hchnhmtirmb prbnhrb rhsˏivimli bmthoricmbf bpirkblicmbf lhdmikˏbmi, siocisma praorinu panhmutbf prhdnhti.

X agvbru svigao paocivcli dit lh praorinsgb gad ìr lhdmh ad nhtadi si bskrp-mbn a` lisˏmlhmlhn. [ a`zbran di lh dha tao nithrblici pbsim u praorinsgan lhzbguEartrim2: , galb lh vrca nmaoa garbsˏḱhm u munhrbkˏgan nadhcbrimlu, u daditgu lhdit gritig prbgiz avao praorinsgao lhzbgi.

@haorid, shpthn`ri 0::0. D. Z. PidumavbḱI. @. [inirdzˏbḱ


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Zrhdoavar trhḱhn bzdimlu

Gmlboi galu bnith prhd saàn lh mistici daridan asmavmao thgsti galb lh DhsimgiPidumavb́k a`livbci 0:::. oadbmh u vbdu \`brgh ziditigi bz munhrbˏkgbf nhtadi. [a`zbran mi dbminbkˏim rizval avh a`cistb nithnitbgh, uscavclhm b bmthmzbvmbn rizva-

lhn bmearnikbambf thfmacaobli, druoa bzdimlh a`livclhma 0::0. oadbmh lh aàoiḱhmapraorinbni mipbsimbn mi lhzbkbni K b Eartrim 2:, galb bnpchnhmtbrilu a`ri -dhmhnhtadh. Avh prakhdurh, gia b daditig pasvh́khm praorinsgan lhzbgu Eartrim 2:mipbsia lh Ichgsimdir [inirdzˏbḱ.

Druoa bzdimlh z`brgh lh rispradita, b tiga sh ugizici prbcbgi di sidrzˏil dapu-mbna prbgizan rhsˏivimli pra`chni saetvhrsgbn pightan Nitci`, galb prhdstivclimhzia`bcizmb praehsbamicmb icit zi munhrbkˏgi b miukˏmi bzrikˏumivimli, b pashdulhvhcbgb `ral bnpchnhmtbrimbf munhrbkˏgbf nhtadi. \i nhtadh a`ri -dhmh u gmlbzb zigalh mh pastalh oatavh bnpchnhmtikblh, prbgizima lh giga `b sh amh naoch bnpch-nhmtbritb u Nitci`-u. Uhcbgh naouḱmastb avao pighti su bsgarbsˏḱhmh di mi da`raadi`rimbn prbnhrbni bcustrulu pra`chnh galb sh naou livbtb prb rhsˏivimlu rhicmbfnadhci (vhcbgbf dbnhmzbli), i orikˏgb prbgizb da`blhmbf rhzuctiti `b trhìca zmikˏilma

di panaomu u rizunhvimlu garbsˏḱhmbf munhrbkˏgbf nhtadi. Avil dha thgsti, gia bdaditig pasvhḱhm praorinsgan pightu Nitci` mipbsia lh Ebcbp Nirbḱ.

Iutarb zifviclulu svbn studhmtbni galb su tagan prhtfadmbf oadbmi, prithḱbgurs bz prhdnhti „Munhrbkgh nhtadh„ mi Nithnitbkgan eigucthtu Xmbvhrzbthtiu @haoridu, svalbn prbnhdìni b garbsmbn suohstblini daprbmhcb gvicbthtu avaoithgsti.

@haorid, shpthn`ri 0::8. D. Z. Pidumavbḱ9

I. @. [inirdzˏbḱ0

E. N. Nirbḱ ;

9dridumJnite.ò.ik.yu0isinirdzbkJnite.ò.ik.yu;ebcbpJnite.ò.ik.yu

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vb


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[idrzˏil

9 Xvad 9

0 Bmthrpacikbli 20.9 Ciorimoh-av pacbman . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.0 Mhwtam-av pacbman si padhclhmbn rizcbgini . . . . . . . . . . . . . . 000.; Zacbmanb si hgvbdbstimtmbn kˏvaravbni . . . . . . . . . . . . . . . . . ;:0.4 Druob vbdavb bmthrpacikblh . . . . . . . . . . . . . . . . . . . . . . . . ;80.8 Fhrnbth-av p acbman . . . . . . . . . . . . . . . . . . . . . . . . . . . 4;0.5 Bmthrpacikbamb spcilm . . . . . . . . . . . . . . . . . . . . . . . . . . . 890.? Munhrbkˏga dbehrhmkbrimlh . . . . . . . . . . . . . . . . . . . . . . . . . 50

; Munhrbkˏgi bmthorikbli 52;.9 Mhwtam‛Kaths-avh gvidriturmh earnuch . . . . . . . . . . . . . . . . ?:;.0 Gvidriturmh earnuch Oiuss-avao tbpi . . . . . . . . . . . . . . . . . 69

4 Ipragsbnikbli eumgkbli 2?4.9 [rhdmlhgvidritmi ipragsbnikbli . . . . . . . . . . . . . . . . . . . . . 9::4.0 Nhtadi milnimlbf gvidriti . . . . . . . . . . . . . . . . . . . . . . . 99;4.; Eaurbhr-av rhd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9924.4 Dbsgrhtmi Eaurbhr-avi trimsearnikbli . . . . . . . . . . . . . . . . . . 90?4.8 Pivmanhrmi ipragsbnikbli . . . . . . . . . . . . . . . . . . . . . . . 94;

8 [bsthnb cbmhirmbf lhdmikˏbmi 9828.9 Oiuss-avh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9828.0 Trauoiami dhganpazbkbli . . . . . . . . . . . . . . . . . . . . . . . . 95?8.; Munhrbkˏgi sti`bcmast . . . . . . . . . . . . . . . . . . . . . . . . . . . 9?:8.4 Bthritbvmh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9?8

5 [apstvhmh vrhdmastb b sapstvhmb vhgtarb nitrbki 9685.9 Obvhms-avi nhtadi ratikblh . . . . . . . . . . . . . . . . . . . . . . . 9655.0 Lika`b-lhvi nhtadi . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92:5.; Faushfacdhr-avi nhtadi . . . . . . . . . . . . . . . . . . . . . . . . . 92?5.4 YP nhtadi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0:;5.8 Dhcbnbkˏim pra`chn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 000

vbb


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vbbb [IDP ˏ\IL

? Mhcbmhirmh lhdmikˏbmh b sbsthnb 0;9?.9 Nhtadi bthrikblh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0;8?.0 Nhtadi Mhwtam-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 42?.; Nhtadi @ibrstaw-i . . . . . . . . . . . . . . . . . . . . . . . . . . . . 058?.4 Oridblhmtmh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . . . 052

6 ADL ‛ Kiukfy-lhvb pra`chnb 0??6.9 Ipragsbnitbvmh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . 0??6.0 Nhtadi Pumoh‛Gutt-i . . . . . . . . . . . . . . . . . . . . . . . . . . 06;6.; Zrhdbgtar-garhgtar nhtadh . . . . . . . . . . . . . . . . . . . . . . . . 02?

2 ADL ‛ orimbkˏmb pra`chnb ;992.9 Nhtadi oi -dimli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ; 992.0 Nhtadi gamikˏmbf rizcbgi . . . . . . . . . . . . . . . . . . . . . . . . . ;98

2.; Uirblikbamh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;0?

9: Bmthoricmh lhdmikˏbmh ;;29:.9 Nhtadi uzistapmbf ipragsbnikbli . . . . . . . . . . . . . . . . . . . . ;4:9:.0 Nhtadi dhohmhrbsimbf lhzoiri . . . . . . . . . . . . . . . . . . . . . . ;409:.; Nhtadi gvidriturmbf earnuci . . . . . . . . . . . . . . . . . . . . . . ;469:.4 Uirblikbamh nhtadh . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;84

99 Zirkblicmh dbehrhmkblicmh lhdmikˏbmh ;5;99.9 Lhdmikˏbmi hcbptbˏkgao tbpi . . . . . . . . . . . . . . . . . . . . . . . . . ;5499.0 Lhdmikˏbmi piriàcbˏkgao tbpi . . . . . . . . . . . . . . . . . . . . . . . ;??99.; Lhdmiˏkbmi fbphràcbkˏgao tbpi . . . . . . . . . . . . . . . . . . . . . . ;64

I Eartrim 2: ;6?I.9 Asmavmb hchnhmtb lhzbgi . . . . . . . . . . . . . . . . . . . . . . . . . ;6?I.0 Tbp avb paditigi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;66I.; Zacli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;29I.4 Bzrizb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;20I.8 Gamtracmh strugturh . . . . . . . . . . . . . . . . . . . . . . . . . . . ;2;I.5 Zrakhdurh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;25I.? Zraorinb b naducb . . . . . . . . . . . . . . . . . . . . . . . . . . . . ;26I.6 Xciz b bzciz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4::

@ Nitci` 4:;@.9 Nitci` gia dbobtram . . . . . . . . . . . . . . . . . . . . . . . . . . . 4:[email protected] Uhgtarb b nitrbkh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4:6

@.; Astich strugturh paditigi . . . . . . . . . . . . . . . . . . . . . . . . [email protected] Zacbmanb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected] Origi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [email protected] Gamtraci tagi bzvrˏsivimli . . . . . . . . . . . . . . . . . . . . . . . . 492@.? n-ditathgh - sgrbptavb b eumgkblh . . . . . . . . . . . . . . . . . . . . . [email protected] [bnàcbˏkgi bzriˏkumivimli . . . . . . . . . . . . . . . . . . . . . . . . 404


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[cbgh

0.9 Ciorimoh-avi bmthrpacikbli eumgkblh 99+08x0

. (i) hgvbdbstimtmi nrhzˏi,

(`) much Kˏh b̀ˏshv-clhvbf pacbmani . . . . . . . . . . . . . . . . . . . . 0:

0.0 Mhwtam-av bmthrpacikbamb pacbman . . . . . . . . . . . . . . . . . . . . 020.; Chva< prbnhmi eumgkblh bmthrp0, Dhsma< prbnhmi eumgkblh orbdditi 4;0.4 Eumgkbli bmthrp9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820.8 Gu`mb spcilmavb (ksiph) . . . . . . . . . . . . . . . . . . . . . . . . . 59

;.9 Erhsmhc-avi spbrici . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24;.0 Chohmdrh-avb b Kˏh b̀ˏshv-clhvb pacbmanb . . . . . . . . . . . . . . . . . . 25

4.9 Orig eumgkblh |k| + |9 ∐ k|. . . . . . . . . . . . . . . . . . . . . . . . 9::4.0 Ipragsbnikbli eumgkblh hx mi bmthrvicu W:, 9^ pacbmanan sthphmi 9:

u admasu mi gimamsgu `izu pacbmani 9, x, x 0, . . . . . . . . . . . . . 9904.; [rhdmlhgvidritmi ipragsbnikbli tikˏigi gruzˏmbkan . . . . . . . . . . 992

4.4 @utthrffy strugturi. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9064.8 [ˏhni bzrikˏumivimli gad `rzh Eaurbhr-avh trimsearnikblh. . . . . . . . 9024.5 Bzrikˏumivimlh dbsgrhtmh Eaurbhr-avh trimsearnikblh zi ditb vhgtar. . . 9;;4.? i) Urhnhmsgb danhm sbomici `) Erhgvhmkblsgb danhm sbomici . . . . . 94:4.6 Nimuicma ctrbrimlh u erhgvhmkblsgan danhmu . . . . . . . . . . . . . 9494.2 Mbsga-erhgvhmkblsga ctrbrimlh . . . . . . . . . . . . . . . . . . . . . . 9404.9: Apshzˏma ctrbrimlh . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9404.99 Zacbman muctao sthphmi mil`aclh rivmanhrmh ipragsbnikblh. . . . . . 9444.90 Pivmanhrmi ipragsbnikbli privan gamgivmh eumgkblh. . . . . . . . . 948

5.9 Hehgit nmazˏhmli nitrbkini ratikblh . . . . . . . . . . . . . . . . . . 92:

?.9 Nhtadi Phouci-eicsb prbnhmlhmi mi eumgkblu e (x) > x;

∐x. . . . . . 0;4

?.0 Orikb eumgkbli e 9(x, y) > :, e 0(x, y) > :. . . . . . . . . . . . . . . 0;6?.; Mhwtam-avi nhtadi prbnhmlhmi mi eumgkblu e (x) > sbom(x)

|x|. . . 05;?.4 Bzrikˏumivimlh nhtadan pagaardbmitmao spusti. . . . . . . . . . . . . 0?:?.8 Nhtadh pagaardbmitmao b mil̀ rˏzho spusti . . . . . . . . . . . . . . . 0?4

6.9 [bnucikbli captbkh gali adsgikˏh . . . . . . . . . . . . . . . . . . . . . 0246.0 Mhsti`bcmast Huchr-avh nhtadh . . . . . . . . . . . . . . . . . . . . . 025

bx


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x [CBGH

99.9 Zadhci a`cistb O mi pada`cistb. . . . . . . . . . . . . . . . . . . . . . ;5899.0 A c̀ist b nrhzˏi u ziditgu ;. . . . . . . . . . . . . . . . . . . . . . . . ;5599.; A c̀ist b nrhzˏi u ziditgu 4. . . . . . . . . . . . . . . . . . . . . . . . ;5699.4 A c̀ist b nrhzˏi u ziditgu 8. . . . . . . . . . . . . . . . . . . . . . . . ;52

@.9 nitci` - rizvalma agruzˏhmlh . . . . . . . . . . . . . . . . . . . . . . 4:[email protected] Xpatrh`i eumgkblh pcat . . . . . . . . . . . . . . . . . . . . . . . . . 49?@.; Xpatrh`i eumgkbli nhsf b sure . . . . . . . . . . . . . . . . . . . . . [email protected] Zirinhtirsgb ziditi grbvi b pavrˏs . . . . . . . . . . . . . . . . . . . . [email protected] Pizcbgh bznh -du sgrbptavi b eumgkbli . . . . . . . . . . . . . . . . . . . 400


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9

Xvad

X uvadman paocivclu `bḱh prhzhmtbrim mikˏbm bzcioimli u aval z`brkb ziditigi miprbnhru pra`chni rhsˏivimli tradblioamicmao sbsthni lhdmikˏbmi. Miocisig z`brghtrh`i di `udh mi pbsimlu praorini, adm. prhkbzmblh prakhduri, galh bnpchnhmtbrilupalhdbmh munhrbˏkgh nhtadh. Zbsimlh gadi zi avil danhm pra`chni padrizunhvipracizig graz svh amh eizh galh su stimdirdmh zi `bca gigva praorinbrimlh, i tasu imicbzi pra`chni, bdhmtbgavimlh ucizi b bzcizi prakhdurh, bnpchnhmtikbli thciprakhdurh b mi grilu thstbrimlh mipbsimao gadi.

Di `b sh usphsˏma prasˏca graz svh mi`ralimh eizh, mhapfadma lh da`ra pazmivimlhmunhrbkˏgh nhtadh gali sh zˏhcb bnpchnhmtbritb. \ita ḱh mi pakˏhtgu riznitrimlisvigh nhtadh, bcb hvhmtuicma oruph nhtadi, `btb mivhdhmh asmavmh earnuch galh shmi tu nhtadu admash. Avil shonhmt thgsti ḱh `btb bstigmut agvbran. Mi prbnhru

nhtadh gali rhsˏivi tradblioamicmb sbsthn cbmhirmbf lhdmikˏbmi, til dha thgsti `b bniaschdhḱb a c̀bg<

Tradblioamicmb sbsthn lhdmikˏbmi lh sbsthn lhdmikˏbmi a`cbgi

k:x: + `:x9 > d:,

ibxb∐9 + kbxb + `bxb+9 > db, b > 9, . . . , m ∐ 0,im∐9xm∐0 + km∐9xm∐9 > dm∐9

Iga lh k: > :, rhsˏhmlh tradblioamicmao sbsthni sh adrh -dulh rhgurhmtmbn earnu-cini

xm∐9 > d

m∐9 ∐im∐9

ΰ m∐9

km∐9 + im∐9ιm∐9 , xb > ιb+9xb+9 + ΰ b+9, b > m ∐ 0, . . . , :,

ι9 > ∐`:k:

, ιb > ∐ `b∐9kb∐9 + ib∐9ιb∐9

ΰ 9 > d:

k:, ΰ b >

db∐9 ∐ ib∐9ΰ b∐9kb∐9 + ib∐9ιb∐9

,

b > 0, . . . , m ∐ 9.

9


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0 9. XUAD

Eizi imicbzh pra`chni trhì di pakˏmh pizˏclbvbn riznitrimlhn avbf earnuci7tiga -dh, vhḱ bz earnuci sh nazˏh uakˏbtb sˏti su ucizb adm. bzcizb u dital nhtadb.Nh -dutbn, gclukˏmb prhduscav pbsimlu gadi galb bnpchnhmtbri mhgu munhrbkˏgu nhtadu lhsth b di sh mhgacbga ziditigi galb sh admash mi tu nhtadu rhsˏh rukˏma. Mi avil mikˏbmsh upazmilh si garikbni ad galbf sh sistalb nhtadi, kˏbnh sh uakˏivi dhganpazbkblimi patpra`chnh pra`chni pbsimli thci prakhdurh gali bnpchnhmtbri nhtadu. Diclh,gia vizˏim rhzuctit rukˏmao rhsˏivimli adrh -dhmao `rali ziditigi astilu nh -durhzuctitbb micmb rhzuctitb zi dith pra`chnh, galb su vhani garbsmb zi thstbrimlh b dhìoavimlhprakhdurh gali bnpchnhmtbri nhtadu. \ita ḱh zi svigu ad nhtadi galh ùdu riz-nitrimh u aval z`brkb `btb dita mhgacbga ziditigi u galbni lh ditb pra`chn rhsˏhmrukˏma garbsˏḱhmlhn adoaviriluḱh nhtadh. Mi prbnhru rhsˏivimli tradblioamicmaosbsthni lhdmikˏbmi, lhdim tigiv ziditig si rhsˏhmlhn `b bnia a`cbg<

9.9 Phsˏbtb tradblioamicmb sbsthn lhdmikˏbmi

;x: + 4x9 > 99x: ∐ 0x9 + ;x0 > 5

0x9 ∐ x0 ∐ x; > ∐;x0 ∐ 0x; > ∐8

Phsˏhmlh< Mi asmavu earnuci zi rhsˏivimlh tradblioamicmao sbsthni lhdmikˏbmida`blilu sh pirinhtrb

ι9 > ∐4;

, ι0 > ∐ ;∐4; ∐ 0 >

2

9:, ι; > ∐ ∐9

0 ¿ 29:

∐ 9 > 8

4,

ΰ 9 > 99; , ΰ 0 >

5∐

99

;∐4; ∐ 0 > ∐ ?9: , ΰ ; > ∐

;∐

0 (∐

?

9:)

0 ¿ 29: ∐ 9 > ∐0,

pi lhx; > 4, x0 > ;, x9 > 0, x: > 9.

\i adrh -dhmb `ral nhtadi u z`brkb, parhd rikˏumsgbf ziditigi, `bḱh uri -dhm b lhdimbcb vbsˏh tharblsgbf ziditigi galb pradu`clulu earnuch galbni lh apbsimi nhtadi bcbbzvadh mhgh panaḱmh rhzuctith galb ḱh `btb ad garbstb zi rikˏumsgh ziditgh.

Gidi sh zivrsˏb si rikˏumsgbn zidikbni zi adrh -dhmu nhtadu, nazˏh sh prbstupbtbpbsimlu gadi galb bnpchnhmtbri nhtadu. \i vh́kbmu a`ri -dhmbf nhtadi uri -dhmi lh adoaviriluḱi prakhduri, prb kˏhnu lh zi bnpchnhmtikblu u prvan dhcu z`brgh

garbsˏḱhm praorinsgb lhzbg K , i u druoan dhcu z`brgh E APTPIM 2:9

. Mi griluvhḱbmh paocivcli, `bḱh dit gritig asvrt mi apbsimh munhrbkˏgh nhtadh graz sbsthnN ITCI@, galb, gia praehsbamicmb icit zi munhrbkˏgi b miukˏmi bzrikˏumivimli, pash-dulh vhcbgb `ral bnpchnhmtbrimbf munhrbˏkgbf nhtadi. Miocisig u bzcioimlu ḱh `btbstivclhm, prh svhoi, mi prbgizbvimlh oatavbf rutbmi b rhsˏivimlh rikˏumsgbf ziditigimlbfavbn garbˏsḱhmlhn.

9avb praorinsgb lhzbkb su a di`rimb zita sˏta sh milkˏhsˏḱh garbsth zi munhrbkˏga praorinbrimlh


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\i praorinbrimlh pad K -an dastupmi lh stimdirdmi cbthrituri (W94^), dag ziupazmivimlh si asmavini praorinbrimli mi EAPTPIM -u trhì gamsuctavitb da-ditig I avh z`brgh. [vh prakhdurh su a`lhdbmlhmh u `b c̀bathgu gali lh mizvimicb`munhrbks b gali lh dastupmi mi _h` strimb prhdnhti (W0?^). @b`cbathgi parhdprakhduri galh bnpchnhmtbrilu palhdbmikˏmh nhtadh sidrzˏb prithḱb Nighebch, gia bpash`mh thst praorinh si adoaviriluḱbn thst prbnhrbni zi svigu nhtadu7 u avalz`brkb `bḱh prhdstivclhm sina thst praorin zi nhtadu zi rhsˏivimlh sbsthni tradblio-amicmbf lhdmikˏbmi, dag ḱh zi astich nhtadh `btb prhzhmtbrimh sina prakhdurh galhbnpchnhmtbrilu nhtadh. Kˏbtivi `b`cbathgi, digch b siv gad galb ḱh `btb prbgizimu aval z`brkb, pbsimi lh bsgclukˏbva si XMBV agruzˏhmlhn mi unu b panaḱu OMX praorinhrsgbf iciti7 zi uputstvi a garbsˏḱhmlu avbf iciti dastupmi lh adoaviriluḱicbthrituri (W0;^).

K X scukˏilu pasnitrimao pra`chni rhsˏivimli tradblioamicmbf sbsthni lhdmikˏbmi

tagan eizh imicbzh pra`chni nazˏh sh zigclukˏbtb di sh bnpchnhmtikbli nhtadh sistalbad prastao prhva -dhmli ditbf earnuci u adoaviriluḱb gad uz dbrhgtma rhsˏivimlhscukˏi li m > 9, ridb bz`hoivimli rhehrhmkbrimli ι: b ΰ :. Gia ucizb u nhtadu naoush uakˏbtb dbnhmzbli sbsthni b, svigiga, gahkblhmtb sbsthni7 ridb usˏthdh prastariàclh lh di gahkblhmtb ùdu prhdstivclhmb gia kˏhtbrb vhgtari (vhgtar gahkblhmitimi ocivmal dblioamicb b dvh da-dblioamich, gia b vhgtar gahkblhmiti mi dhsmal strimb lhdmikˏbmi) mhoa di sh u nhnarblb drzˏb kˏbtivi nitrbki sbsthni. Bzciz bz nhtadh lhvhgtar rhsˏhmli sbsthni. [ a`zbran di mi K lhzbgu prakhdurh mh naou di vriḱiluvhgtarh, adoaviriluḱu prakhduru trhì dhgcirbsitb tiga di vriḱi vabd tbp, i vhgtaru ga lb ḱh `btb snhsˏthmi rhsˏhmli trhì prhmhtb gia daditmb irounhmt prakhdurh. Miasmavu svhoi avaoi, nazˏh sh mipbsitb dhgcirikbli prakhdurh gali ḱh bnpchnhmtbritbnhtadu (prakhduru ḱhna mizvitb trbdbio(), i trudb́khna sh di graz diclb thgstbdhmtbgitarbni u gadu daschdma dilhna dhsgrbptbvmi bnhmi mi hmochsgan lhzbgu)<

vabd

trbdbio (bmt m, dau`ch *i, dau`ch *`, dau`ch *k, dau`ch *d, dau`ch *x)7

Avdh lh m dbnhmzbli sbsthni, i i, `, k b d vhgtarb adoaviriluḱbf gahkblhmitisbsthni, dag lh x vhgtar u galb prakhduri trbdbio() trhì di snhstb rhsˏhmli sbsthni.X gadu galb ḱh `btb prhzhmtbrim u aval z`brkb, zi prhdstivclimlh vhcbkˏbmi galh ucizh ubzrikˏumivimlh daschdma lh garbsˏḱhm rhicmb tbp dvastrugh tikˏmastb, bz prastao rizcaoisˏta lh zi tbp ipcbgikblh gigvi lh munhrbkˏgi imicbzi tikˏmast rhzuctiti mi prvan nhstu.

[ini bnpchnhmtikbli prakhdurh trbdbio() sista lb sh, giga lh oarh vhḱ panhmuta,ad prhva -dhmli earnuci galh apbsulu nhtadu u adoaviriluḱh bzrizh7 parhd avaoi, mipakˏhtgu svigh prakhdurh dadiḱhna adoaviriluḱu pravhru vicbdmastb irounhmitiprakhdurh sˏta ḱh u aval `b`cbathkb `btb uri -dhma prastbn isshrt() pazbvbni0. Mitil mikˏbm, thca prakhdurh trbdbio() bni schdhḱb a`cbg (zi praorinsgb gad ḱh grazavu z`brgu tiga -dh b̀tb garbsˏ́khm pashìm eamt)<

0u rhicmbn uscavbni `b svigiga trhìca asnbscbtb mhgb raùstmblb mikˏbm rhioavimli mi hvhm-tuicmh orhsˏgh u irounhmtbni ad misbcmao prhgbdimli praorini, sˏta sh pastbzˏh avan eumgkblan


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#bmkcudh 3stdcb̀ .f=

#bmkcudh 3isshrt.f=

/* Eumgkbli trbdbio() rhsivi tradblioamicmb sbsthn lhdmikbmi a`cbgi<

iWb^*xWb-9^+kWb^*xWb^+`Wb^*xWb+9^>dWb^

odh lh m=:, b>:,...,m-9, iW:^>>:, `Wm-9^>>: b kWb^!>:. Irounhmtb eumgkblh su<

m - dbnhmzbli sbsthnii, `, k, d - pacli si gahebkblhmtbni sbsthni lhdmikbmi

x - paclh u galh kh `btb snhsthma rhshmlh sbsthni

Zrhtpastivcli sh di su svi pacli icakbrimi bzvim eumgkblh trbdbio(). */

vabdtrbdbio (bmt m, dau`ch *i, dau`ch *`, dau`ch *k, dau`ch *d, dau`ch *x)

{

dau`ch *icpfi, *`hti7 /* Zacli panakmbf gahebkblhmiti. */

bmt b7 /* @ralik u phtclini. */

/* Zravhrivi sh di cb su bspumlhmb uscavb sbsthni. */

isshrt (m = :)7

isshrt (iW:^ >> : && `Wm - 9^ >> :)7ear (b > :7 b 3 m7 b++)

isshrt (kWb^ !> :)7

/* Phsivi sh sphkblicmb scukil zi m lhdmiga 9. */be (m >> 9) {

xW:^ > dW:^ / kW:^7

rhturm7

}

/* Icakbrilu sh pacli panakmbf gahebkblhmiti. */

icpfi > (dau`ch *) niccak ((m + 9) * sbzhae (dau c̀h))7`hti > (dau`ch *) niccak ((m + 9) * sbzhae (dau`ch))7

/* Pikumilu sh vrhdmastb panakmbf gahebkblhmiti. */

icpfiW9^ > -`W:^ / kW:^7`htiW9^ > dW:^ / kW:^7

ear (b > 07 b 3 m7 b++)

{

icpfiWb^ > -`Wb - 9^ / (iWb - 9^ * icpfiWb - 9^ + kWb - 9^)7`htiWb^ >

(dWb - 9^ - iWb - 9^ * `htiWb - 9^) / (iWb - 9^ * icpfiWb - 9^ +

kWb - 9^)7

}

/* Pikumilu sh rhshmli sbsthni. */

xWm - 9^ > (dWm - 9^ - iWm - 9^ * `htiWm - 9^) / (iWm - 9^ * icpfiWm - 9^ +

kWm - 9^)7ear (b > m - 07 b => :7 b--)

xWb^ > icpfiWb + 9^ * xWb + 9^ + `htiWb + 9^7

/* Asca`idlilu sh pacli panakmbf gahebkblhmiti. */erhh (icpfi)7

erhh (`hti)7

}

Mistalica sh, u sgcidu si hchnhmtirmbn praorinhrsgbn privbcbni, di ganhm-tiri `udh sta vbˏsh, i pash`ma di svigi prakhduri bni ganhmtir u ziocivclu galbmizmikˏivi sˏti ami tikˏma ridb b galb su lal ucizb b bzcizb. Diclh, di dhgcirikblu svigh


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pranhmclbvh pritb ganhmtir galb apbsulh kˏhnu scuzˏb ti pranhmclbvi. Mi grilu, disvigb mhtrbvblicmb shonhmt gadi ùdh prapriḱhm ìr lhdman rhkˏhmbkan ganhmtirigali palisˏmlivi mi sˏti sh bstb admasb.

Trhì uakˏbtb giga prbcbgan aphrikbli dhclhmli mblh pravhrivima di cb lh dhclhmbg lhdmig :. Pizcao lh sˏta sh prhtpastivcli di sh bzvrikˏumivimli u pagrhtman zirhzubzvrsˏivilu prhni BHHH ?84 stimdirdu, galb prhdvb -di pastalimlh vrhdmastb zi prhd-stivclimlh rhzuctiti tigvbf aphrikbli, gia b mistivig bzrikˏumivimli mid tigvbn vrhd-mastbni. Avigiv prbstup lh daschdma spravhdhm b u svbn asticbn prbnhrbni u avanthgstu.

Migam sˏta lh oarh prbgiziman prakhduran trbdbio() bnpchnhmtbrimi nhtadizi rhsˏivimlh tradblioamicmao sbsthni lhdmikˏbmi, patrh`ma lh mipbsimb gad thstbritb.X tu svrfu, nazˏh sh mipbsitb lhdim nicb praorin galb, rhkbna, kˏbti pastivgu pra`-chni si stimdirdmao ucizi, patan pazbvi prakhduru trbdbio() di rhsˏb pra`chn, bmi gri lu bspbsulh rhsˏhmli mi stimdirdmb bzciz. X avan scukˏilu, til praorin `b `ba

a`cbgi<

#bmkcudh 3stdba.f=#bmkcudh 3stdcb̀ .f=

/* Zraorin thstbri rhsivimlh tradblioamicmao sbsthni lhdmikbmi. Zraorin kbti

pastivgu pra`chni si stimdirdmao ucizi b bspbsulh rhzuctith mi stimdirdmbbzciz. Mi ucizu sh prva umasb dbnhmzbli sbsthni, i zitbn rhd pa rhd

gahebkblhmtb lhdmikbmi sbsthni iWb^, `Wb^, kWb^ b dWb^. Mi bzcizu sh u prvb

rhd upbsulh dbnhmzbli sbsthni, i zitbn rhd pa rhd rhshmli sbsthni. Zraorin

mhni uoridlhmu pravhru sbmtigsmbf b shnimtbkgbf orhsigi prb umasu pra`chni. */bmt

nibm ()

{

bmt m7 /* Dbnhmzbli sbsthni. */

dau`ch *i, *`, *k, *d7 /* Zacli si gahebkblhmtbni sbsthni. */dau`ch *x7 /* Zaclh si rhshmlhn sbsthni. */

bmt b7 /* @ralik u phtclini. */

/* Xkbtivi sh dbnhmzbli sbsthni. */

skime ("%d", &m)7

/* Icakbrilu sh pacli si gahebkblhmtbni sbsthni. */i > (dau`ch *) niccak (m * sbzhae (dau`ch))7

` > (dau`ch *) niccak (m * sbzhae (dau`ch))7

k > (dau`ch *) niccak (m * sbzhae (dau`ch))7

d > (dau`ch *) niccak (m * sbzhae (dau`ch))7

/* Xkbtivilu sh gahebkblhmtb sbsthni. */

ear (b > :7 b 3 m7 b++)skime ("%ce%ce%ce%ce", &iWb^, &`Wb^, &kWb^, &dWb^)7

/* Icakbri sh paclh zi rhshmlh sbsthni. */

x > (dau`ch *) niccak (m * sbzhae (dau`ch))7

/* Phsivi sh sbsthn. */

trbdbio (m, i, `, k, d, x)7

/* Ascaìdlilu sh pacli si gahebkblhmtbni sbsthni. */

erhh (i)7


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5 9. XUAD

erhh (`)7

erhh (k)7

erhh (d)7

/* Bspbsulh sh rhshmlh sbsthni. */

prbmte ("%dSm", m)7

ear (b > :7 b 3 m7 b++)prbmte ("%oSm", xWb^)7

/* Ascaìdli sh paclh si rhshmlhn sbsthni. */

erhh (x)7

hxbt (HVBTQ[XKKH[[)7

}

[trugturi thst praorini lh lhdmastivmi b dhticlma palisˏmlhmi ganhmtirbni, iu uvadman ganhmtiru lh prhkbzbrim b earnit ucizmbf adm. bzcizmbf paditigi. Thstpraorin mh pravhrivi umhshmh paditgh u snbscu sbmtigsmbf (mi prbnhr, iga sh miucizu unhsta `ra li galb prhdstivcli dbnhmzblu sbsthni palivb mhsˏta sˏta mblh `ral,rhkbna iii) admasma shnimtbkˏgbf (mi prbnhr, iga sh mi ucizu zi dbnhmzblu sbsthnipalivb -8) orhsˏigi u ucizman earnitu. Lhdmastivmastb ridb, prhtpastivgi lh grazsvh thst praorinh u `b`cbathkb di ḱh ucizmb padikb `btb bsprivmb7 u scukˏilu pbsimli rh-icmbf praorini trhì sh mirivma zmitma dhticlmblh paziìvbtb ambn dhcan vicbdikblhumasi galb lh mhapfadma uridbtb ridb privbcmao ukˏbtivimli ucizmbf paditigi ;.

Thst praorinb avigvao a`cbgi su paoadmb zi mhbmthrigtbvma thstbrimlh vhḱbn `ralhn thst prbnhri. Tiga -dh sh da`ra ugcipilu u XMBV cazalu pbsimli praorinigalb a`ri -dulu paditgh si stimdirdmao ucizi b bspbsulu rhzuctith mi stimdirdmb bzciz,galb sh amdi naou gan`bmavitb tiga di sh panaḱu vbsˏh avigvbf nicbf praorinirhsˏivi mhgb vhḱb pra`chn. Mi prbnhr, iga lh ziditig adrhdbtb marnu vhgtari rhsˏhmli

mhgao sbsthni tradblioamicmbf lhdmikˏbmi, nmaoa lh àclh mipbsitb pash`mh praorinhgalb rhsˏivilu tradblioamicmb sbsthn adm. rikˏumilu marnu, i patan mlbfavbn gan-`bmavimlhn (tikˏmblh prhusnhrivimlhn bzcizi prvao praorini mi uciz druoao) rhsˏbtbpacizmb pra`chn, mhoa pbsitb lhdim vhcbgb praorin galb rhsˏivi kˏbtiv pra`chn.

Za usphsˏman prhva -dhmlu avao sidi vh́k nicao pralhgti, trhì prbstupbtb th-stbrimlu si adrh -dhmbn `ralhn thst prbnhri. Tiga sh nazˏh prbprhnbtb eilc u oarhapbsiman ucizman earnitu galb adoaviri ziditgu 9,

4

: 4 ; 9 9

9 ; - 0 5

0 -9 -9 -;

9 : - 0 - 8

b zitbn pagrhmutb oarmlb praorin si stimdirdmbn ucizan prhusnhrhmbn bz avao ei- lci. Xgacbga mhni orhsˏigi u prakhdurb gali bnpchnhmtbri nhtadu, mi stimdirdmanbzcizu ḱh `btb bspbsim tikˏim rhzuctit (apht u oarh apbsiman earnitu)<

;svh prakhdurh galh sikˏbmlivilu `b`cbathgu bmikˏh rboarazma pravhrivilu ucizmh paditgh usnbscu zidavaclivimli pakˏhtmbf uscavi munhrbkˏgh nhtadh galu rhsˏivilu


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?

4

9

0

;

4

Xgacbga rhzuctit mblh bdhmtbkˏim amanh sˏta lh da`blhma rukˏmbn rhsˏivimlhn ditaopra`chni, amdi sh panaḱu dhìohri b garbsˏḱhmlhn nh -durhzuctiti da`blhmbf rukˏmbnrhsˏivimlhn ciga nazˏh ustimavbtb bzvar orhsˏgh u gadu.

NITCI@ Lisma lh di mblh naouḱh akˏhgbvitb di nitci` rispacizˏh oatavbnbnpchnhmtikblini svbf nhtadi galh sh u agvbru avh z`brgh panbmlu. X mhgbnscukˏilhvbni, u mhdastitgu vhḱ oatavh bnpchnhmtikblh adrh -dhmh munhrbkˏgh nhtadh,`bḱh prbgizima giga `b sh ami naoci bnpchnhmtbritb. Mi avil garig sna sh adcukˏbcbgiga `bsna prbgizicb mikˏbm gadbrimli sphkbkˏim zi sbsthn nitci` b sgrhmucb pizˏmlu

mi rizcbgh u stbcu gadbrimli bznh -du nitci`-i b asticbf praorinsgbf lhzbgi galbsh u z`brkb panbmlu. Uhḱbmi rhsˏhmli lh diti u a`cbgu nitci` sgrbptavi, i mh ua`cbgu eumgkbli. Asmavmb rizcao zi ta lh sˏta mbsna zˏhchcb di apthrhḱulhna z`brguapshzˏmbn pravhrini garhgtmastb irounhmiti eumgkbli, gia mb mhapfadman stim-dirdman dagunhmtikblan galh bnpchnhmtikbli nitci` eumgkbli padrizunhvi. [idruoh strimh, trudbcb sna sh di sgrbptavb ùdu mipbsimb mi mikˏbm galb anaouḱivimlbfava lhdmastivma prbcioa -divimlh b hvhmtuicma prhtvirimlh u eumgkblh.

Zrbcbgan bzàri ziditigi galb su rhsˏivimb garbsˏḱhmlhn nitci`-i, trudbcb snash di amb ùdu bcustritbvmb b di ugiˏzu mi pra`chnh da galbf nazˏh daḱb prbcbganmunhrbkˏgao rhsˏivimli ziditigi, i galb sh nazˏdi mh naou uakˏbtb prbcbgan rukˏmaorhsˏivimli ziditigi gad galbf lh dbnhmzbli pra`chni rhcitbvma nici. Trudbcb sna shdi vhḱbmu ziditigi b orikˏgb bcustrulhna u midb di ḱh ava daprbmhtb lasˏ àclhn sioch-

divimlu panhmutbf pra`chni. Zadstbkˏhna kˏbtiakh di sinasticma hgsphrbnhmtbsˏunhmlimlhn ucizmbf pirinhtiri sgrbptavi galb su u z`brkb ditb b rhsˏivimlhn vhḱho`rali ziditigi bstan nhtadan. Asnitrimlh orikˏgao prbgizi da`blhmbf rhzuctiti`b trhìca zmikˏilma di panaomh rizunhvimlu garbsˏḱhmbf munhrbkˏgbf nhtadi.

[viga paocivclh a nitci`-u pakˏbmlh rhzbnhan eumgkbli galh su zi ta paocivclhgirigthrbstbkˏmh. X scukˏilu rhsˏivimli tradblioamicmao sbsthni, rhzbnh `b `ba<

Phsˏivimlh sbsthni Ix > ` x > I`Phsˏivimlh sbsthni xI > ` x > `/IHgisma kˏuvimlh rhtgbf nitrbki P > spirsh(I)7

Grhbrimlh pumh nitrbkh mi asmavu rhtgh I > eucc(P)7

Phtgh dblioamicmh nitrbkh P > spdbios(kacs, dbiomuns, n, m)

Migam rhzbnhi, schdb apbs eumgkbamicmastb galbni nitci` rispacizˏh, i ga lh su rhch-vimtmh zi rhsˏivimlh ziditigi bz a`cistb gali sh u paocivclu a`ri -dulh. Mi prbnhr,zi rhsˏivimlh sbsthni cbmhirmbf lhdmikˏbmi, nitci` rispacizˏh uori -dhmbn aphritarbniS b /. [bsthn a`cbgi Ix > `, odh lh I gvidritmi nitrbki, i x b ` vhgtarb gacamh, shrhsˏivi ganimdan x > IS`. [bsthn a`cbgi xI > `, odh su x b ` vhgtarb vrsth, shrhsˏivi ganimdan x > `/I. Nmhnathfmbkˏga privbca galh acigsˏivi pinḱhmlh aph-ritari galb sh garbsth zi rhsˏivimlh sbsthni lh schdhḱh< rhsˏhmlh prvao sbsthni lh


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6 9. XUAD

x > I∐9`, i druoao lh x > Ì∐9. Ava isakbri mi dhclhmlh nitrbkan I, prbkˏhnu sh u prvan scukˏilu ami micizb chva ad vhgtari `, i u druoan dhsma ad `.Zasˏta lh u aì scukˏili I u ,,bnhmbaku„ ama sh nari miḱb ,,bspad„ `.

Asmavmi girigthrbstbgi tradblioamicmbf sbsthni lh pastalimlh vhcbgao `rali muciu nitrbkb sbsthni. Giga `b paàclsˏia hgismast prb ridu si nitrbkini galh bniluvhcbgb `ral muci, nitci` bnpchnhmtbri pash`mu strugturu rhtgbf nitrbki (spirsh nitrbkhs ). A`bkˏmi nitrbki sh u rhtgu rhprhzhmtikblu prhvadb garbsˏḱhmlhn eumgkblhspirsh, dag sh rhtgi nitrbki prhvadb u ua`bkˏilhmu, pumu, rhprhzhmtikblu garbˏs-ḱhmlhn eumgkblh eucc. \i grhbrimlh rhtgbf dblioamicmbf nitrbki, nitci` mudbeumgkblu spdbios. A`cbg garbsˏḱhmli avh eumgkblh galb sh nazˏh garbstbtb prb rhsˏivimlutradblioamicmao sbsthni lh I > spdbios(kacs, dbiomuns, n, m). Irounhmtkacs lh nitrbki kˏblh su gacamh vhgtarb galb sh pastivclilu mi dblioamich rhzuctuluḱhnitrbkh. Irounhmt dbiomuns sidrzˏb azmigh dblioamici mi galh sh tb vhgtarb pastiv-clilu, prb kˏhnu sh garbstb gamvhmkbli di sh ocivmi dblioamici azmikˏivi mucan, db-

lioamich bspad ocivmh, rhdan, mhoitbvmbn `ralhvbni, i dblioamich bzmid ocivmh,rhdan, pazbtbvmbn `ralhvbni. Irounhmtb n b m azmikˏivilu dbnhmzblu rhzuctuluḱhnitrbkh. Xgacbga lh vhgtar gacami galb sh pastivcli mi dblioamicu bzmid ocivmh duzˏbad dblioamich, bomarbsˏu sh mlhoavb pakˏhtmb hchnhmtb. Xgacbga sh pastivclimlh avigvaovhgtari vrsˏb mi dblioamicu bspad ocivmh, bomarbsˏu sh mlhoavb grilmlb hchnhmtb.

Uhḱbmi uori -dhmbf eumgkbli b aphritari, ugclukˏuluḱb b aphritarh zi rhsˏivimlh sbs-thni lhdmikˏbmi, su umiprhd prbcioa -dhmb ridu si rhtgbn nitrbkini.

Migam avaoi, schdb mhgacbga ziditigi galb bcustrulu garbsˏḱhmlh upriva apbsimbfeumgkbli. Gia prbnhr, garbsˏḱhmlhn nitci`-i rhsˏbḱhna sbsthn zidit ziditgan 9.9.

Zaˏsta lh sbsthn tradblioamicmb, grhbriḱh sh rhtgi nitrbki sbsthni, i zitbn shbsgarbstbtb ua`bkˏilhmb aphritar zi rhsˏivimlh sbsthni. Avil aphritar lh prbcioa -dhmridu si rhtgbn nitrbkini, tiga di ḱh rhsˏivimlh sbsthni `btb prbcbkˏma hgisma.

I > spdbios(W9 0 9 :7 ; -0 -9 -07 : 4 ; -9^‒, W-9 : 9^, 4, 4)7

` > W99 5 -; -8^‒7

x > IS`

Mikˏbm bzcioimli munhrbkˏgbf nhtadi galb lh u avan paocivclu dhnamstrbrim miprbnhru rhsˏivimli tradblioamicmao sbsthni lhdmikˏbmi, `bḱh daschdma spravhdhm grazastitig thgsti. Mirivma, tagan kˏbtimli trhì sticma bnitb mi unu di sh pumbhehgit u snbscu avcidivimli thfmbgan praorinsgao rhsˏivimli pra`chni munhrbkˏghnithnitbgh nazˏh pastbḱb sina ugacbga sh uvhg rikˏumsgb zidikb b praorinhrsgi bn-pchnhmtikbli nhtadh prva pagusˏilu uridbtb sinasticma.


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0

Bmthrpacikbli

Eumgkbli o (x) bmthrpacbsˏh ziditu eumgkblu e (x) iga lh

o(xg) > e (xg), g > :, . . . , m .

Tikˏgh x:, . . . , xm mizbvilu sh kˏvaravb bmthrpacikblh. Iga lh eumgkbli o(x) pacbmansthphmi m, ami sh mizbvi bmthrpacikbamb pacbman eumgkblh e (x).

0.9 Ciorimoh-av pacbman

Ciorimoh-av bmthrpacikbamb pacbman sthphmi m eumgkblh e (x) adrh -dhm kˏvaravbnibmthrpacikblh x:, . . . , xm, zi xb > xl , lh

Cm(x) >m

b>:

m

l>:l>b

x ∐ xlxb ∐ xl

e (xb) > m

b>:

ςm+9(x)

(x ∐ xb) ςm+9(xb) e (xb),

odh lh

ςm+9(x) >m

b>:

(x ∐ xb)

Orhsˏgi bmthrpacikblh eumgkblh lh

Pm(x) > e (x) ∐ Cm(x) > 9(m + 9)!

e (m+9)(ν ) ςm+9(x), ν ∆ Wy9, y0^,

odh lh y9 > nbm(x:, . . . , xm), y0 > nix(x:, . . . , xm), b akhmlulh sh bzrizan

|Pm(x)| ≤ 9(m + 9)!

N m+9|ςm+9(x)|, N m+9 > nixWy9,y0^

|e (m+9)|

2


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9: 0. BMTHPZACIKBLI

0.9 Iga lh

cg(x) >

ml>:l>g

x ∐ xlxg ∐ xl , g > :, . . . , m ,

dagizitb di lh m

g>:

cg(x) ≩ 9.

Phsˏhmlh< \`ao lhdbmstvhmastb Ciorimoh-avao bmthrpacikbamao pacbmani schdb di lh zi svigb pacbman Z (x) sthphmi mh vhḱho ad m

Z (x)

≩

m

g>: cg(x)Z (xg).Tvr -dhmlh schdb gidi sh stivb di lh Z (x) ≩ 9.

0.0 Mipbsitb Ciorimoh-av bmthrpacikbamb pacbman m-tao sthphmi ( m ≨ 0) zi eumgkblu

e (x) < x ∐← 9 +m

g>:

xgx ∐ xg

ς(x)

ς(xg),

odh lh

ς(x) >

mb>:

(x ∐ xb), xb 3 xl zi b 3 l.

Phsˏhmlh< Ciorimoh-av bmthrpacikbamb pacbman eumgkblh gali lh sini pacbman lhbdhmtbkˏim eumgkblb ugacbga lh sthphm pacbmani vhḱb bcb lhdmig sthphmu eumgkblh. [taoi lh

mg>:

xgx ∐ xg

ς(x)

ς(xg) ≩ x,

pi lh e (x) > 9 + x b mlhm bmthrpacikbamb pacbman lh

Cm(x) >m

g>:

9 + xg

x ∐ xgς(x)

ς(xg).

0.; Gamstrubsitb Ciorimoh-av bmthrpacikbamb pacbman zi eumgkblu e (x) > ∜

x, iga su kˏvaravb bmthrpacikblh x: > 9::, x9 > 909 b x0 > 944. Bzrikˏumitb vrhdmastda`blhmao pacbmani u tikˏgb V > 998 b akhmbtb orhsgu rhzuctiti garbsthḱb earnucu zi orhsˏgu bmthrpacikblh. Xparhdbtb si stvirman orhsˏgan.


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0.9. CIOPIMOH-AU ZACBMAN 99

Phsˏhmlh< \iditi ti`hci

x 9:: 909 944

e (x) 9: 99 90

dhmbsˏh Ciorimoh-av bmthrpacikbamb pacbman druoao sthphmi

C0(x) > (x ∐ 909)(x ∐ 944)(9:: ∐ 909)(9:: ∐ 944) 9: +

(x ∐ 9::)(x ∐ 944)(909 ∐ 9::)(909 ∐ 944) 99

+ (x ∐ 9::)(x ∐ 909)(944 ∐ 9::)(944 ∐ 909) 90 > ∐

0

09080 x0 +

9484

09080 x +

6?90:

09080.

Orhsˏgi bmthrpacikblh lh

|e (998)∐C0(998)| ≤ 9;!

N ;|(998∐9::)(998∐909)(998∐944)| > :.95 ¿9:∐0 ≆ 9:∐;,

lhr lh N ; > nixW9::,944^ |e (x)| > :.;?8 ¿ 9:∐8.[ a`zbran mi rizcbkˏbtb rhd vhcbkˏbmh gahkblhmiti pacbmani C0(x), nazˏh sh akˏhgbvitb

vhcbgi orhsˏgi rhzuctiti iga gahkblhmtb pacbmani mbsu bzrikˏumitb si zidavacliviluḱantikˏmasˏḱu, tl. iga mbsu ditb si davaclmbn `ralhn dhkbnici. Adrh -dulhna dapusˏthmuipsacutmu orhsˏgu ∅ gahkblhmiti pacbmani, tiga di lh rikˏumsgi orhsˏgi rhzuctiti∅

C0(998)

> 9:∐4 (di `b ami `bci zimhnirclbvi u admasu mi orhsˏgu nhtadh),

∅9980 + ∅ 998 + ∅ > 9:∐4 >⇐ ∅ > :.?8 ¿ 9:∐6.Digch, gahkblhmth pacbmani trh`i bzrikˏumitb `ir mi asin dhkbnici (tidi lh ipsa-

cutmi orhsˏgi tbf `ralhvi :.8 ¿ 9:∐6),C0(x) > ∐:.::::2499x0 + :.:56409?9x + 4.:22;?666.

Phzuctit zipbsim sina sbourmbn kberini, s a`zbran mi prakhmlhmu tikˏmast, lh

C0(998) > 9:.?0;0?: ¿ ¿ ¿ > 9:.?0;∜

998 > 9:.?0;6: . . .

0.4 Mhgi su hgvbdbstimtman ti`hcan si garigan f dith vrhdmastb eumgkblh e (x)ziagruoclhmh mi r dhkbnici. Cbmhirman bmthrpacikblan sh panaḱu ti`hch rikˏumi e (x). Mhgi su ipskbsh u ti`hcb dith tikˏma, ipskbsi x lh ziagruoclhmi mi s dhkbnici,i bzrikˏumiti vrhdmast e (x) ziagruoclhmi mi t dhkbnici. Iga lh e (x) mhprhgbdmi eumgkbli mi ti`hcbriman bmthrvicu i δ taticmi orhsˏgi bmthrpacikblh, pagizitb di lh

|δ | ≤ 96

N 0f0 + 8N 9 ¿ 9:∐s∐9 + 8 ¿ 9:∐r∐9 + 8 ¿ 9:∐t∐9,

odh lh N 9 > nix

Wx:,xm^|e (x)|, N 0 > nix

Wx:,xm^|e (x)|.


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90 0. BMTHPZACIKBLI

Phsˏhmlh< Bmthrpacikbamb pacbman prvao sthphmi eumgkblh e (x) adrh -dhm kˏvaravbnixg

b xg+9

u tikˏgb x lh

C9(x) > x ∐ xg+9xg ∐ xg+9 e (xg) +

x ∐ xgxg+9 ∐ xg e (xg+9).

Orhsˏgi nhtadh (cbmhirmh bmthrpacikblh) sh akhmlulh bzrizan

|P9(x)| > |e (ν )|0

|(x ∐ xg)(x ∐ xg+9)| ≤ f0

6 N 0.

Akhmbna sidi rikˏumsgu orhsˏgu. Urhdmastb eumgkblh e su dith si ipsacutman orhsˏganI(e ) > 8 ¿ 9:∐r∐9, i I(x) > 8 ¿ 9:∐s∐9. Cbmhirmi akhmi ipsacutmh orhsˏgh vrhdmastbpacbmani C9 u tikˏgb x lh amdi

I(C9(x)) > ∀C9(x)

∀x I(x) + ∀C9(x)

∀e g I(e ) + ∀C9(x)

∀e g+9 I(e )>

e g+9 ∐ e gfI(x) +

xg+9 ∐ x

f +

x ∐ xgf

I(e ) >

|∅e g|f

I(x) + I(e )

Bz vhzh gamikˏmh rizcbgh b bzvadi eumgkblh ∅e g > f e (ν ), ν ∆ Wxg, xg+9^ schdb di lhI(C9(x)) > |e (ν )|I(x) + I(e ) ≤ 8N 9 ¿ 9:∐s∐9 + 8 ¿ 9:∐r∐9.

Taticmi orhsˏgi rhzuctiti cbmhirmh bmthrpacikblh lh z`br orhsˏgh bmthrpacikblh, rikˏumsghorhsˏgh b orhsˏgh ziagruoclbvimli gamikˏmao rhzuctiti,

δ > |P9(x)| + I(C9(x)) + 8 ¿ 9:∐t∐9,sˏta dagizulh tvr -dhmlh ziditgi.

0.8 Adrhdbtb u bmthrvicu (9, 9.0) tikˏgh x: b x9 tiga di cbmhirmi bmthrpacikbli zi eumgkblu e (x) > cm x graz tikˏgh 9, x:, x9 b 9.0 bni orhsˏgu mh vhḱu ad 8 ¿ 9:∐4.

Phsˏhmlh< Orhˏsgi Ciorimoh-avao bmthrpacikbamao pacbmani prvao sthphmi zieumgkblu cm x gamstrubsimao graz tikˏgh i b ` lh

P9(x) > ∐ 90ν 0

(x ∐ i)(x ∐ `), ν, x ∆ (i, `).

Giga lh 9/ν 0 ≤ 9/i0 b |(x ∐ i)(x ∐ `)| ≤ (` ∐ i)0/4, ta lh

|P9(x)

| ≤ 9

6 `

i ∐9

0

.

Bz zifthvi di lh |P9(x)| ≤ 8 ¿ 9:∐4 schdb di lh `/i ≤ 9.:5;08. X prvan bmthrvicu lhi > 9, pi lh ` > x: > 9.:5;08. x: lh pakˏhtig schdhḱho bmthrvici, i mlhoav gril lhx9 > 9.:5;08 ¿x: > 9.9;:42. [chdhḱb kˏvar bmthrpacikblh `b `ba 9.:5;08 ¿x9 > 9.0:922,sˏta lh vhḱh ad 9.0. [taoi lh b u bmthrvicu (x9, 9.0) tikˏmast zidavaclhmi, pi lh

x: > 9.:5;08, x9 > 9.9;:42.


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0.9. CIOPIMOH-AU ZACBMAN 9;

0.5 Mhgi lh eumgkbli

e (x) > x:

sbm0 t dt

ti`hcbrimi mi bmthrvicu x ∆ W:, 9^ u rivmanhrma risparh -dhmbn tikˏgini si gariganf. Zad prhtpastivgan di su vrhdmastb eumgkblh uzhth si davaclma sbourmbf kbeiri di sh nazˏh zimhnirbtb orhsˏgi ziagruoclbvimli, adrhdbtb ι gia eumgkblu ad f b θ tiga di orhsˏgi cbmhirmh bmthrpacikblh mh prhnisˏulh ditu vrhdmast θ mi bmthrvicu Wι, 9^.Gigva f trh`i di `udh di `b sh pastboci tikˏmast :.::8 zi ι > :.91

Phsˏhmlh< Giga lh e (x) > sbm0 x b e (x) > sbm(0x), ta lh

nixWι,9^

|e (x)| >

9, ι ∆ W:, ϊ/4^sbm (0ι), ι ∆ Wϊ/4, 9^

lhr lh sbm (0x) ristuḱi eumgkbli gidi x ∆ W:, ϊ/4^ b apidiluḱi gidi x ∆ Wϊ/4, 9^.[taoi lh orhsˏgi cbmhirmh bmthrpacikblh

|P9(x)| ≤

f0/6, ι ∆ W:, ϊ/4^(f0/6)sbm(0ι) ι ∆ Wϊ/4, 9^ x ∆ Wι, 9^.

X scukˏilu gidi orhsˏgi bmthrpacikblh P9 zivbsb ad ι, orhsˏgi ḱh b̀tb nimli ad θ iga lh

ι ≨ 90

irksbm 6θ

f0.

Gidi lh ι > :.9 orhsˏgi mh zivbsb ad ι b `bḱh nimli ad :.::8 iga lh

f0/6 ≤ :.::8, tl. f ≤ :.0.

0.? Iga uznhna much Kˏh`bsˏhv-clhvao pacbmani T m+9(x) > kas((m + 9) irkkas x) zi kˏvaravh xg u Ciorimoh-aval bmthrpacikbamal earnucb

Cm(x) >m

g>:

cg(x)e (xg), cg(x) >m

b>:b>g

x ∐ xbxg ∐ xb ,

dagizitb di lh

cg(x) >

(

∐9)g 9 ∐

x0g

m + 9

T m+9(x)

x ∐ xg ,b akhmbtb orhsˏgu bmthrpacikblh.

Phsˏhmlh< Iga su kˏvaravb bmthrpacikblh much Kˏh`bsˏhv-clhvao pacbmani T m+9(x),

xg > kas 0g + 9

0(m + 9)ϊ, g > :, . . . , m ,


24/437

94 0. BMTHPZACIKBLI

amdi lh

ςm+9(x) >

mb>:

(x ∐ xb) > 09∐(m+9)T m+9(x).

Giga lh

T m+9(x) >

kas

(m + 9) irkkas x

> m + 9∜

9 ∐ x0 sbm

(m + 9) irkkas x

,

b

T m+9(xg) > m + 9

9 ∐ x0gsbm(0g + 9)

ϊ

0 >

(∐9)g(m + 9) 9 ∐ x0g

,

tvr -dhmlh mhpasrhdma schdb lhr lh

cg(x) > ςm+9(x)(x ∐ xg)ςm+9(xg)

> T m+9(x)(x ∐ xg)T m+9(xg)

.


|e (x) ∐ Cm(x)| ≤ 9(m + 9)!

nixWx:,xm^

|e (m+9)(ν )| nixWx:,xm^

|0∐mT m+9(ν )|

> 9

0m(m + 9)! nixWx:,xm^

|e (m+9)(ν )|.

0.6 Garbsˏḱhmlhn bmthrpacikbamao pacbmani druoao sthphmi mi bmthrvicu W∐9, 9^ bzri-kˏumitb h:.2 iga su

i) kˏvaravb rivmanhrma risparh -dhmb,`) kˏvaravb much adoaviri luḱho Kˏh`bsˏhv-clhvao pacbmani.

Akhmbtb orhsˏgu bmthrpacikblh u a`i scukˏili.

Phsˏhmlh< Zacbman druoao sthphmi adrh -dhm lh vrhdmastbni eumgkblh u trb tikˏgh.i) X scukˏilu rivmanhrmao risparhdi kˏvaravi, kˏvaravb su x: > ∐9, x9 > : b x0 > 9,

x ∐9 : 9e (x) :.;5?6?2 9.:::::: 0.?96060

Trizˏhmi vrhdmast lh

C0(:.2) > :.2(∐:.9)(∐9)(∐0) :.;5?6?2 + 9.2(∐:.9)∐9 9.:::::: + 9.2(:.2)0 0.?96060

> 0.42?8?5,

bzrikˏumiti si orhsˏgan

| h:.2 ∐ C0(:.2)| ≤ h;!

|9.2 ∟ :.2 ∟ (∐:.9)| > :.6 ¿ 9:∐9.


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\idrzˏiviluḱb sina sbourmh kberh, rhzuctit lh

h:.2 > 0.8: ¹ :.:6,

prb kˏhnu lh 8 sbourmi kberi u sˏbrhn snbscu.

`) Iga su kˏvaravb bmthrpacikblh much Kˏh`bsˏhv-clhvao pacbmani T 0(x) > kas(0 irkkas x),ti`hci lh

x ∐:.655:08 : :.655:08e (x) :.40:50: 9 0.;??44;

Trizˏhmi vrhdmast lh

C0(:.2) > 0.44??42,

bzrikˏumiti si orhsˏgan

| h:.2 ∐ C0(:.2)| ≤ N ;;!

|9.?55:08 ∟ :.2 ∟ :.:;;2?8)| > :.00 ¿ 9:∐9,

odh lh N ; > nixWx:,x0^ | hx| > hx0 > 0.;??44;.\idrzˏiviluḱb sina sbourmh kberh, rhzuctit lh

h:.2 > 0.48 ¹ :.:0,

prb kˏhnu lh 4 sbourmi kberi u uzˏhn snbscu. Tikˏmi vrhdmast lh h :.2 > 0.4825....

0.2 i) Adrhdbtb nbmbnicmb sthphm m bmthrpacikbamao pacbmani Z m(x) galb mi bm-thrvicu W:, 9^ ipragsbnbri eumgkblu e (x) > h

∐x si tikˏmaˏsḱu 9:

∐5, iga su kˏvaravb

bmthrpacikblh much Kˏh`bsˏhv-clhvao pacbmani. Kˏh`bsˏhv-clhv pacbman sthphmi m mi bm-thrvicu W∐9, 9^ lh pacbman T m(x) > kas(m irkkas x).`) Iga sh garbstb dha pa dha cbmhirmi bmthrpacikbli si kˏvaravbni xb > b/m,b > :, . . . , m, adrhdbtb nbmbnicmu vrhdmast zi m gali a`hz`h -dulh tikˏmast ipragsb-nikblh 9:∐5.

Phsˏhmlh< i) Kˏh`bsˏhv-clhv pacbman T m(t) > kas(m irkkas t) lh dhmbsim zi t ∆W∐9, 9^ b much su nu tikˏgh tg > kas((0g + 9)ϊ/(0m)), g > :, . . . , m ∐ 9. Cbmhirmansnhman t > 0x∐9 avil pacbman dhmbsˏhna mi bmthrvicu W:, 9^, T W:,9^m (x) > T m(0x∐9),i much su nu xg > (9 + tg)/0. Marnbrimlhn avao pacbmani (nmaˏzhmlhn rh-

kbprakˏman vrhdmasˏḱu gahkblhmti uz milvbsˏb sthphm) da`blina pacbman T W:,9^m (x) >

09∐

0m

T

W:,9^

m (x).Bmthrpacikbamb pacbman sthphmi m adrh -dhm lh vrhdmastbni u (m + 9)-an kˏvarubmthrpacikblh. Iga su kˏvaravb bmthrpacikblh much Kˏh`bsˏhv-clhvao pacbmani sthphmi

m + 9 mi bmthrvicu W:, 9^, xb > 90

9 + kas (0b+9)ϊ

0(m+9)

, b > :, . . . , m, pacbman

ςm+9(x) >m

b>:

(x ∐ xb) ≩ T W:,9^m+9(x) ≩ 0∐9∐0mT m+9(0x ∐ 9),


26/437

95 0. BMTHPZACIKBLI

Kˏh`bsˏhv-clhv pacbman T m(x) > kas(m irkkas x) aorimbkˏhm lh si |T m(x)| ≤ 9. Giga lh

nixx∆W:,9^

|e (g)(x)| > 9, b nixx∆W:,9^

|T W:,9^m+9(x)| > 0∐9∐0m,

orhsˏgi bmthrpacikblh pacbmanan sthphmi m si avbn bz`aran kˏvaravi lh

|e (x) ∐ pm(x)| ≤ 9(m + 9)!

9

00m+9,

b zidavaclivi ziditu tikˏmast zi m ≨ 8.`) Giga lh

|e (x) ∐ pb9(x)| ≤ 9

0 nixx

∆W b∐9

m , bm

^ x ∐ b ∐ 9

m )(x ∐ b

m≤ 9

6m0,

nbmbnicmb `ral padbmthrvici zi pastbzimlh zifthvimh tikˏmastb lh m > ;84.

0.9: Mhgi lh Cm(x) > m

g>: cg(x) e (xg) Ciorimoh-av bmthrpacikbamb pacbman sthphmi m eumgkblh e (x) b

f > nix:≤g≤m∐9

(xg+9 ∐ xg), f > nbm:≤g≤m∐9

(xg+9 ∐ xg).

Dagizitb di vizˏb akhmi m

g>:

|cg(x)| ≤ 0m

f

f

m.

Phsˏhmlh< [ a`zbran mi prhtpastivgu di lh x b+9 ∐ xb ≨ f schdb di lh zi b = l

xb ∐ xl > (xb ∐ xb∐9) + ¿ ¿ ¿ + (xl+9 ∐ xl) ≨ (b ∐ l)f,pi lh

|(xg ∐ x:) . . . (xg ∐ xg∐9) (xg ∐ xg+9) . . . (xg ∐ xm)| ≨ g! (m ∐ g)! fm.Diclh, prhtpastivbna di x ∆ Wxl∐9, xl^, pi lh x: 3 ¿ ¿ ¿ 3 xl∐0 3 xl∐9 3 x 3 xl 3xl+9 3 ¿ ¿ ¿ 3 xm adigch schdb di lh

|x ∐ xl∐9| ≤ f, |x ∐ xl∐0| ≤ 0f, ¿ ¿ ¿ |x ∐ x:| ≤ lf,

|x ∐ xl | ≤ f, |x ∐ xl+9| ≤ 0f, ¿ ¿ ¿ |x ∐ xm| ≤ (m ∐ l + 9)f,tl.

|(x ∐ x:) . . . (x ∐ xg∐9) (x ∐ xg+9) . . . (x ∐ xm)| ≤ l! (m ∐ l + 9)! fm.X scukˏilu di lh l > 9 bcb l > m lh

|(x ∐ x:) . . . (x ∐ xg∐9) (x ∐ xg+9) . . . (x ∐ xm)| ≤ m!fm,


27/437

0.9. CIOPIMOH-AU ZACBMAN 9?

sˏta lh b nigsbnicmi vrhdmast avao prabzvadi `bca odh di sh u bmthrvicu Wx:, xm^micizb irounhmt x.

Gamikˏma lh

|cg(x)| > (xg ∐ x:) . . . (xg ∐ xg∐9) (xg ∐ xg+9) . . . (xg ∐ xm)(x ∐ x:) . . . (x ∐ xg∐9) (x ∐ xg+9) . . . (x ∐ xm)

≤ m!g!(m ∐ g)! fm

fm,

pi lh

mg>:

|cg(x)| ≤m

g>:

m!

g!(m ∐ g)!

f

f

m>

f

f

m mg>:

m

g

> 0m

f

f

m.

0.99 Garbsˏḱhmlhn Ciorimoh-avh bmthrpacikbamh earnuch dagizitb di lh g

l>∐g(∐9)l

0g

g + l

e (x + lf) > (∐9)gf0ge (0g)(t), x ∐ gf ≤ t ≤ x + gf.

Phsˏhmlh< Ciorimoh-av bmthrpacikbamb pacbman zi eumgkblu e (x) adrh -dhm kˏvara-vbni x ∐ g f , . . . , x ∐ f, x + f , . . . , x + gf lh

C0g∐9(x) >g

l>∐gl>:

gb>∐gb>:,l

x ∐ (x + bf)

gb>∐gb>:,l

(x + lf) ∐ (x + bf)

e (x+ lf)

tl.

C0g∐9(x) > ∐g

l>∐gl>:

(∐9)l(g!)0(g + l)!(g ∐ l)! e (x + lf)


e (x) ∐ C0g∐9(x) > e (0g)(t)

(0g)!

gb>∐gb>:

x ∐ (x + bf), x ∐ gf ≤ t ≤ x + gf,

pi lh, uzbniluḱb u a`zbr bzvhdhmb bzriz zi C0g∐9(x),

e (x) +g

l>∐gl>:

(∐9)l(g!)0(g + l)!(g ∐ l)! e (x + lf) >

(∐9)g(g!)0(0g)!

f0g e (0g)(t).

Nmazˏhmlhn paschdmlho bzrizi si (0g)!/(g!)0 schdb tvr -dhmlh.


28/437

96 0. BMTHPZACIKBLI

0.90 Mhgi su _ g > hxp 0ϊgm+9

Ĺ, g > :, . . . , m , garhmb (m + 9)-ao rhdi lhdbmbkh,

_ m+9

g > 9. Garbsthḱb Ciorimoh-av bmthrpacikbamb pacbman adrh -dhm kˏvaravbni _ g,dagizitb di lh zi prabzvaclim pacbman Z m(x) > i: + i9x + ¿ ¿ ¿ + imxm sthphmi m

(9) i: > 9

m + 9

mg>:

Z m(_ g),

(0) im > 9

m + 9

mg>:

_ g Z m(_ g).

Mi asmavu prhtfadmao dagizitb di lh zi sviga n > 9, . . . , m

(;)m

g>: _ ng > :.

Phsˏhmlh< Ciorimoh-av bmthrpacikbamb pacbman zi eumgkblu Z m(x) adrh -dhm kˏvara-vbni _ g lh

Cm(x) >m

g>:

ςm+9(x)

(x ∐ _ g)ςm+9(_ g) Z m(_ g),

odh lh ςm+9(x) >m

g>:(x∐_ g) > xm+9 ∐9 b ς m+9(_ g) > (m+ 9)_ mg > (m+9)/_ g lhr lh _ m+9g > 9. Eumgkbli gali sh bmthrpacbsˏh Z m(x) lh tiga

-dh pacbman sthphmi m, piz`ao lhdbmstvhmastb pacbmani sthphmi m adrh -dhmao svalbn vrhdmastbni u (m +9)-alrizcbkˏbta l tikˏgb schdb di lh Z m(x) ≩ Cm(x). [taoi lh

Z m(x) > xm+9

∐ 9m + 9

mg>:

_ gx ∐ _ g Z m(_ g).

\i x > : lh

i: > Z m(:) > 9

m + 9

mg>:

Z m(_ g),

i gidi x ← ∝

im > cbnx←∝

Z m(x)

xm >

9

m + 9 cbnx←∝

mg>:

xm+9 ∐ 9xm(x ∐ _ g) _ g Z m(_ g)

> 9m + 9

mg>:

_ g Z m(_ g).

[idi lh mi asmavu bzrizi (9)

(m + 9)i: >m

g>:

Z m(_ g) > (m + 9)i: + i9

mg>:

_ g + i0

mg>:

_ 0g + ¿ ¿ ¿ + imm

g>:

_ mg ,


29/437


tl.

i9

mg>:

_ g + i0

mg>:

_ 0g + ¿ ¿ ¿ + imm

g>:

_ mg > :.

Giga lh Z m(x) prabzvaclim pacbman, paschdmli lhdmigast vizˏb zi svigb bz`ar gahkblhmiti i:, . . . , im, i ta lh naouḱh sina iga lh

mg>:

_ ng > :, n > 9, . . . , m .

NITCI@

Ciorimoh-av pacbman< p > pacyebt(x, y, m)sthphm m lh zi lhdim nimlb ad duzˏbmh vhgtari x b y

Ciorimoh-av bmthrpacikbamb pacbman lh dbrhgtma naouḱh gamstrubsitb garbsˏḱhmlhneumgkblh p > pacyebt(x, y, m). Irounhmtbni x b y lh prhdstivclhmi ti`hcibmthrpacikblh, dag lh m sthphm rhzuctuluḱho pacbmani. Eumgkbli vrsˏb bmthrpacikbluu scukˏilu di lh m zi lhdim nimlb ad dbnhmzblh vhgtari x, admasma y. Zrbgizˏbnagarbsˏḱhmlh avh eumgkblh graz schdhḱb ziditig.

0.9; Garbˏśkhmlhn nitci`-i, bzvrsˏbtb bmthrpacikblu eumgkblh 99+08x0 mi bmthrvicuW∐9, 9^ Ciorimoh-avbn bmthrpacikbambn pacbmanan sthphmi 2.

i) Bmthrpacikblu bzvsˏbtb mi hgvbdbstimtmal nrhzˏb.

`) \i kˏvaravh bmthrpacikblh uzhtb much Kˏh`bsˏhv-clhvbf pacbmani.

Phsˏhmlh<

% Eumgkbli gali sh bmthrpacbshe > bmcbmh(‒9./(9+08*x.]0)‒)7

% @ral kvaravi bmthrpacikblh

m > 9:7

% Nrhzi zi bskrtivimlhV > cbmspikh(-9,9)7

% Hgvbdbstimtmi nrhzi

xhqd > cbmspikh(-9, 9, m)7% Nrhzi muci Kh`bshv-clhvbf pacbmani

xkh` > kas((0*(m-9 ehvic(e, xkh`)7

% Ursbna bmthrpacikblu

Chqd > pacyebt(xhqd, yhqd, m-9)7

Ckh` > pacyebt(xkh`, ykh`, m-9)7

% Ursbna bskrtivimlh eumgkblh, pacbmani b kvaravi nrhzh


30/437

0: 0. BMTHPZACIKBLI

% zi scukil bmthrpacikblh mi hgvbdbstimtmal nrhzb

su`pcat(9,0,9)7 orbd am7

pcat(V, e(V), ‒`‒, V, pacyvic(Chqd, V), ‒r‒, xhqd, yhqd, ‒ra‒)tbtch(‒e(x) > 9/(9+08*x]0) - hgvbdbstimtmi nrhzi‒)7

chohmd(‒e(x)‒, ‒Chqd(x)‒)7

% Ursbna bskrtivimlh eumgkblh, pacbmani b kvaravi nrhzh% zi scukil bmthrpacikblh mi Kh`bshv-clhval nrhzb

su`pcat(9, 0, 0)7 orbd am7

pcat(V, e(V), ‒`‒, V, pacyvic(Ckh`, V), ‒r‒, xkh`, ykh`, ‒ra‒)

tbtch(‒e(x) > 9/(9+08*x]0) - much Kh`bshvclhvbf pacbmani‒)7chohmd(‒e(x)‒,‒Ckh`(x)‒)7

Phzuctit ridi sgrbpti lh prbgizim mi scbkb 0.9.

∐9 ∐:.8 : :.8 9∐:.4

∐:.0

:

:.0

:.4

:.5

:.6

9

e(x) > 9/(9+08x0) ∐ hgvbdbstimtmi nrhzi

e(x)

Chqd(x)

∐9 ∐:.8 : :.8 9∐:.4

∐:.0

:

:.0

:.4

:.5

:.6

9

e(x) > 9/(9+08x0) ∐ much Kh`bshvclhvbf pacbmani

e(x)

Ckh`(x)

[cbgi 0.9< Ciorimoh-avi bmthrpacikbli eumgkblh 99+08x0 . (i) hgvbdbstimtmi nrhzˏi,

(`) much Kˏh`bsˏhv-clhvbf pacbmani

Zrbnhtbna di uri -dhmb prbnhr pagizulh giga sh askbcavimlh bmthrpacikbamao pacb-mani b orhsˏgi naou zmikˏilma snimlbtb ugacbga sh zi bmthrpacikblu garbsth uprivamuch Kˏh`bsˏhv-clhvbf pacbmani.

Mi avan nhstu mivadbna ganpchtim gad eumgkblh gali vrsˏb Ciorimoh-avu bm-thrpacikblu. Kbcl lh di graz prbnhr prbgiˏzhna mikˏbm gadbrimli galb lh pazˏhclim zi

sbsthn nitci`.

% BMTHPZ9CIOPIMOH bmthrpacikbli Ciorimzhvbn bmthrpacikbambn pacbmanan

% C > bmthrp9Ciorimoh(x,y)

% Ti`cbki bmthrpacikblh nari di sidrzb `ir lhdim kvar b svb kvaravb narilu `btb

% rizcbkbtb%

% Irounhmtb eumgkblh su


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% x - m-dbnhmzbamb vhgtar galbn sh zidilu kvaravb bmthrpacikblh

% y - m-dbnhmzbamb vhgtar galbn sh zidilu vrhdmastb eumgkblh u kvaravbni

% Eumgkbli vriki <% C - gahebkblhmth Ciorimohavao bmthrpacikbamao pacbmani

eumktbam C>bmthrp9ciorimoh(x,y)

% Zravhri garhgtmastb irounhmiti

be mirobm~>0 | nbm(sbzh(x))~>9 | nbm(sbzh(y))~>9 | chmotf(x)~>chmotf(y)

hrrar(‒Akhgbvimi su 0 irounhmti gali prhdstivclilu ti`cbku bmthrpacikblh‒)

hmd

be (~icc(ebmbth(x)) | ~icc(ebmbth(y)) | ~icc(bsrhic(x)) | ~icc(bsrhic(y)))

hrrar(‒Ti`cbki bmthrpacikblh nari di sidrzb bsgclukbva gamikmh, rhicmh `ralhvh‒)

hmd

be (chmotf(x)~>chmotf(umbquh(x)))

hrrar(‒[vb kvaravb bmthrpacikblh narilu `btb rizcbkbtb‒)

hmd

% Bmthrpacikbamb pacbman oridbna dbrhgtma mi asmavu dhebmbkbamh

% gamstrugkblh. Phzuctulukb pacbman prhdstivcli z`br pacbmani pb galb

% bnilu much u svbn kvaravbni bmthrpacikblh asbn u xb u galhn bnilu% vrhdmast yb.

% @ral kvaravi bmthrpacikblh

m > chmotf(x)7

% vhgtar C prhdstivcli gahebkblhmth Ciorimzhvao b.p.

C>zhras(9, m)7

% Zracizbna graz svh kvaravh bmthrpacikblh

ear b>9pacy(x(9 b))7

% mi rhzuctulukb pacbman dadilhna pacbman pb > p*yb/p(xb)

C> C + p*y(b)/pacyvic(p,x(b))7

hmd

Gia sˏta sh vbdb, vhḱbmi gadi adcizb mi stimdirdmu dagunhmtikblu. \itbn schdb`cag u ganh sh vrˏsb bskrpmi pravhri garhgtmastb irounhmiti eumgkblh. [ini bn-pchnhmtikbli nhtadh zi bmthrpacikblu sh bmthmzbvma `izbri mi eumgkbamicmastb vhḱoatavbf rutbmi zi rid si pacbmanbni b mi caobkˏgan bmdhgsbrimlu vhgtari. Gclukˏmanhsta u gamstrugkblb bmthrpacikbamao pacbmani lh bzoridmli pacbmani p > l>b(x∐xl). Avb pacbmanb sh oridh garbsthḱb eumgkblu pacy galal sh gia irounhmtb prhdiluvhgtarb galb sidrzˏb tikˏgh xl zi l > b. \i earnbrimlh avigvbf vhgtari garbsˏḱhma lh caobkˏga bmdhgsbrimlh admasma gamstrugkbli x(9 b). Bzriz 9


32/437

00 0. BMTHPZACIKBLI

0.0 Mhwtam-av pacbman si padhclhmbn rizcbgini

Zadhclhmh rizcbgh<

e Wxg^ > e (xg) rhdi :

e Wxg, xg+9^ > e (xg+9) ∐ e (xg)

xg+9 ∐ xg rhdi 9

e Wxg, xg+9, xg+0^ > e Wxg+9, xg+0^ ∐ e Wxg, xg+9^

xg+0 ∐ xg rhdi 0¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿

e Wxg, . . . , xg+m^ > e Wxg+9, . . . , xg+m^ ∐ e Wxg, . . . , xg+m∐9^

xg+m∐

xgrhdi m

e Wx:, . . . , xm^ >m

b>:

e (xb)ml>:l>b

(xb ∐ xl)

Mhwtam-av bmthrpacikbamb pacbman si padhclhmbn rizcbgini lh

Cm(x) > e (x:) + e Wx:, x9^(x ∐ x:) + e Wx:, x9, x0^(x ∐ x:) (x ∐ x9) + ¿ ¿ ¿+ e Wx:, . . . , xm^(x ∐ x:) . . . (x ∐ xm∐9)

Orhsˏgi bmthrpacikblh bzrizˏhmi panaḱu padhclhmbf rizcbgi

Pm(x) > e Wx, x:, . . . , xm^ςm+9(x)

0.94 Mhgi lh Cm(x) > m

g>: cg(x) e (xg) Ciorimoh-av bmthrpacikbamb pacbman sthphmi m eumgkblh e (x). Dagizitb di lh

c:(x) > 9 + x ∐ x:x: ∐ x9 +

(x ∐ x:)(x ∐ x9)(x: ∐ x9)(x: ∐ x0) + ¿ ¿ ¿ +

(x ∐ x:) . . . (x ∐ xm∐9)(x: ∐ x9) . . . (x: ∐ xm) .

Phsˏhmlh< Eumgkbli c:(x) lh pacbman m-tao sthphmi tigiv di lh c:(x:) > 9 bc:(xb) > :, b = :. Zadhclhmh rizcbgh avh eumgkblh su

c:Wx:, x9^ > c:(x9) ∐ c:(x:)

x9 ∐ x: > 9

x: ∐ x9

c:Wx:, x9, x0^ > 9x0 ∐ x: c:(x0) ∐ c:(x9)x0 ∐ x9 ∐ c:(x9) ∐ c:(x:)x9 ∐ x: > 9(x: ∐ x9)(x: ∐ x0)

¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿

c:Wx:, . . . , xm^ >m

b>:

m

l>:l>b

9

xb ∐ xl

c:(xb) > m

l>9

9

x: ∐ xl


33/437

0.0. MH_TAM-AU ZACBMAN [I ZADHCLHMBN PI\CBGINI 0;

Mhwtam-av bmthrpacikbamb pacbman sthphmi m zi eumgkblu c:(x) bdhmtbkˏim lh sinaleumgkblb lhr lh pacbman sthphmi m si (m + 9)-bn uscavan bmthrpacikblh lhdbmstvhmaadrh -dhm. [taoi lh

c:(x) > c:(x:) + (x ∐ x:)c:Wx:, x9^ + ¿ ¿ ¿ + (x ∐ x:) . . . (x ∐ xm∐9)c:Wx:, . . . , xm^,

adigch schdb tvr -dhmlh.

0.98 Bzrikˏumitb e (9.95) zi eumgkblu ditu ti`hcan

x 9.90?8 9.98:; 9.9?;8 9.92?0

e (x) :.992?9 :.9;28? :.982;9 :.9?2:0

Phsˏhmlh< Trizˏhmu vrhdmast ḱhna bzrikˏumitb panaḱu Mhwtam-avao bmthrpaci-kbamao pacbmani si padhclhmbn rizcbgini

C;(x) > e (x:) + (x ∐ x:)e Wx:, x9^ + (x ∐ x:)(x ∐ x9)e Wx:, x9, x0^+ (x ∐ x:)(x ∐ x9)(x ∐ x0)e Wx:, x9, x0, x;^

adrh -dhmao schdhḱan ti`hcan padhclhmbf rizcbgi

x e e W9^ e W0^ e W;^

9.90?8 :.992?9:.6?9:8;

9.98:; :.9;28? ∐:.4;62;8:.68:650 :.4962?69.9?;8 :.982;9 ∐:.4:2?;0

:.6;95459.92?0 :.9?2:0

Phd padhclhmh rizcbgh lh mizmikˏhm `ralhn u uocistal zioridb u ziocivclu ti`hch, ikˏvaravb galbni lh ohmhrbsimi adrh -dhmb su pacazˏilhn vrhdmastb padhclhmh rizcbgh uti`hcb.

Gahkblhmtb pacbmani su prvh padhclhmh rizcbgh u svigal gacamb ti`hch. Tiga shda`bli di lh

e (9.95) ≆ C;(9.95) > :.94?66.

0.95 Dagizitb di sh orhsˏgi bmthrpacikblh eumgkblh e (x) > 9/(9 + x) Ciorimoh-avbnbmthrpacikbambn pacbmanan Cm(x) adrh -dhmbn kˏvaravbni xg > g, g > :, . . . , m , nazˏh akhmbtb bzrizan 99 + x ∐ Cm(x)

≤ 9(m + 9)! |x(x ∐ 9) . . . (x ∐ m)|.


34/437

04 0. BMTHPZACIKBLI

Phsˏhmlh< Dagizˏbna bmdugkblan di lh zi ditu eumgkblu

e Wx:, . . . , xn^ > (∐9)n(x: + 9) . . . (xn + 9)

.

\i n > 9 lh

e Wx:, x9^ > 9

x9 ∐ x:

9

x9 + 9 ∐ 9

x: + 9

>

∐9(x: + 9) (x9 + 9)

.

Mhgi tvr -dhmlh vizˏb zi n > m,

e Wx:, . . . , xm^ > (∐9)m

(x: + 9) . . . (xm + 9).

Dagizˏbna di vizˏb zi n > m + 9<

e Wx:, . . . , xm+9^ > 9

xm+9 ∐ x:

e Wx9, . . . , xm+9^ ∐ e Wx:, . . . , xm^

> 9

xm+9 ∐ x:

(∐9)m

(x9 + 9) . . . (xm+9 + 9) ∐ (∐9)

m

(x: + 9) . . . (xm + 9)

> (∐9)m+9

(x: + 9) . . . (xm+9 + 9).

Iga sidi u earnucb zi orhsˏgu Ciorimoh-avao bmthrpacikbamao pacbmani

Pm(x) > e (x) ∐ Cm(x) > e Wx:, . . . , xm, x^ (x ∐ x:) . . . (x ∐ xm)zinhmbna xg > g, g > :, . . . , m, b bsgarbstbna bzvhdhmb bzriz zi padhclhmh rizcbghrhdi m + 9 u ganh lh xm+9 > x, da`blina di lh

Pm(x) > (∐9)m+9

(m + 9)!(x + 9) x(x ∐ 9)(x ∐ 0)...(x ∐ m).

Giga lh lasˏ nixW:,m^ 9/(x + 9) > 9 schdb di lh

|Pm(x)| ≤ 9(m + 9)!

|x(x ∐ 9)...(x ∐ m)|,

kˏbnh lh dagizima tvr -dhmlh ziditgi.

0.9? Iga lh e (x) > 9/(i∐

x) dagizitb di lh

e Wx:, x9, . . . , xm^ > 9

(i ∐ x:)(i ∐ x9) . . . (i ∐ xm)b, mi asmavu taoi,

9

i ∐ x > 9

i ∐ x: + x ∐ x:

(i ∐ x:)(i ∐ x9) + ¿ ¿ ¿ + (x ∐ x:) . . . (x ∐ xm∐9)

(i ∐ x:) . . . (i ∐ xm) + P(x),


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0.0. MH_TAM-AU ZACBMAN [I ZADHCLHMBN PI\CBGINI 08

odh lh

P(x) > ςm+9(x)(i ∐ x)ςm+9(i) , ςm+9(x) >m

b>:

(x ∐ xb).

Phsˏhmlh< Nithnitbˏkgan bmdugkblan sh mhpasrhdma dagizulh bzriz zi padhclhmurizcbgu. \inhman da`blhmao bzrizi u Mhwtam-av bmthrpacikbamb pacbman si padh-clhmbn rizcbgini b bzriz zi orhsˏgu,

e (x) > e (x:) + e Wx:, x9^(x ∐ x:) + ¿ ¿ ¿ + e Wx:, . . . , xm^(x ∐ x:) . . . (x ∐ xm∐9)+ e Wx:, . . . , xm, x^(x ∐ x:) . . . (x ∐ xm)

da`bli sh trizˏhmb bzriz zi eumgkblu 9/(i ∐ x).

0.96 Dagizitb di lh zi eumgkblu e (x) > 9/(x∐f), odh lh f ganpchgsim `ral, orhsˏgi Ciorimoh-avh bmthrpacikblh pacbmanan Cm(x) sthphmi m

9

x ∐ f ∐ Cm(x) > 9

x ∐ fx ∐ x:f ∐ x: ¿ ¿ ¿

x ∐ xmf ∐ xm .

Phsˏhmlh< Tvr -dhmlh sh dagizulh gia u prhtfadman ziditgu.

0.92 Iga lh e (x) > u(x) v(x), dagizitb di lh

e Wx:, x9^ > uWx:^ vWx:, x9^ + uWx:, x9^ vWx9^,

b, uapsˏth,

e Wx:, . . . , xm^ >m

g>:

uWx:, . . . , xg^ vWxg, . . . , xm^.

Phsˏhmlh< \iditig ḱhna dagizitb nithnitbkˏgan bmdugkblan. \i m > 9 tvr -dhmlhschdb bz dhmbkblh padhclhmh rizcbgh prvao rhdi,

e Wx:, x9^ > 9

x9 ∐ x:

u(x9)v(x9) ∐ u(x:)v(x:)

> u(x9) ∐ u(x:)

x9 ∐ x: vWx9^ + uWx:^ v(x9) ∐ v(x:)

x9 ∐ x: .

Zrhtpastivbna di tvr -dhmlh vizˏb zi m > n,

e Wx:, . . . , xn^ >ng>:

uWx:, . . . , xg^ vWxg, . . . , xn^.

Dagizˏbna di vizˏb zi m > n + 9. Zrhni dhmbkblb padhclhmh rizcbgh rhdi n + 9 lh

e Wx:, . . . , xn+9^ > 9

xn+9 ∐ x:

e Wx9, . . . , xn+9^ ∐ e Wx:, . . . , xn^

,


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05 0. BMTHPZACIKBLI

th lh, uzbniluḱb u a`zbr bmdugkblsgu fbpathzu,

e Wx:, . . . , xn+9^ > 9

xn+9 ∐ x:n+9

g>9

uWx9, . . . , xg^ vWxg, . . . , xn+9^

∐ng>:

uWx:, . . . , xg^ vWxg, . . . , xn^

> 9

xn+9 ∐ x:

ng>:

uWx9, . . . , xg+9^ vWxg+9, . . . , xn+9^

∐ng>:

uWx:, . . . , xg^ vWxg, . . . , xn^

Garbsthḱb vhzuuWx9, . . . , xg+9^ > (xg+9 ∐ x:) uWx:, . . . , xg+9^ + uWx:, . . . , xg^

diclh da`blina di lh

e Wx:, . . . , xn+9^ > 9

xn+9 ∐ x:

ng>:

(xg+9 ∐ x:) uWx:, . . . , xg+9^ vWxg+9, . . . , xn+9^

+ng>:

(xn+9 ∐ xg) uWx:, . . . , xg^ vWxg, . . . , xn+9^

tl.

e Wx:, . . . , xn+9^ > 9xn+9 ∐ x:

n+9g>9

(xg ∐ x:) uWx:, . . . , xg^ vWxg, . . . , xn+9^

+ng>:

(xn+9 ∐ xg) uWx:, . . . , xg^ vWxg, . . . , xn+9^

Tvr -dhmlh mhpasrhdma schdb pasch si`brimli adoaviriluḱbf si`brigi u paschdmlh dvhsunh.

0.0: Iga lh xg > x: + gf, g > :, . . . , m, dagizitb di lh

e Wx:, . . . , xm^ >

9

fmm!

mg>:

(∐9)m∐gmge (xg).

Phsˏhmlh< Tvr -dhmlh mhpasrhdma schdb bz bzrizi zi padhclhmh rizcbgh

e Wx:, . . . , xm^ >m

g>:

mb>:b>g

(xg ∐ xb)∐9

e (xg),


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0.0. MH_TAM-AU ZACBMAN [I ZADHCLHMBN PI\CBGINI 0?

b

mb>:b>g

(xg ∐ xb)∐9

> 9g!fg (∐9)m

∐g

(m ∐ g)!fm∐g > (∐9)m

∐g

m!fmmg.

Tvr -dhmlh sh nazˏh dagizitb b garbsˏḱhmlhn vhzh padhclhmh b gamikˏmh rizcbgh

e Wx:, . . . , xm^ > ∅me :

m!fm

b bzrizi

∅me : >m

g>:

(∐9)m∐g

m

g

e (xg).

0.09 Iga lh e (x) > de

dx dagizitb di lh ddxe Wx:, x^ > e Wx:, x^, asbn iga lh e (x)cbmhirmi eumgkbli.

Phsˏhmlh< Giga lh prhni dhmbkblb padhclhmh rizcbgh

d

dxe Wx:, x^ >

d

dx

e (x) ∐ e (x:)

x ∐ x:

>

9

x ∐ x:

e (x) ∐ e Wx:, x^

b

e Wx:, x^ > 9

x ∐ x:

e (x) ∐ e (x:)

akˏbochdma lh di lh u apsˏthn scukˏilu e Wx:, x^ > e (x:). \i cbmhirmu eumgkblu e (x) >ix + ` lh e Wx:, x^ > e

(x:) > i. Tbnh lh tvr -dhmlh dagizima.

0.00 Iga lh e (x) > (ix + `)/(kx + d), bzrikˏumitb e Wx, y^, e Wx,x,y^ b e Wx,x,y,y^.

Phsˏhmlh< Zrhni dhmbkblb lh

e Wx, y^ > 9

y ∐ x

iy + `

ky + d ∐ ix + `

kx + d

>

id ∐ `k(kx + d)(ky + d)

.

Bz vhzh padhclhmh rizcbgh b bzvadi eumgkblh

e W x , . . . , x

m+9^ >

9

m! e (m)(x)

schdb di lh

e Wx,x,y^ > ∀

∀xe Wx, y^ >

(`k ∐ id)k(kx + d)0(ky + d)

b

e Wx,x,y,y^ > ∀

∀ye Wx,x,y^ >

(id ∐ `k)k0(kx + d)0(ky + d)0

.


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06 0. BMTHPZACIKBLI

NITCI@

Gamikˏmh rizcbgh dy > dbee(y)

Zadhclhmh rizcbgh dbeehrhmkhs > y7

ear b>0 dbee(dbeehrhmkhs) ./ (x(b e (x9) + e Wx9, x0^(x ∐ x9) + . . . + e Wx9, . . . , xm^(x ∐ x9)(x ∐ x0)(x ∐ xm∐9)Avil pacbman oridbna pastupgan scbkˏmbn Farmhr-aval sˏhnb

Z m∐9 > e Wx9, x0, . . . , xm^Z m∐0 > (x ∐ xm∐9)Z m∐9 + e Wx9, x0, . . . , xm∐9^

. . .

Z 9 > (x ∐ x0)Z 0 + e Wx9, x0^Z > (x ∐ x9)Z 9 + e (x9)


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0.0. MH_TAM-AU ZACBMAN [I ZADHCLHMBN PI\CBGINI 02

Ava oavarb di ḱhna pacbman Z bzoridbtb pakˏhvsˏb ad pacbmani Z m∐9, nmaˏzh́kb usvigan garigu thguḱb pacbman pacbmanan (x

∐xb) b dadiluḱb adoaviriluḱu padh-

clhmu rizcbgu e Wx9, x0, . . . , xb^.

% Ti`cbki bmthrpacikblh

x > W: :.; :.? 9 9.; 9.? 0 0.; 0.? ;^7

y > W: :.; :.58 :.64 :.25 9 :.2 :.?8 :.4; :.94^7

% @ral patrh`mbf kvaravi zi gamstrugkblu pacbmani trhkho sthphmi

m > ; + 9 7

dbeehrhmkhs > y(90


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;: 0. BMTHPZACIKBLI

0.; Zacbmanb si hgvbdbstimtmbn kˏvaravbni

Pivmanhrma risparh -dhmb (hgvbdbstimtmb) kˏvaravb, si garigan f, su

xg > x: + gf, g > :, ¹9, ¹0, . . . , x > x: + qf, |q | 3 9

Gamikˏmh rizcbgh sh dhmbsˏu rhgurhmtmbn earnucini

∅e g > e (xg+9) ∐ e (xg) rhdi 9∅0e g > ∅e g+9 ∐ ∅e g > e (xg+0) ∐ 0e (xg+9) + e (xg) rhdi 0¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿ ¿

∅me g

> ∅m∐9e g+9 ∐

∅m∐9e g

>m

l>:

(∐

9)lm le (xg+m∐l) rhdi mUhzi padhclhmbf rizcbgi, gamikˏmbf rizcbgi b bzvadi eumgkblh lh

e Wxg, . . . , xg+m^ > ∅me g

fmm! >

e (m)(ν )

m! , ν ∆ Wxg, xg+m^

Mhwtam-av pacbman zi bmthrpacikblu umiprhd

Cm(x: + qf) > e (x:) +m

g>9

q (q ∐ 9) . . . (q ∐ g + 9)g!

∅ge :

Mhwtam-av pacbman zi bmthrpacikblu umizid

Cm(x: + qf) > e (x:) +m

g>9

q (q + 9) . . . (q + g ∐ 9)g!

∅ge ∐g

[tbrcbmo-av bmthrpacikbamb pacbman

C0m+9(x: + qf) > e : + q ∅e ∐9 + ∅e :

0 +

q 0

0!∅0e ∐9

+ ¿ ¿ ¿ + q (q 0 ∐ 9) . . . (q 0 ∐ m0)

(0m + 9)!

∅0m+9e ∐(m+9) + ∅0m+9e ∐m0

@hsshc-av bmthrpacikbamb pacbman

C0m+9(x: + qf) > e : + e 9

0 + (q ∐ :.8)∅e : + q (q ∐ 9)

0!

∅0e ∐9 + ∅0e :0

+ ¿ ¿ ¿ + q (q 0 ∐ 9) . . . (q 0 ∐ (m ∐ 9)0)(q ∐ m)(q ∐ :.8)

(0m + 9)! ∅0m+9e ∐m


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0.;. ZACBMANB [I HGUBDB[TIMTMBN KˏUAPAUBNI ;9

0.04 Mhgi lh e (x) eumgkbli tigvi di lh e (gf) > :, g > ¹9, ¹0, . . . , b e (:) > 9.Dagizitb di lh Mhwtam-av bmthrpacikbamb pacbman th eumgkblh

Cm(x) > 9 ∐ x9!f

+ x(x ∐ f)

0!f0 + ¿ ¿ ¿ + (∐9)m x(x ∐ f) . . . (x ∐ (m ∐ 9)f)

m!fm

Phsˏhmlh< Tvr -dhmlh mhpasrhdma schdb gidi sh uznh di lh x: > :, tl. q > x/f uearnucb

∅ge : >g

b>:

(∐9)g∐b

g

b

e (xb) > (∐9)g,

0.08 Bmthoric vhravitmaḱh

e (x) > 0ϊ x

:

h∐t0/0 dt

dit lh schdhḱan ti`hcan

x 9.:: 9.:8 9.9: 9.98 9.0: 9.08

e (x) :.560562 :.?:5060 :.?06556 :.?42685 :.?52659 :.?66?::

Bzrikˏumitb e (9.0;8) b akhmbtb orhsˏgu rhzuctiti.

Phsˏhmlh< Ti`hci gamikˏmbf rizcbgi lh

x e ∅e ∅0e ∅;e

9.:: :.560562

:.:0;82;9.:8 :.?:5060 ∐:.::90:?:.:00;65 :.:::::2

9.9: :.?06556 ∐:.::9926:.:09966 :.::::98

9.98 :.?42685 ∐:.::996;:.:0:::8 :.::::9?

9.0: :.?52659 ∐:.::9955:.:966;2

9.08 :.?66?::

Tiˏkgi u galal sh trizˏb prb`cbzˏmi vrhdmast bmthorici vhravitmaḱh prbpidi bmthrvicuadrh -dhman si paschdmli dvi kˏvari u ti`hcb. [taoi ḱhna garbstbtb Mhwtam-av pacbman

zi bmthrpacikblu umizid

C;(x: + qf) > e : + q ∅e ∐9 + q (q + 9)

0 ∅0e ∐0 +

q (q + 9)(q + 0)

5 ∅;e ∐;,

odh lh q > (x ∐ x:)/f. [tivcliluḱb di lh x: > 9.08, x > 9.0;8 b f > :.:8 b garbsthḱbpadvukˏhmh gamikˏmh rizcbgh u ti`hcb, da`blina di lh

e (9.0;8) ≆ C;(9.0;8) > :.?6;9?:.


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;0 0. BMTHPZACIKBLI

Orhsˏgi bmthrpacikblh sh oru`a nazˏh akhmbtb tiga sˏta sh u Ciorimoh-avan bzrizuzi orhsˏgu bzvad eumgkblh ipragsbnbri srhdmlan vrhdmasˏḱu adoaviriluḱh gamikˏmhrizcbgh,

|P;(x)| > | 904

q (q + 9)(q + 0)(q + ;)f4e (4)(ν )|

≆ | 904

q (q + 9)(q + 0)(q + ;)∅4e sr| > 9.5 ∟ 9:∐?

0.05 Dit lh dha dblioamicmh ti`hch zi eumgkblu e (x) > sbm x

x e (x) ∅0e ∅4e

:.2 :.?6;;; ∐:.:;90; :.::9089.9 :.62909 ∐:.:;88; :.::9499.; :.25;85 ∐:.:;640 :.::986

Bzrikˏumitb e (9) si tikˏmaˏsḱu 8 ¿ 9:∐8.

Phsˏhmlh< Giga 9 ∆ (:.2, 9.9), uzhḱhna di lh x: > :.2. Tidi lh q > :.8, pi lh orhsˏgi bmthrpacikblh eumgkblh e (x) u tikˏgb x > 9 @hsshc-avbn bmthrpacikbambnpacbmanan trhḱho sthphmi

|P;(9)| ≤ 94!

nix |e (4)| |:.8(:.80 ∐ 9)(:.8 ∐ 0)| :.04 ≤ 4 ¿ 9:∐8,

prb kˏhnu lh uzhta di lh nix |e (4)(x)| > 9. Digch, @hsshc-avbn bmthrpacikbambn pacb-manan C;(x) nazˏhna bzrikˏumitb si zidavaclivi luḱan tikˏmasˏḱu trizˏhmu vrhdmast.

Giga lh q > :.8, zi bzrikˏumivimlh su min patrh`mh sina gamikˏmh rizcbgh pirmaorhdi galh su dith u ti`hcb. Zrbnhman earnuch

C;(x: + qf) > 9

0(e : + e 9) + (q ∐ :.8)∅e :

+ q (q ∐ 9)

0!

∅0e ∐9 + ∅0e :0

+ q (q ∐ :.8)(q ∐ 9)

;! ∅;e ∐9

da b̀lina rhzuctit si trizˏhman tikˏmasˏḱu

e (9) ≆ C;(9) > :.6498.

0.0? Dagizitb di orhsˏgi gvidritmh bmthrpacikblh, adrh -dhmh rivmanhrma risparh -dhmbnkˏvaravbni x:, x9 b x0 si garigan f, mblh vhḱi ad

|P0(x)| ≤ f;

2∜

;N ;, N ; > nix

Wx:,x0^|e (x)|.

Tiga -dh pagizitb di sh nigsbnicmi orhsˏgi nazˏh akˏhgbvitb mi ristalimlu f/∜

; ad khmtricmh ipskbsh.


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0.;. ZACBMANB [I HGUBDB[TIMTMBN KˏUAPAUBNI ;;

Phsˏhmlh< Orhsˏgi gvidritmh bmthrpacikblh lh

P0(x) > 9;!

e (ν ) ς;(x), ν ∆ Wx:, x0^,

odh lh ς;(x) > (x ∐ x:)(x ∐ x9)(x ∐ x0). [nhman x > x9 + qf pacbman ς;(x)sh nazˏh mipbsitb u a`cbgu ς(x9 + qf) > f

;q (q 0 ∐ 9). Akˏbochdma lh di am dastbzˏhhgstrhnmh vrhdmastb

ς(xh) > ¹ 0;∜

;f; zi xh > x9 ¹ f∜

;.

0.06 Mhgi lh hx/(x + 0) ≤ e (4)(x) ≤ hx/(x + 9) b e (9) > :.?54;0. [i galbngarigan f trhì ti`hcbritb eumgkblu e (x) mi bmthrvicu W9, 0^ tiga di ti`hci dazvaclivi

gvidritmu bmthrpacikblu si tikˏmasˏḱu θ > 8 ¿ 9:∐?.Phsˏhmlh< Zrhni prhtfadman ziditgu orhsˏgi gvidritmh bmthrpacikblh adrh -dhmh

rivmanhrma risparh -dhmbn kˏvaravbni sh akhmlulh bzrizan

|P0(x)| ≤ f;

2∜

;N ;, N ; > nix

Wx:,x0^|e (x)|.

Di `bsna adrhdbcb gamstimtu N ; pa -dbna ad tharhnh a srhdmlal vrhdmastb

e (x) > e (9) + (x ∐ 9)e (4)(ν ), 9 ≤ ν ≤ x ≤ 0,

gali, garbsthḱb prhtpastivgu ziditgi, dilh akhmu

e (9) + (x ∐ 9) hν

ν + 0 ≤ e (x) ≤ e (9) + (x ∐ 9) h

ν

ν + 9, tl. N ; > ;.0;.

\inhman u bzriz zi akhmu orhsˏgh da`blina uscav zi f

|P0(x)| ≤ ;.0;2∜

;f; ≤ 8 ¿ 9:∐? zi f ≤ :.:9;.

0.02 Gacbgb trhì di ùdh garig f u ti`hcb zi eumgkblu e (x) > x9/; di `b mi bmthrvicu W9, 9:::^ orhˏsgi< i) cbmhirmh, `) gvidritmh bmthrpacikblh, `bci nimli ad 8

∟9:∐4. Imicbzbritb dvh naouḱmastb b) Mi khcan bmthrvicu W9, 9:::^ garbstb sh bstb garig f7bb) Mi bmthrvicu W9, 9::^ garbstb sh garig f9, i mi bmthrvicu W9::, 9:::^ garig f0.

Phsˏhmlh< i) Orhsˏgi cbmhirmh bmthrpacikblh mi bmthrvicu Wxg, xg+9^ sh akhmlulhbzrizan

|P9(x)| ≤ 90

(xg+9 ∐ xg)04

N 0, N 0 > nixWxg,xg+9^

|e (x)|.


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;4 0. BMTHPZACIKBLI

(b) N 0 > nixW9,9:::^ |0x∐8/;/2| > 0/2 pi lh

|P9| ≤ 90

02

f0

4 ≤ 8 ¿ 9:∐4 zi f ≤ :.9;.

(bb) X bmthrvicu W9, 9::^ astilh bsti prakhmi garigi f9 ≤ :.9;, i u bmthrvicu W9::, 9:::^ lh N 0 > 0 ¿ 9::∐8/;/2, pi lh f0 ≤ 5.0;.`) Orhsˏgi gvidritmh bmthrpacikblh mi bmthrvicu Wxg, xg+0^ sh akhmlulh, prhni prhtfad-man ziditgu, bzrizan

|P0| ≤ f;

2∜

;N ;, N ; > nix

Wxg,xg+0^|e (x)|.

(b) N ; > nixW9,9:::^ |9:x∐6/;/0?| > 9:/0? pi lh

|P0| ≤ 9:

0?

9

2∜ ; f;

≤ 8 ¿ 9:∐4

zi f ≤ :.0?.(bb) X bmthrvicu W9, 9::^ astilh bsti prakhmi garigi f9 ≤ :.0?, i u bmthrvicu W9::, 9:::^ lh N ; > 9: ¿ 9::∐6/;/0?, pi lh f0 ≤ 95.88.

0.;: Iga lh ∅ azmigi zi aphritar gamikˏmh rizcbgh umiprhd, ∅e g > e g+9 ∐ e g, i δ azmigi zi aphritar khmtricmh gamikˏmh rizcbgh δe g > e g+9/0 ∐ e g∐9/0, dagizitb di vizˏb

∅ > 9

0δ 0 + δ

9 +

9

4δ 0.

Phsˏhmlh< Trimsearnbsˏbna prva ditb bzriz gvidrbrimlhn

∅ ∐ 9

0δ 00

> δ 0

9 + 9

4δ 0

, tl. ∅0 > δ 0 + ∅ δ 0 > (9 + ∅) δ 0.

[idi lh

(9 + ∅)δ 0e g > (9 + ∅) (e g+9 ∐ 0e g + e g∐9)> e g+9 ∐ 0e g + e g∐9 + e g+0 ∐ e g+9 ∐ 0(e g+9 ∐ e g) + e g ∐ e g∐9> ∅0e g.

0.;9 Mhgi lh f > 9/m, xb > bf, b > :, . . . , m, b mhgi lh

υ(t) > 9 ∐ |t

|,

|t

| ≤9,

:, |t| = 9Dagizitb di lh zi sviga e (x) ∆ K0(:, 9)

∅0e g∐9 > e (xg+9) ∐ 0e (xg) + e (xg∐9) > 9

:

υg(t)e (t) dt,

odh lh υg(t) > fυ

(t ∐ xg)/f

.


45/437

0.4. DPXOB UBDAUB BMTHPZACIKBLH ;8

Phsˏhmlh< Eumgkbli υg(t) lh

υg(t) >

f ∐ xg + t, t ∆ Wxg∐9, xg^,f + xg ∐ t, t ∆ Wxg, xg+9^:, t ∆ Wxg∐9, xg+9^

th lh 9:

υg(t)e (t) dt >

xgxg∐9

(f ∐ xg + t)e (t) dt + xg+9xg

(f + xg ∐ t)e (t) dt,

sˏta bmthorikblan mhpasrhdma dilh tvr -dhmlh ziditgi.

0.4 Druob vbdavb bmthrpacikblh

X avan adhclgu su sina bmearnitbvma, graz prbnhrh, dith bmthrpacikblhtrboamanhtrblsgbn b rikbamicmbn eumgkblini, bmvhrzmi bmthrpacikbli b bmthrpa-cikbli eumgkbli vbsˏh pranhmclbvbf.

0.;0 Bmthrpacikblan phrbadbkˏmh eumgkblh e (x) trboamanhtrblsgbn pacbmanan

Y(x) > k: + k9 kas (0x) + k0 sbm (0x),

galb zidavaclivi uscavh bmthrpacikblh Y(xg) > e (xg), g > :, 9, 0, bzvhstb Fhrnbth-avuearnucu zi trboamanhtrblsgu bmthrpacikblu.

Phsˏhmlh< Xscavb bmthrpacikblh adrh -dulu sbsthn cbmhirmbf lhdmikˏbmi pa mhpazmitbngahkblhmtbni pacbmani,

k: + k9 kas(0xg) + k0 sbm (0xg) > e (xg), g > :, 9, 0.

Dhthrnbmimti sbsthni lh

D >

9 kas ( 0x:) sbm ( 0x:)9 kas ( 0x9) sbm ( 0x9)

9 kas ( 0x0) sbm ( 0x0)

> 4sbm(x9 ∐ x:) sbm(x0 ∐ x:) sbm(x0 ∐ x9) > :

zi : ≤ x: 3 x9 3 x0 3 ϊ, th am bni mhtrbvblicma rhsˏhmlh. Iga avan sbsthnu da-dina trboamanhtrblsgb pacbman mipbsim u prabzvaclmal tikˏgb x rizcbˏkbtal ad kˏvaravi,da`blina fanaohmb sbsthn lhdmikˏbmi

∐Y(x) + k: + k9 kas (0x) + k0 sbm(0x) > :,∐e (xg) + k: + k9 kas (0xg) + k0 sbm (0xg) > :, g > :, 9, 0,


46/437

;5 0. BMTHPZACIKBLI

kˏblh lh rhsˏhmlh (∐9, k:, k9, k0) mhtrbvblicma. [taoi dhthrnbmimti sbsthni nari `btb lhdmigi mucb,

Y(x) 9 kas (0x) sbm (0x)e (x:) 9 kas (0x:) sbm ( 0x:)e (x9) 9 kas (0x9) sbm ( 0x9)e (x0) 9 kas (0x0) sbm ( 0x0)

numericke metode - zbirka zadataka kroz c, fortran i matlab

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