Numerical Computation of a Polynomial GCD and Extensions
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. NUMERICAL COMPUTATION OF A POLYNOMIAL GCD AND EXTENSIONS Victor Y. Pan. N˚ 2969 Aouˆt 1996 THE`ME 2. ISSN 0249-6399. apport de recherche.
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(135) 3
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(137). 1. . -Z[6!. G GaV G G* G G. G G* !]. -+3 . . QvC%uB\NV E Y.LBwQv\NVhuB\NV . 1 " .Q C uN\NV EY.LBw*Qv\NVhuB\NV V ?NExTWQnE: WT MOEDL{V ]9Q$ Vh?BEu Q PO^`LBQvC MRY P. 6Q*lZlZSZS* ]5 )3 ! . Uv / d E .. . . . . G G*. ". E. . . . 5. . J "! .. . . . -Zd_Q! \ _S%. % \& !c*. . X?NMR>Ty?*Q{MR+.]tEDYf_M,$ ^[1 3 ? ! 0 Sv TW6?w*u Q4$ 6;:;: % "!@ Y LB >Yw @#"}= < " !XY MLB w %`?QvY.aB]t]_EDY fa+vwBEpEDSvV f_?BEDY ExV QE$r+vYEWf_VX^ C%]_QvMOC ]_V unJPREK;5 uY[V ? a2EWVa0 & EWEDL Y LBw %¬& MOVh? Y PbPXEDwBSvEW] Pb^dMRLBS MRLÁ),Y TDQvC%u2QvLBEDL{V Q$.-GV ?BE*aNMOuY.f VhMbV E SvQ.Vf ?BY.uNEWfl? ? Y LB% wPTDQv% LNED]_Y.MOTy]_? V ]xEDQwB$ S.E4Y VxTDQvC%LNQvLB]_EDVKTUV _ ]l V& \FQ6%+vEWEWf_wBVhSvMOTDEDEW]=]) Qa{$ ^ Vh?BPbE%^dMRwBLNES "YLnVKMOV_MRY QvL wnMR]QVy$ Y.LBTW-E 5=Y IpVlL C VhQ.?B]_EV _ 3i$ fhQvC EZY Ty? Q.V ?BEW;f 5l>@?BEDf_E $ Qvf_E % ,M $= ! ) kM $= ! % f_ED]_u EWTWVhM,+vEUPO^'-@% Vh?BEDL~Y.L{^ u Q.MOL{VG4Q $ ) f_ED]_u EWTWVh,M +vEUPO^ %7Q $ -pPOMOED] Y[V4Y*wNMR] VyY. LNTDE Y VC%Qv]_V _ _ \$64!53(V 1 3 $ fhQvC 5xo MOLY PbPO^% LBQ.VhEGV ?Y[V 3 V B 1 .+e\:6]% a{^ V ?BE wBE"LNMbVhMOQvL Q$ % Y.LBw E V PO?EZY[Y Vf_Pb%^]TD% Q.VhLB?B]hEEWc`ED\N] EDVhL{MOC VhPb]^Y[%V ED1 ] . $ Qvf4-+3V ?N "E ? \
(138) S_ V -Mk\*?4''5 :@Q$9u EWfW$ Qvf_C%MOLBSk`VyY SvED] Y.LBw *Q $ m9POSvQvf MOVh?BC' 5b Y PR]_Q~Y.uNuNPO^ V Q V ?BE TZY.]_E 4Q $ mtPRSvQ.f_MbV ?BC '5O] %&aB\NVGk`V Y.SvE F uN]_),MOY.POQvMOLBEDPL bw ]=5*%=fhEW>@uB]hY.?Bu2f_E EDV_TUMRTWTWVhQv\NM,+vC%P EpY uBf_] Pb\NMR]^ C%V % Y uNMbVhV PbMO] MkQv"BLuBTZY YY.P V_f MRYTDQvPbQ.LPO]_EWV4GP Q$ 4Q+vm9$EWfhu PO] SvMREWQvQvWf lLf $ MOQv-4Vhf_?BC MOC]%MRLB]h'\BS~5baN]k`]_%dVyVyuYY Y.LdSvf V_E VhM MOY[FTDPO\NPb^Q PRY.$ f]_mtPOMO^C%PR%nSvuNQQ.PO$ f_,M "MbMOVVhEW?B] Pw C u%¬Y &X f Y?N5b PbMOPRTyEUMO? Pl]+vMOwNEWiC 4Q fa000 C%Svf MOY.LuNY ? VhED w~57a{>@^*?nV MR?B] ETWQvTDQvC%] uBVp\NQ Vy$ Y[TDVhMOQvQvC%L uBQv\NLNV_PbMR^ LNS f_EDV cd?B\NEMOfhTDEWQv] LN LBED , TU V EDw TW! QvC%uY.u fhQ.Y LBPOPOEDEWL{P V TD]lQvC%Q $ uV Y ?Nf_EGMO]hQvaNL MOu]Y.V f EDVhuNMbV ];E % O &X?NMRTy?*u2EDWf $ Q.fhC , ! TWQvC%uY.f_MO]hQ.LB]XQ $ MRL{V EWSvEDf_]Xf Y.LBS MRLBi S $ f_QvC VhQ 1 * / $#v|{F %]_Q V ?N{ E \ Q`Q.POEZY.L ) anMO?V - TWQvC%uNPOE (`MOV ^ Q $ru EWWf $ Qvf_C%MOLBS%V ?NMO]t]_VyY SvExMO]ta Qv\NLBwBEDw a{^ -
(139) !
(140) *4 , !
(141) !+Z &. . . . . . . . . . . . . O. Û<Û¸GÞÜÅÜ. . . O. O. . O. O. O. . . . .
(142) }.
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(144) . . . . . . . . . . . . . . É}º<ÛÉ>È.
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(146) / O 5 * !GY V%kdVyY.SvE FY%7Y.LBw , O * !Y V CtE(`V %gPOEWV%\B] C%Q`wNM,$ ^mtPRSvQ.f_MbV ?BC 5 ~a{^ Y.uBuNPb^dMRLBS aNMOLY.f ^]_EZY.f_Ty? Y[V k`VyY SvE F MRY.L LBw Qvf_ wBED ,fK V Q wB*EW TDf_EZ! Y.)]_f E Y LBMbV wB]KQvTWC QvMk]_2DVxEDwla2-Qv5 \BLB w EWV V Q V ?N E4wBPREWELB+vQ.EWV PO]ExV ?B ,Ex EDO wNSv EG* ]_ EW Vp!iQ4) $ wNVEW?BV EWExfhC aNMOMOuLNY.MR]f VhVhMOMb/TV ESvSvff Y.Y.uNuN??P%¬ $ QvO f #:Y.LN. w 9 PREU*lV 6Q* T*wBED LB\$Q V E~_ Z¬V ?NIpE~aB]_LdED\BfWCG+.E a2EDVhf?Y QV $TWQvLB LBMREW] TWVhY EDw]_\BTDaBQv]_C%EWV u2Qv4Q LB$ EDL{Vh] C Q $K VY.?NLBMOw ] VFY?%Y &XVk?N6 MOTy?7V & EC & " MbPOV P TWMbV E V Y.]% k`\B1 aB]_ V Y.\SvE +[SF 5O$ Q.f%Q $Y POVhP0?B#E 5 EWLdm@V_V MR f_E Vh?BY E POSv"Qvfhf_] MbV V ?N]_
(147) C \BaB%~] $ VyQ.Y.fGSvE*E +. QED$pf "^ k`VyuY Y SvMOEf. V ,-?B* EK* !+MOLd* ,VhED. SvEWf6 * * 9 ]h\BTy*E?*. V ?6QY *V T* *n& E4Y.uBuNPb^ anMRLY f_^ ]hEZY fhTy?~MOL QvfhwBEWf9VhQ TDQvC%uB\nV E
(148) 3
(149) H ! ". . . . . . . . . . / G C. . _ _ \*4i! 3 C *hg !. C \]g aG V:_ _ \ i4 ! C 3. ". . .QvY LBLNPbw%^ MRLBMk$ TU#PR\NwBEtVh5 ?BEXEDwBS.E rMOL{V QV ?BEXEDwNSvEX]hEWV Q$2V ?BEtaNMOuY.f VhMbV EXSvf Y uB? Mk$ Y.LBw @ > N ? D E P%=Y VxV ?BE ]hEDTWQvLBw ]h\NaB]_V Y.SvE Q$9kdVyY.SvE FY%&X?NMOTy? & E& MbPbP@TWMbV E Y.] kn\NaB]_V Y.SvE L YF 5 FGQ4$ Vh?BEEDL{VhMOfhExY POSvQvf MOVh?BC3%'& ETWQvC%uB\NV E 9Y.LBw BC ` %BQvaB]_EDfa+vEV ?Y V 9 V =\ C ` / 1 W\ +'%dY.LBw Y uBuNPb^GaNMOLY.f ^G]_EZY.f_Ty?4MRLGQvfhwBEWfrV QlTDQvC%uB\NVhE V ?NE C MRLnMRC\BC %G]h\BTy?%V ?Y[~V $ Qvfr]_QvC%E #P%]& E ? Y +.7E 9V \ C / %"ZNE POEZY.f PO^ % % 06%{Y.LBw ,M $ % . 6% VC ?BY^EW(nPL MO%UCG$ Qv\NfXC V ?N )3EG1 TDQ.!fh0 fhSvEWTD]hw u2QvQLB$ wNMO %LB "2S !5!g#PY.%MLB& w E#"? " Y +v !E%N$ Qvf 3. 1 . C " %2-]h Q \ Vh?4QY ! V 3SJ'* 5V "6!;E. & JMO* PbP28TW Mb V "Et!X]hMO\B]pTy? Y YQ.$WL EDL{ % "VhMO!5fh!xExY.C%LN@wQnwN#", M " % "TDY!%rVhMOaQvL*Y.]hQEWw$rmtQvPRL S.QvVhf_?BMbVE ?BanC MRL Y 5f_Y ^ %'$]hQvEZf Y fhTDTy Q.P? C %guNY \N] VhMOLBSz Yh C t}^ Y r (`s MOCG\BC )3 1q Vh]G!0 TDS.Q.TD]_w V ) MOL V EWfhC%]tQ$ V ?NELd\BCa EWflQ$ Y.f_MbV ?BC%EUVhMOTQvu2EDfhY VhMOQvLB]9Y.LBw TDQvC%uY f_MO]hQvLB]tMRL!+.Q.PREWwn-XMO] , O * !tY Vk`V Y.SvE ] % , O * !9Y[VKk`V Y.SvE4_F ) MOLBTWPO\BwNMOLBS*kn\NaB]_V Y.SvED]KYF 5O4Y.LBw YF 5 F -%Y.LNw , O * ! Y V9k`V Y.SvE' % &X?BEWfhE /'5 & o R M L [ Y O P b P ]%& E C YZ^ f_EDuNPRY.TWE TWQvC%uB\NVhMOLBS*V ?BE TDQvLNLBEDTUV EDw TDQvC%u2QvLBEWLdVh]GQ$ a{^ ^ TDLBQvE C%+.EDuBf \NwNV_EDMRLNTDf_S EZY.Y ]_EDC ] ^Y MO(nL MOC Vh?BY EXPgV C f Y Y LBVh]_TyMb?NVhMOMRQvLN=L S $ f_QvC MOL V Q Z >@?BC E TD)Y.] fhMRwnLBMRTWLEpY[YPOMbPbV PH^ V ?NG E9TWQv C% Gu Q Qv$
(150) LN EDL{V ] ". QC $ ^Y (nMOCG \NV C\NfhLC MOY[L{V V TyQG?NMOanLBMRS uYMRL f_Vh MbV E9C TWPbMR cd,M \B$EWG ] MOL V G?N0MO] G Vhf Y.LNC ]_MbVh MOG¦Qv%[LnY.LB-5wKo;V \N?Bf_EWVPL ?N% EDY fhC%[ -Qv?f_E\ % 4i ! C C 3" 1 0 S.MO]@TDwY " " ". MR]9TDQvC%uB\NVhEDw~MOC%C EWwNMRY V EUPO^ 5>@?BEDf_E $ Qvf_E %b& E4C YZ^ \B]_eE G GBY.]9V ?BEaNMRLBY.f_^ ]hEDY.fhTy? . . . . . . . . . Û<Û¸GÞÜÅÜ. .
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