SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL

Transcription

1 Joual of athmatcal Sccs: Advacs ad Applcatos Volum, 05, ags 5-8 SIULTANEUS ETHDS FR FINDING ALL ZERS F A LYNIAL JUN-SE SNG ollg of dc Yos Uvsty Soul Rpublc of Koa -mal: usopsog@yos.ac. Abstact Th pupos of ths pap s to pst th w mthods fo fdg all smpl os of polyomals smultaously. Fst, w gv a w mthod fo fdg smultaously all smpl os of polyomals costuctd by applyg th stass mthod to th o th tapodal Nwto s mthod, ad pov th covgc of th mthod. also pst two modfd Nwto s mthods combd wth th dvatv-f mthod, whch a costuctd by applyg th dvatv-f mthod to th o th tapodal Nwto s mthod ad th mdpot Nwto s mthod, spctvly. Fally, w gv a umcal compaso btw vaous smultaous mthods fo fdg os of a polyomal.. Itoducto th a typcal tato mthod such as Nwto s mthod, a tal appoxmato of a o covgs to a spcfc o, but th stass mthod (o Duad-K mthod appoxmats all smpl (al o complx os of polyomal smultaously (s [, 4]. 00 athmatcs Subct lassfcato: 65H04, 65H05. Kywods ad phass: polyomal os, smultaous mthods, stass mthod, covgc, Nwto s mthod. Rcvd Apl 8, Sctfc Advacs ublshs

2 6 JUN-SE SNG Lt a a a b a polyomal of dg havg smpl os wth costats a, a,, a. Lt,, b th dstct os of ad lt dstct complx umbs,, b th appoxmatos. Th stass mthod (Duad-K mthod s dfd as m m m ( m m (, ( fo m 0, ad ths mthod s o of th most fqutly usd tatv mthods whch gv smultaous computato of all os of. If a fucto s dfd by (, th has th sam os as th polyomal, ad so th poblm of fdg th os of ducs to that of os of th fucto. If w dot ( fo,,, th cas of, ( ca b wtt as, ( wh s a cut appoxmato ad ẑ s a w appoxmato to a o of polyomal. Th mthod costuctd by ( s calld th stass-l mthod (bfly, L. Th am of ths pap s to pst th w mthods fo fdg all smpl os of polyomals smultaously. Ths w mthods a basd o th Fot-Soma s mdpot Nwto s mthod ([7] ad th aoo s tapodal Nwto s mthod ([8], whch w modfcatos of th Nwto s mthod though tatv appoxmatos.

3 SIULTANEUS ETHDS FR FINDING ALL 7 f ( x It s wll-ow that Nwto s mthod s dfd by x x f ( x wth a appoxmato x ad a w appoxmato x of a o, ad s ffct to fd a o of a quato f ( x 0 fo a dfftabl fucto f wth pop codtos ad a suffctly clos tal valu (s [8]. I [8], aoo ad Fado poposd th tapodal Nwto s mthod dfd by Thy appld Nwto s mthod to th f ( x x x f ( x f ( x. ( x of th domato. Alog wth (, th mdpot Nwto s mthod that Fot-Soma poposd [7] s costuctd as x x f ( x. f ( x ( x x Thy also appld Nwto s mthod to th f ( x st x x f ( x. (4 x of th domato, ad so Both th tapodal Nwto s mthod ad th mdpot Nwto s mthod a of cubc od, whl th ogal Nwto s mthod was of quadatc od. A vaty of mthods ca b appld to th x addto to Nwto s mthod. tovć t al. [5] dvd th followg smultaous mthod fo fdg all smpl os of polyomals by applyg th stass mthod to th x th mdpot Nwto s mthod: (, (5 (

4 8 JUN-SE SNG whch s calld Nwto-stass mthod (o N. Also, tovć ad tovć [6] foud th followg dvatv-f mthod (o DF dfd as: ( (, (6 whch has a smla fom wth th o abov ad ths mthod s of cubc od. I ths pap, w pst th w mthods fo th smultaous appoxmato of all smpl os of polyomals by applyg th stass-l mthod ad th dvatv-f mthod to x th tapodal Nwto s mthod ad th mdpot Nwto s mthod. Thoughout ths pap, th covgc of os wll b dscussd ad th od wll b calculatd fo w costuctd mthods. wll us th otato a ( b fo two complx umbs a ad b, whos modul a of th sam od, that s, ( b. max { } wth fo,,.,, a I addto, th o s dfd as I all dscussos, th od latd to, whch s a o of th pvously appoxmatd os, s psumd to b th sam. Aft that, w wll show that th od latd to, whch s a o of th appoxmatd os cocg ach mthod, s dtcal. Fo th sam bg, th od latd to th alady appoxmatd os ê s hypothsd to b dtcal as follows: fo all. I Scto, w gv a w mthod fo fdg smultaously all smpl os of polyomals costuctd by applyg th stass mthod to th x th tapodal Nwto s mthod, ad pov th covgc of th mthod. I Scto, w pst two modfd

5 SIULTANEUS ETHDS FR FINDING ALL 9 Nwto s mthods combd wth th dvatv-f mthod. Thy a costuctd by applyg th dvatv-f mthod to th x th tapodal Nwto s mthod ad th mdpot Nwto s mthod, spctvly. I Scto 4, w gv a umcal compaso btw vaous smultaous mthods fo fdg os of a polyomal. Fally, w coclud that th covgc of all w costuctd mthods ths pap a smla o supo tha oth tatv mthods of cubc od.. stass-l Tapodal Nwto s thod I ths scto, w costuct a w mthod fo fdg smultaously all smpl os of polyomals of cubc od. By applyg th stass mthod (5 to th x th tapodal Nwto s mthod (, w dv a w mthod costuctd as follows: ( ( (. (7 call (7 th stass-l tapodal Nwto s mthod, ad fom ths, smply, call t thod. Th calculato ad dscusso of th od of thod a smla to thos of th Nwto-stass mthod, whch s a altato of tovć s mdpot Nwto s mthod (s [5]. Fom (7, w hav th followg: Lmma. Fo a polyomal, w hav ( ( ( ( ( ( (. oof. By th Taylo s xpaso aoud, w hav that ( ( (, ( ( ( (, ( (8 (

6 0 JUN-SE SNG ( ( (. ( ( ( Fom (8, w obta ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (. Fom Lmma, w hav th followg thom: Thom. If th appoxmat o x goudd fom thod s clos ough to ad th od of s th sam, th th od of ê s dtcal, ad ( s fomd. oof. asly s that th followg quato s satsfd: ( ( ( ( (. That s, ( ( (. (9 If Q (, th Q s a polyomal of od, ad Q ( fo all. Thfo, Q s th Lagag tpolato of pots,,,, ad so w hav

7 SIULTANEUS ETHDS FR FINDING ALL Q. Thfo, w obta. (0 Fom (0, t follows that. ( Substtutg (, w obta. Thfo, w hav (. (

8 JUN-SE SNG Now w wll fd th od of thod. If th Taylo s xpaso s appld to, th w hav (. 4 By (9, (, ad Lmma, w hav that (. Th hbyshv s mthod s dfd by, x f x f x f x f x f x x ad of cubc od (s [7, Subscto 5.]. Accodg to th hbyshv s mthod, w s that.

9 SIULTANEUS ETHDS FR FINDING ALL Thfo, th od of ê s calculatd as follows: ( ( (. ( (. odfd Nwto s thods ombd wth Dvatv-F thod I ths scto, w pst two modfd Nwto s mthods combd wth th dvatv-f mthod (6 fo fdg all smpl os of a polyomals smultaously. Th o s a fom that th dvatv-f mthod s appld to th x th tapodal Nwto s mthod ( as follows: (, ( ( ( whch s calld th dvatv-f tapodal Nwto s mthod, o smply, thod. Fom (, w hav th followg thom: Thom. If th appoxmat o x goudd fom thod s clos ough to ad th od of s th sam, th th od of ê s dtcal, ad ( s fomd. oof. Sc tovć s dvatv-f mthod (6 s of cubc od (, ( ( (4 (s [6]. Usg (4 ad th Taylo s xpaso, ẑ s calculatd as follows. (I ths cas, ( (.! (

10 JUN-SE SNG 4 ( ( ( ( 6 ( ( 4 (. Thfo, th od of ê s calculatd as follows:. Now w apply th dvatv-f mthod to th x th mdpot Nwto s mthod (4 ad costuct th tato as follows:, (5 whch s calld th dvatv-f mdpot Nwto s mthod. Fom ths, w call t thod smply. Fom (5, w hav th followg thom: Thom. If th appoxmat o x goudd fom thod s clos ough to ad th od of s th sam, th th od of ê s dtcal, ad s fomd.

11 SIULTANEUS ETHDS FR FINDING ALL 5 oof. By usg (4 ad Taylo s xpaso, ẑ s calculatd as follows. (I ths cas,!. ( ( 6 ( ( 4 4 ( 4. Thfo, th od of ê ca b calculatd as follows:.

12 6 JUN-SE SNG 4. Numcal ompaso I ths scto, w gv umcal xpmts ad compasos btw vaous smultaous mthods fo fdg os of a polyomal. Ths mthods a all of cubc od. Thy clud thod, thod, thod, th dvatv-f mthod (DF, th tovć s Nwto- stass mthod (N, ad th stass-l mthod (L. Fo a polyomal a a a, w choos tal appoxmatos as Abth s appoach (s []: ( 0 a xp π R. I ths cas, R s a adus of a ccl, wh th tal os by Abth s appoach a locatd complx umb pla. us th followg Hc s fomula to slct R (s []: R max a. Accodg to Hc s fomula, a ds { : < R} ctd at th og cotas all os of polyomal. Th polyomals that w usd o umcal compaso a as follows: ( x ( x ( x ( x 4, ( x ( x ( x ( x 4 ( x 5, (6 ( x ( x ( x ( x 4 ( x 5 ( x 6, x 5x x 7x 6x 8x x x 7. H ( x, ad ( x a lso s polyomals wh 4, 5, 6,, x spctvly.

13 SIULTANEUS ETHDS FR FINDING ALL 7 appoxmatd th os utl t satsfy th followg codto: ( 0 max < 0. m (7 I Tabl, w gv a umcal compaso btw sval mthods to fd all os of thos polyomals (6. It cotas th tato m umb m ad th valu max ( of tatv mthods, aft w appoxmatd (7 to a satsfyg labl. Th small th m, th fast appoxmatd o th os. h m s th sam, t ca b tptd m that a small max ( lads to a hgh accuacy of appoxmato. All computatos hav b do usg ATLAB. Tabl. Th umb of tatos (th o of tatv mthods oly. thod thod thod DF N L (6 (7 ( (5 (6 (5 ( 9(8-4 8(9-4 7(-4 9(9-4 8(-0 (- (7- (4-9(4- (- (- 7(- 4(- (5- (8- (8- (7- (- 4 4(- (- 0(- 4(- (- (- 5. ocluso I ths pap, th w mthods fo th smultaous appoxmato of all smpl os of polyomals by utlg th tapodal Nwto s mthod ad th mdpot Nwto s mthod w poposd. It was pov that ach mthod was of thd od. By smultaously appoxmatg all smpl os of polyomals ad by compag umcal xpmts wth vaous mthods that a of thd od, w obtad that th sults of thod ad thod a smla wth that of pvous mthods. But, w foud out that th sult of thod a supo tha that of ay oth mthods. All mthods w costuctd ths pap a w ad catv. It sms that ths mthods ca b appld to vaous flds, ad th study o th applcatos of thod s ow pogss.

14 8 JUN-SE SNG Rfcs []. Abth, Itato mthods fo fdg all os of a polyomal smultaously, ath. omput. 7 (97, [] E. Duad, Soluto Numéqus ds Équatos Algébaqus, Tom. I: Équatos du Typ F ( x 0, Racs d u olyôm, asso, as, 960. []. Hc, Appld ad omputatoal omplx Aalyss, Vol., Joh ly ad Sos Ic., Nw Yo, 974. [4] I.. K, E Gsamtschttvfah u Bchug d Nullstll vo olyom, Num. ath. 8 (966, [5]. S. tovć, D. Hcg ad I. tovć, a smultaous mthod of Nwto- stass' typ fo fdg all os fo a polyomal, Appl. ath. omput. 5 (009, [6]. S. tovć ad L. D. tovć, a cubcally covgt dvatv f oot fdg mthod, It. J. omput. ath. 84 (007, [7] J. F. Taub, Itatv thods fo th Soluto of Equatos, tc-hall, Eglwood lffs, Nw Jsy, 964. [8] S. aoo ad T. G. I. Fado, A vaat of Nwto s mthod wth acclatd thd-od covgc, Appl. ath. Ltt. (000, g