Beyond the Quadratic Equations and the N-D Newton-Raphson Method

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1 IOSR Jornl o Mthmts (IOSR-JM) -ISSN: - p-issn: 9-X. Volm Iss Vr. IV (Jl. - A.) PP - Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Wn Opornprsrt (Dprtmnt o Cl Ennrn Coll o Ennrn/ Rnst Unrst Thln) Abstrt: Th objt o ths thnl ppr ws to propos n pproh or soln polnomls o r hhr thn two. Th mn onpts wr th omposton o polnoml o hhr r to th prot o two polnomls o lowr rs n th n-d Nwton-Rphson mtho or sstm o nonlnr qtons. Th ont o h trm n n ornl polnoml o orr m wll b qt to th orrsponn trm rom th ollt-pn prot o th two polnomls o th lowr rs bs on th onpt o ntrmn onts. Consqntl sstm o nonlnr qtons ws orm. Thn th nknown onts o th ompos polnoml o th lowr r o th two ompos polnomls wol b lmnt rom th sstm o th nonlnr qtons. Atr tht th nknown onts n th ompos polnoml o th hhr r wol b obtn b th n-d Nwton-Rphson. Fnll th nknown onts or th ompos polnoml o th lowr r wol b obtn b bk sbstttons. In ths thnl ppr th ormltons or th ompos polnomls wol b r or th polnomls r thr to nn. Srl nmrl mpls wr lso n to r th pplblt o th propos pproh. Kwors: Roots o Polnoml o Hh Dr th n-d Nwton-Rphson Mtho Untrmn Conts Jobn o th Fntons Mtr Inrson I. Introton Fnn soltons to polnoml o orr hhr thn two hs bn nobl n nnrn works. Mostl onl rl roots wr rqr. In ths ss th rphl mtho ol oo ntl sss or som nt nmrl mthos sh s th Nwton-Rphson mtho. Howr th trmnton o ll possbl roots hs bn r hllnn. Thr r nrl soltons or b- n qrt polnomls []. Bon th qrt polnomls som spl orms n snt ontons or solbl polnomls h bn st [-]. Th prpos o ths thnl ppr ws to propos mthmtl tool or soln or ll possbl roots o polnoml o r hhr thn two. It nl th omposton thnq n th Nwton-Rphson mtho or sstm o nonlnr qtons. Th omposton thnq ws ppl or rwrtn th ornl polnoml nto th orm o prot o two polnomls o lowr rs. Bs on th onpt o ntrmn onts h ont o n powr n th ornl polnoml wol b qt to th orrsponn ollt-pn on o th prot o th two ompos polnomls. Th nknown onts n ompos polnomls o th lowr r wol b lmnt. Bs on ths sstm o nonlnr qtons o nknown onts n th ompos polnoml o th hhr r ws obtn. Thn th n-d Nwton-Rphson mtho ws s to sol or th nknown onts rom th sstm o nonlnr qtons. Th lmnt onts wr obtn b bk sbstttons. Th ormltons n th onpts wol b sss n Ston. In Ston th pplblt o th propos mthmtl tool wol b monstrt n srl nmrl mpls. From whh rtl onlson ol b rwn n Ston. II. Domposton o th Ornl Polnoml Eqton. Domposton o Polnoml Eqton o Orr m A polnoml qton o orr m m b nrll prss n orm o (): m m... m m () Whr... m r th onts o n m. Wthot losn nrlt m m b s throhot ths thnl ppr. Ths: m m... m () Gn r... m r ll possbl roots o th polnoml qton. Th polnoml qton o () n b rwrttn s: r r r r () m DOI:.99/- P

2 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho In th s tht root r s on r wll b on tor o th ornl polnoml qton. Th lt qton m b obtn b th rt lon son o th ornl qton o () b r s: m m m m r m m r r r r r () Frthr ltons n b on s soon s tonl roots wll b on. Rl roots o polnoml qtons ol b obtn n Brnstn orm b nmrl nlss []. Th rphl mtho n b sr s powrl tool or trmnn th nmbr o rl roots ppromt ls o rl roots n ntr o th rl roots.. obl rl roots or trpl rl roots t. Usll n l o rl root rom th rphl mtho n b s s n ntl ss l o nmrl mtho.. th Nwton-Rphson mtho. Th rst o ths thnl ppr wll b ot or th omposton o n ornl polnoml qton nto two ompos polnoml qtons o lowr rs.. Domposton o Polnoml Eqton.. Propos Domposton An ornl polnoml qton o orr m s n th orm o () n lws b ompos nto two polnoml qtons o lowr rs. For n ornl polnoml qton o n o r th qton wll b ompos nto two ompos qtons o lowr rs.. on ompos polnoml qton wth n o r n th othr ompos polnoml qton wth n n r. For n ornl polnoml qton o n r howr th nton wll b ompos to two ompos qtons o n rs. Th rson bhn s tht polnoml qton o o r wll lws h t lst on rl root. Howr no rl root s rnt or polnoml qton o n n r. Thror or th s o n ornl polnoml qton o n r t s onsrt to ssm both two ompos polnoml qtons o n rs sn qrt qton n lws b sol n los orm orml. Or sssons wll b os on th polnoml qtons o r hhr thn two.. r thr onwrs. Th orrs o th two ompos qtons r smmr n Tbl. Tbl Drs o Two Dompos Eqtons or Ornl Eqtons o Dr rom Dr o Eqton Ornl Eqton st Dompos Eqton n Dompos Eqton 9.. Brstow s Domposton Brstow [] propos omposn polnoml o orr m n orm o () nto prot o two lowr rs.. qrt nton n polnoml o r m pls rmnr trm n orm o (). P P m m m m... () m m b b b b b m m whr b... m r th onts o b obtn b lon n... () P b r onts n b. Th lttr polnoml n n th trm DOI:.99/- P m s th rmnr. On th ls o n r ssm All b s s wll s n n b trmn. B trt prors th tl n s wll s ll b s n b obtn s soon s pprohs ro. Ths () s r to (). m m b m bm... b b b () For th sk o rthr sssons m n b m n b st to wthot losn n nrlt. Ths () m b rwrttn s: m m m... () Bs on () polnoml qton n b ompos to th prot o two polnoml qtons.. qrt qton n polnoml qton o orr m. Thr n b srl prs o th ompos polnomls pnn on th ntl ss o th trt prors. Sn n qrt qton n lnr qton s lws sol ll possbl roots o polnoml qton n b trmn b th Brstow s mtho. Th onrn o th mtho s qrt onl th ros r ompl onjt

3 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho prs o mltplt on or r rl o mltplt t most two. For hhr mltplts t s mprtll slow or sbjt to lr. Th motons o th Brstow s mtho wr propos [-] bt th tls r ot o th sop o ths thnl ppr... Complt Brstow s Domposton Th ompos polnoml qton o () n b rthr ompos ntl th r o th lst ompos qton s two or on. For n ornl polnoml qton o n n r or n o r th omplt Brstow s solton n b rwrttn n orm o (9) or () rsptl.... Q (9)... Q () whr Q s qrt nton.. Domposton o Cb Fnton Consr nrl b qton: () Th b qton m b ompos nto prot o two qtons.. on lnr qton n on qrt qton s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (.) n (.) n b rwrttn n trm o (.b) (.) n s: () Ths n b lmnt rom (.) n (.b) sh tht sstm o two nonlnr qtons n two smltnos nknowns n s orm. (.) (.b) On n r obtn rom nmrl mtho.. th Nwton-Rphson mtho n two mnsons n b obtn b bk sbsttton (). Th propos omposton or b qton hs tl th sm orm s th Brstow s mtho n t s lr omplt.. Domposton o Qrt Fnton Consr nrl qrt qton: () Th qrt qton m b ompos nto prot o two qtons.. two qrt qtons s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () ls: (9) Eqtn h ont n (9) to th orrsponn trm n th ornl qton n () ls to qtons: (.) DOI:.99/- P

4 n (.) n Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho (.b) (.) (.) n (.) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) Ths n n b lmnt rom (.b) n (.) sh tht sstm o two smltnos nonlnr qtons n two nknowns n s orm. (.) (.b) On n r obtn b th Nwton-Rphson mtho n two mnsons n n b obtn b bk sbsttton (.) n (.b) rsptl. Th propos omposton or qrt qton hs tl th sm orm s th Brstow s mtho n t s lr omplt.. Domposton o Qnt Fnton.. Th Propos Domposton Consr nrl qnt qton: () Th qnt qton m b ompos nto prot o two qtons.. on qrt qton n on b qton s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (.) n (.) n (.b) (.) (.) (.) n (.) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) Ths n n b lmnt rom (.b) (.) n (.) sh tht sstm o thr smltnos nonlnr qtons n thr nknowns n s orm. (.) (.b) (.) On n r obtn b th Nwton-Rphson mtho n thr mnsons n n b obtn b bk sbsttton (.) n (.b) rsptl. DOI:.99/- P

5 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P.. Brstow s Domposton Th propos omposton or qnt qton hs tl th sm orm s th Brstow s mtho n th b qton obtn n b rthr ompos b sn th propos omposton s sss n Ston.... Complt Brstow s Domposton Th qnt qton o () n b rwrttn n orm o omplt Brstow s omposton s: (9) Epnn th prot n (9) ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) Ths sstm o smltnos nonlnr qtons n nknowns n s orm.. Domposton o St Fnton.. Th Propos Domposton Consr nrl st qton: () Th st qton m b ompos nto prot o two qtons.. on qrt qton n on qrt qton s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) (.) n (.) n n (.) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) Ths n n b lmnt rom (.b) (.) (.) n (.) sh tht sstm o or smltnos nonlnr qtons n thr nknowns n s orm. (.)

6 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- 9 P (.b) (.) (.) On n r obtn b th Nwton-Rphson mtho n or mnsons n n b obtn b bk sbsttton (.) n (.b) rsptl... Brstow s Domposton Th propos omposton or st qton hs tl th sm orm s th Brstow s mtho n th qrt qton obtn n b rthr ompos b sn th propos omposton s sss n Ston.... Complt Brstow s Domposton Th st qton o () n b rwrttn n orm o omplt Brstow s omposton s: () Epnn th prot n () ls: (9) Eqtn h ont n (9) to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) (.) Ths sstm o s smltnos nonlnr qtons n s nknowns n s orm.. Domposton o Spt Fnton.. Th Propos Domposton Consr nrl spt qton: () Th spt qton m b ompos nto prot o two qtons.. on b qton n on qrt qton s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.)

7 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.) (.) (.) n (.) n (.b) n n (.) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) (.) Ths n n b lmnt rom (.) (.) (.) n (.) sh tht sstm o or smltnos nonlnr qtons n or nknowns n s orm. (.) (.b) (.) (.) On n r obtn b th Nwton-Rphson mtho n or mnsons n n b obtn b bk sbsttton (.) (.b) n (.) rsptl... Brstow s Domposton Th spt qton o () n b rwrttn n orm o th Brstow s omposton s: () Epnn th prot n () ls: () Eqtn h ont n () to th orrsponn trm n th ornl qton n () ls to qtons: (9.) (9.b) (9.) (9.) (9.) (9.) (9.) Ths sstm o sn smltnos nonlnr qtons n sn nknowns n s orm... Complt Brstow s Domposton Th spt qton o () n b rwrttn n orm o omplt Brstow s omposton s: () Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.)

8 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.) (.) (.) (.h) Ths sstm o sn smltnos nonlnr qtons n sn nknowns n s orm.. Domposton o n Ot Fnton.. Th Propos Domposton Consr nrl ot qton: () Th ot qton m b ompos nto prot o two qtons.. two qrt qtons s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) (.) (.) (.h) n (.) n (.b) n (.) n n (.h) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) (.) (.) Ths n n b lmnt rom (.) (.) (.) n (.) sh tht sstm o or smltnos nonlnr qtons n or nknowns n s orm. (.) (.b) (.) (.) On n r obtn b th Nwton-Rphson mtho n or mnsons n n b obtn b bk sbsttton (.) (.b) (.) n (.) rsptl.

9 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P.. Brstow s Domposton Th ot qton o () n b rwrttn n orm o th Brstow s omposton s: () Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) (.) (.) (.h) Ths sstm o ht smltnos nonlnr qtons n ht nknowns n s orm... Complt Brstow s Domposton Th ot qton o () n b rwrttn n orm o omplt Brstow s omposton s: (9) Epnn th prot n (9) n qtn h ont to th orrsponn trm n th ornl qton n () ls to qtons: (.) (.b) (.) (.) (.) (.) (.) (.h) Ths sstm o ht smltnos nonlnr qtons n ht nknowns n s orm..9 Domposton o Non Fnton.9. Th Propos Domposton Consr nrl non qton: 9 () Th non qton m b ompos nto prot o two qtons.. on qrt qtons n on qnt qton s shown blow: () whr n r th nknown onts o th qtons. Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to 9 qtons: (.) (.b) (.)

10 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.) (.) (.) (.) (.h) (.) n (.) n (.b) n (.) n n (.) rsptl n b rwrttn n trm o th othr trms s: (.) (.b) (.) (.) Ths n n b lmnt rom (.) (.) (.) (.) n (.h) sh tht sstm o smltnos nonlnr qtons n nknowns n s orm. (.) (.b) (.) (.) (.) On n r obtn b th Nwton-Rphson mtho n mnsons n n b obtn b bk sbsttton (.) (.b) (.) n (.) rsptl..9. Brstow s Domposton Th non qton o () n b rwrttn n orm o th Brstow s omposton s: () Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to 9 qtons: (.) (.b) (.) (.) (.) (.) (.) (.h) (.) Ths sstm o nn smltnos nonlnr qtons n nn nknowns n s orm.

11 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P.9. Complt Brstow s Domposton Th non qton o () n b rwrttn n orm o omplt Brstow s omposton s: () Epnn th prot n () n qtn h ont to th orrsponn trm n th ornl qton n () ls to 9 qtons: (9.) (9.b) (9.) (9.) (9.) (9.) (9.) (9.h) (9.) Ths sstm o nn smltnos nonlnr qtons n nn nknowns n s orm. III. N-D Nwton-Rphson Mtho n th Jobn o th Fntons. Nwton-Rphson Mtho or Nonlnr Fnton Th prton o th Nwton-Rphson mtho ws bs on rst orr Tlor srs pnson: ' () whr s th ntl ss t th root or th pros stmt o th root n s th pont t whh th slop ntrpts th s. At ths ntrpt b nton n () n b rrrn to: ' () whh s th Nwton-Rphson mtho or nonlnr qton.. Nwton-Rphson Mtho n mor thn on Dmnson.. Nwton-Rphson Mtho n two Dmnsons Th Nwton-Rphson mtho or two smltnos nonlnr qtons n b r n th smlr shon. Howr mltrt Tlor srs hs to b tkn nto ont or th t o mor thn on npnnt rbls ontrbtn to th trmnton o th root. For th two-rbl s rst orr Tlor srs n b wrttn or h nonlnr qton s: (.) (.b)

12 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P Jst s or th snl-qton s th root stmt orrspons to th ponts whh n. (.) n (.b) n b rrrn to: (.) (.b) In () onl n r nknown. Ths () s st o two smltnos lnr qtons wth two nknowns. Consqntl wth smpl lbr mnpltons.. Crmr s rl n n b sol s: (.) (.b) Th nomntor o h o () s th trmnnt o th Jobn o th sstm. () s th qton or th Nwton-Rphson mtho n two mnsons. For th bnts o rthr sssons on th mtho or mor thn two mnsons () shol b rwrttn n trm o th mtr notton.. th Jobn o th nton. () () whr s th Jobn o th nton n s th nrs o... Nwton-Rphson Mtho n Mor Thn Two Dmnsons Consr sstm o n smltnos nonlnr qtons: n n n n () Th solton o ths sstm onssts o st o ls tht smltnosl rslt n ll th qtons qln ro. Jst or th s o two nonlnr qtons Tlor srs pnson s wrttn or h qton n l n n l l l l () whr th sbsrpt l rprsnts th qton or nknown n th son sbsrpt nots whthr th l o nton nr onsrton s t th prsnt l ( ) or t th nt l ( ). Eqtons n th orm o () r wrttn or h o th ornl nonlnr qtons. All l trms r st to ro n () n b rwrttn s: n l n l l n l n l l l (9) Not tht onl th n j j... trms on th rht-hn s r nknowns. As rslt st o n lnr smltnos qton s obtn.

13 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P Th prtl rts n b prss n trm o mtr notton s: n n n n n n () Th prsnt n th nt ls n b prss n th tor orm s n T X () n T X () Th nton ls t n b prss s n T F () Usn ths rltonshps (9) n b rwrttn s X W () X F W () Assm tht th nrs o n b obtn. Thn X n () n b sol. X F X F W X () In () I s nt mtr. Ths () n b rwrttn s: F X X (). Fntons rom th Dompos Eqtons n th Jobn o th Fntons For th bnts o ppltons th sstm o smltnos nonlnr qtons r or th ompos qtons sss n Ston. Ston.9 r smmr n th ollown sbstons... Ornl Cb Eqtons F () (9).. Ornl Qrt Eqtons F (9) (9)

14 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P.. Ornl Qnt Eqtons... Th Propos Domposton F (9) (9)... Brstow s Domposton F (9) (9)... Complt Brstow s Domposton F (9) (9)

15 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P.. Ornl St Eqtons... Th Propos Domposton n Brstow s Domposton F (9) (99)... Complt Brstow s Domposton F () ().. Ornl Spt Eqtons... Th Propos Domposton F ()

16 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- 9 P ()... Brstow s Domposton F () ()... Complt Brstow s Domposton F () () (.) (.b) (.) (.) (.) (.) (.)

17 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.h) (.) (.j) (.k) (.l) (.m) (.n) (.o) (.q) (.r) (.s) (.t) (.) (.).. Ornl Ot Eqtons... Th Propos Domposton F (9) () (.) (.b) (.) (.) (.) (.) (.)

18 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P... Brstow s Domposton F () ()... Complt Brstow s Domposton F () () (.) (.b) (.) (.) (.) (.) (.) (.h) (.)

19 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.j) (.k) (.l) (.m) (.n) (.o) (.p) (.q) (.r) (.s) (.) (.) (.w) (.) (.) (.) (.b) (.) (.) (.) (.) (.) (.h).. Ornl Non Eqtons... Th Propos Domposton F ()

20 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P () (9.) (9.b) (9.) (9.) (9.) (9.) (9.) (9.h) (9.) (9.j) (9.k) (9.l) (9.m)... Brstow s Domposton F ()

21 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P ()... Complt Brstow s Domposton F () () (.) (.b) (.) (.) 9 (.) (.)

22 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.) (.h) (.) (.j) (.k) (.l) (.m) 9 (.n) (.o) (.p) (.q) (.r) (.s) (.t) (.) (.) 9 (.w) (.) (.) (.) (.) (.b) (.) (.) (.) 9 (.) (.)

23 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (.h) (.) (.j) (.k) (.l) (.m) (.n) 9 (.o) (.p) (.q) (.r) (.s) (.t) (.) (.) (.w) 9 (.) (.) (.) (.b) (.bb) (.b) (.b) (.b) (.b) 9 (.b) 9 (.bh) 9 (.b) 9 (.bj) 9 (.bk) 9 9 (.bl)

24 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho IV. Nmrl Empls Sn nmrl mpls wll b monstrt n ths ston to r th pplblt o th propos prors. Th rst or mpls r slt rom rl ppltons n l nnrn. Th r th b- qrt- qnt- n st qtons rsptl. Th lst thr mpls r bon th st qton monstrt n thnl pprs o th othr thor []. Th ntnton s jst to sr th rsrhrs to rs som hllnn problms n th ormls n hrn shol b sl or som stn ppltons n. All lltons n ths thnl ppr wr on n th nronmnts o Sotwr Mth Prm.. Onl rslts rom th propos omposton wll b shown rthr n tls. Th rslts rom th othr two ltrnt ompostons wr lso llt to r th orrtnss o th qtons n to ompr th n wth th propos mtho bt th rslts rom th ltrnt mthos wll b l bs o th sp lmtton. Empl : Roots o Cb Eqton Th rqr pth o sqr tmbr ston ol b trmn b soln th b qton Ths qton n b ompos to th prot o lnr qton n qrt qton s shown n (). Frstl th two nknown n m b obtn b th Nwton- Rphson mtho n mnsons (). In ths s F n r llt () n (9) rsptl. Th rslts o llton r smmr s shown blow.. Intl ss. Itrton : Itrton : Itrton : Itrton : F F F F DOI:.99/- P

25 Itrton : Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho F Thn th stmts o n r.9 n.99 rsptl. Thn n b obtn rom (9) Th ompos qton boms Howr or th sn prpos onl th post rl root. m s tkn. Empl : Roots o Qrt Eqton Th mnmm nhor lnth o sht pl n b obtn rom th qlbrm o th ltrl rth prssr tn on sht pl n orm o qrt qton Ths qton n b ompos nto th prot o two qrt qtons s shown n (). Frstl th two nknown n m b obtn b th Nwton-Rphson mtho n mnsons (). In ths s F n r llt (9) n (9) rsptl. Th rslts o llton r smmr s shown blow. Intl ss Itrton : F....9 Itrton : F Itrton : Itrton :..9 F F DOI:.99/- P

26 .. Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Ths th stmts o n r.9 n. rsptl. Thn n rom n b obtn rom (.) n (.b) rsptl.. n.. Th ompos qton boms Howr or th sn prpos onl th post rl root.9 m s tkn. Empl : Roots o Qnt Eqton Th mtr o rlr stl olmn sbjt to n l or m b trmn b soln qnt qton Ths qton n b ompos nto th prot o two qtons.. on qrt qton n on b qton s shown n (). Frstl th thr nknown n m b obtn b th Nwton-Rphson mtho n mnsons (). In ths s F n r llt (9) n (9) rsptl. Th rslts o llton r smmr s shown blow. Intl ss Itrton : F Itrton : F DOI:.99/- 9 P

27 Itrton : F..99. Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P Atr ls o trton th stmts o n r n rsptl. Thn. n. n b obtn rom (.) n (.b) rsptl. Th ompos qton boms prpos th lrr l o th post rl roots. 9 m s tkn Howr or th sn Empl : Roots o St Eqton Th rtl wtr hht (nt n m) n n opn hnnl o trpol ston m b onsr b soln st qton.9.9. Ths qton n b ompos nto th prot o on qrt qton n on qrt qton s shown n (). Frstl th or nknown n m b obtn b th Nwton-Rphson mtho n mnsons (). In ths s F n r llt (9) n (99) rsptl. Th rslts o llton r smmr s shown blow. Intl ss Itrton : F

28 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Itrton : F Atr ls o trton th stmts o n r n -.9 rsptl. Thn.9 n. n b obtn rom (.) n (.b) rsptl. Th ompos qton boms An th qrt qton n b ompos rthr to two qrt qtons s sss rlr n Empl Howr or th sn prpos th lrr l o th post rl roots. 9 m s tkn. Empl : Roots o Spt Eqton Lt s onsr spt qton. Ths qton n b ompos nto th prot o on b qton n on qrt qton s shown n (). Frstl th or nknown n m b obtn b th Nwton-Rphson mtho n mnsons (). In ths s F n r llt () n () rsptl. Th rslts o llton r smmr s shown blow. Intl ss Itrton :. -. F DOI:.99/- P

29 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Itrton : F Atr ls o trton th stmts o n r.. -. n.99 rsptl. Thn..9 n. n b obtn rom (.) (.b) n (.) rsptl. Th ompos qton boms An th b qton n b ompos rthr to on lnr qton n on qrt qton s sss rlr n Empl. Whrs th qrt qton n b ompos rthr to two qrt qtons s sss rlr n Empl Empl : Roots o n Ot Eqton Lt s onsr n ot qton. Ths qton n b ompos nto th prot o two qrt qtons s shown n (). Frstl th or nknown n m b obtn b th Nwton-Rphson mtho n mnsons (). In ths s F n r llt (9) n () rsptl. Th rslts o llton r smmr s shown blow. Intl ss DOI:.99/- P

30 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Itrton : F Itrton : F Atr ls o trton th stmts o n r n. rsptl. Thn n. n b obtn rom (.) (.b) (.) n (.) rsptl. Th ompos qton boms An h o th qrt qtons n b ompos rthr to two qrt qtons s sss rlr n Empl Empl : Roots o Non Eqton 9 Lt s onsr non qton. Ths qton n b ompos nto th prot o on qrt qton n on qnt qton s shown n (). Frstl th nknown n m b obtn b th Nwton-Rphson mtho n mnsons DOI:.99/- P

31 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho DOI:.99/- P (). In ths s F n r llt () n () rsptl. Th rslts o llton r smmr s shown blow. Intl ss Itrton : F Itrton : F

32 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho Atr ls o trton th stmts o n r n 9.9 rsptl. Thn n 9. n b obtn rom (.) (.) (.) n (.) rsptl. Th ompos qton boms An th qrt qton n b ompos rthr to two qrt qtons s sss rlr n Empl. Whrs th qnt qton n b ompos rthr to on qrt qton n on b qton s sss rlr n Empl. Fnll th b qton n b ompos to on lnr qton n on qrt qton s sss rlr n Empl V. Conlson ) An pproh or soln polnoml qtons o r hhr thn two ws propos. ) Th mn onpts wr omposton o polnoml o hhr rs to th prot o two polnomls o lowr rs n th n-d Nwton-Rphson mtho or sstm o nonlnr qtons. ) Th ont o h trm n n ornl polnoml o orr m wll b qt to th orrsponn trm rom th ollt-pn prot o th two polnomls o th lowr rs bs on th onpt o ntrmn onts. Consqntl sstm o m nonlnr qtons ws orm. Thn th nknown onts o th ompos polnoml o th lowr r o th two ompos polnomls wol b lmnt rom th sstm o nonlnr qtons. Thror th nmbr o nonlnr qtons wol b r to th nmbr o nknown onts n th ompos polnoml qton o hhr r. ) Th nknown onts n th ompos polnoml o th hhr r wol b obtn b th Nwton-Rphson mtho or smltnos nonlnr qtons. Thn th nknown onts or th ompos polnoml o th lowr r wol b obtn b bk sbstttons. ) Th ormltons or th ompos polnomls wol b r or th ornl polnomls o r rom thr to nn. ) Th sstm o nonlnr qtons n spplmntr qtons or trmnn th nknown onts o th ompos polnomls wr lso smmr or th ornl polnoml or r rom thr to nn. ) For th s o n ornl polnoml qton o n o r th ornl polnoml qton wll b ompos to two polnoml qtons.. on qton o n o r n th othr qton o n n r. Whrs or th s o n ornl polnoml qton o n n r two ompos polnoml qtons o n rs wr propos to rnt obtnn ll possbl roots.. ompl onjts stnt rl roots obl rl root trpl rl root t. ) Two ltrnt orms o ompos polnoml qtons wr lso n.. Brstow s omposton n omplt Brstow s omposton. For th polnoml qton o r or hhr th tor o nonlnr qtons n th orrsponn Jobn mtr rom th Brstow s omposton wr smplr thn th propos omposton. Whrs thos rom th omplt Brstow s omposton wr mor ompl thn th propos omposton. Both ltrnt orms nol lrr sstms o smltnos nonlnr qtons. 9) Sn nmrl mpls wr lso n to r th pplblt o th propos pproh. For nmrl mpls or polnoml qtons o r thr or n s wr monstrt. Ths problms wr slt rom th rl ppltons n l nnrn. Th othr thr problms or polnoml qtons o r sn ht n nn wr n to hlln to th rsrhrs. Nmrl rslts wr n rthr n tls so tht th rrs n kp trk or ll stps o lltons. ) For n polnoml qton thr st srl possbl prs o ompos polnoml qtons bt n pr o ompos qtons wll lws th sm nl rslts. ) Th Nwton-Rphson mtho n two n mor mnsons wr pro to b r nt tool or DOI:.99/- P

33 Bon th Qrt Eqtons n th N-D Nwton-Rphson Mtho soln sstm o smltnos nonlnr qtons t lst n th tnt o nmrl mpls shown n ths thnl ppr n thor s prn on nn roots o polnoml qton. ) Th ompos polnoml qtons n lws ompos rthr to th qtons o lowr rs. Fnll th ornl polnoml qtons n b rwrttn n orm o prot o lnr qtons n qrt qtons. Thror ll possbl roots n lws b trmn. ) Th mtho propos n b tn or polnoml qton o n r bon nn bt wth lonr qtons n lrr sstm o smltnos nonlnr qtons. Frthr tnson ws not shown n ths thnl ppr bs o th lmtton o th ppr sp bt t n b on sstmtll n orm o mtr nottons n omptr prormmn. Aknowlmnts Fll nnl spport n Grntn Prossonl Lns o Sotwr Mth Prm. or Clltons throhot th thor rsrh projts b As Grop (999) Compn Lmt ws orll knowl. Rrns [] R.B. Kn Bon th qrt qton (Boston-Bsl-Brln: Brkhäsr 99). [] R.G. Klkrn Un mthos or soln nrl polnoml qtons o r lss thn s Albm Jornl o Mthmts (Sprn/Fll) -. [] R.G. Klkrn Soln rtn qnts Annls Mthmt t Inormt () 9. [] M. A. Fl n D. Lr Soln qnts n spts b rls Intrntonl Jornl o Snt n Rsrh Pbltons () -. [] C. Boswll n M.L. Glssr Solbl st qtons Cornl Unrst Lbrr Mthmtl Phss rx:mthph/ Aprl -. [] S.Y. Hn n S.S. Chn Absn o post roots o st polnomls Twns Jornl o Mthmts () 9-. [] R.G. Klkrn Etrtn th roots o spts b polnom omposton Ltrs Mtmáts 9 -. [] R.G. Klkrn On th solton to ot qtons Th Montn Mthmts Enthsst () 9. [9] R.G. Klkrn On th solton to non qtons Albm Jornl o Mthmts (Sprn/Fll) -. [] S.C. Chpr n R.P. Cnl Nmrl Mthos or Ennrs th E. (Nw York NY: MGrw-Hll ). [] M.R. Spnr Polnoml rl root nn n Brnstn orm otorl ss. th Dprtmnt o Cl Ennrn o Brhm Yon Unrst Proo UT 99. [] L. Brstow Insttons rltn to th stblt o th ropln Rports n Mmorn # Asor Commtt or Aronts Otobr 9 -. [] F.M. Crrno A mo Brstow mtho or mltpl ros o polnoml Mthmts o Comptton () OCTOBER 9. [] L.B. Rll Conrn o th Nwton pross to mltpl soltons Nmr. Mth. 9(MR #9) 9 -. DOI:.99/- P