A General Solver Based on Sparse Resultants: Numerical Issues and Kinematic Applications

Texte intégral

(1)A General Solver Based on Sparse Resultants: Numerical Issues and Kinematic Applications Ioannis Z. Emiris. To cite this version: Ioannis Z. Emiris. A General Solver Based on Sparse Resultants: Numerical Issues and Kinematic Applications. RR-3110, INRIA. 1997. �inria-00073580�. HAL Id: inria-00073580 https://hal.inria.fr/inria-00073580 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A General Solver Based on Sparse Resultants: Numerical Issues and Kinematic Applications Ioannis Z. Emiris. N˚ 3110 January 1997 THE`ME 2. ISSN 0249-6399. apport de recherche.

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(18) ÖØ×'Ù'ÖØÚuÛiÜ.Ý ÞSÝjßàÛcáâNãjäåçæèØÚméUêÓåçãcâ8ëìåîíðïNñßïòiÛcójô±æNédõ~ö÷øjù8úiùfûNüiûýþfÿù ù úù újÝ . . . É_ÊmË~Ì±ÍiÎË¥Ï'Ð¼Ì±ÑÓÒUÔ. . Unite´ de recherche INRIA Sophia Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA ANTIPOLIS Cedex (France) Te´le´phone : 04 93 65 77 77 – Te´lećopie : 04 93 65 77 65.

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(64)

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(118)

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(183) t?aradgl<^_] ; ="! ¦ P<QSVlhdt?lVt?VUlk1`pg8hd]g¬ad2gQ1^_ll?nBlgVUT^_lh¹kS]1^pµdhdt$^ hjgVº ad`pn1]SacT;^ hj`Y^_] ¦§¨V l8QSai¤¿o%Vf`_ai¤¿QSai¤gQ1^_lØact?Tk1` hjg?^_ac]t?VUSkSeUVUlgQSV<lad`_k1g?^_ac]0ad gQSV<act$^_bd^_]hj`S¤¥Vf`p`¼¯ eUac="]S! l?g thj^_]SVU lnBl?gVUT7gahd]Vf^_bcVU]SSt?aco1`_VUT³¦ ÃrÇ Ã Qacb !!% M & I &o , 2 !" /!!% &( )&@

(184) & , hf- i , & I & I

(185) . , S ", &*F -, , -& ! = ", ; (= !% I & I & ="! " ; ="! , > - % & , gf ) ) W d I , < , % % &( + * / , , ="! & I < , I &k-! " gfV ="! ="! i I / > % ! % EF1$ = /F& I h )& &( , h , & I % Jgf ÇJÇ zactYl8 VUef^ hj`p^_»UVU¡ ad`pn1]SacT;^ hj`¥eUaBVfÀef^_VU]gli¸

(186) ^_ll?^_]Sbck1` hdt2onSVf°]1^pg?^_ac]J¦ BHneUac]Sl?gt?kSefg?^_ac]J¸ct$^_bcQg'T2k1`pg?^_1`p^_eihjg?^_ac];on2h<µVUefgactmad1gQSV¥"= eU! ad`_ kST0]2T0ac]SacT;^ hj`_l|l%VUef^ hj`p^_»UVU . . . . . . jh gxh; ad^_]g4St?aBSkSeUVUl4h;µVUefgactad.gQSVµchj`_kSVUl

(187) ad.gQSN VYt?ai¤ ad`pn1]SacT;^ hj`_l4hjg

(188) gQ1^_l

(189) %ad^_]gi¦ Vfg gQSV ¯±gQLt?ai¤ ad eUac]g8hj^_]gQSVYeUarVfÀef^_VU]gl

(190) ad^ ! ) SçSgQSVU] "! ) 5 ="! W ¦¦ = ! ="! ¦¦ ¦¦ . . ! ) 5 =! . W. ¤QSVUt?V " >*W ! 0\E . ªHacT0Sk1g8hjg?^_ac]hj`p`pn2^pgm^_l'St?VfØVUthdo1`_V¥ga

(191) QhiµVuga4SVihj`¤<^pgQ;hdlmlThj`p`1hThjgt$^¼hdl. acl8l?^_o1`_Vc¦ 'P a³gQ1^_lYVU]S¤HVhdt$g?^pg?^_ac] ^_]ga³áckSto1`_areN©rl2la¶gQhjgYgQSV0kSS%VUt`_VfEgYlkSoSThjgt$^¼ ^_lxlqBkhdt?Vç ]Sac]Sl^_]Sbck1` hdthd]S¬^_]SSVU%VU]SSVU]gad ="! ¦

(192) }

(193) a¤:hd]S·eUad`_kST0]· VUt?Tk1g8hjg?^_ac]SlSa ]Sadg

(194) h ,VUefggQSVYThjgt$^¼LSt?ac%VUt$g?^_VUlla;¤HVhdS1`pngQSVUT gaaco1g8hj^_]h;Th1^_Thj` ¦ ZhdkSll?^ hd];Vf`p^_T;^_]hjg?^_ac]²ad gQSV9`_VfEgT0aclglVfgad eUad`_kST0]Sl^_l]Sa¤¿ acl8l?^_o1`_Vhd]S0V1St?VUl8lVU hdl

(195) Thjgt$^¼³Tk1`pg?^_1`p^_eihjg?^_ac]J¸¤QSVUt?V ^_l<gQSV^_SVU]g?^pg nLThjgt$^¼LadhdSSt?acSt$^ hjgV2l?^_»UV . ss ="! W ="! q 3 2 ¤QSVUt?V s = ! s = ! s ="! - s ="! Vfg, F ^_]SSV ¦'§¨VSa"]Sadg<QhUµV $) , , ©B]Sai¤<`_VUSbcVadmgQSVxl^_»UVUlHad hd]S ¤QSVfgQSVUtgQSVYlkSo%1^pµr^_l?^_ac]³actgQSV^_]SeUt?VUT0VU]g8hj`hj`_bcact$^pgQST QhdleUac]Sl?gt?kSefgVU ¦ . s W s ="! - s ="! s ="="!! . . . ÖØ×.ÙmÖ´Ú.

(196)

(197) !"$#!% '&( )&* Ç Ç ( (ES Ä aw_m -&1 @ S gf ) . . ! = 6"8 ; (= % I & I & = ! "m ; (= ! , ) xW d I ="! W i f- <

(198) &( Z + / i I / %

(199) !F= , i s ="! q H W . mP a¶t?VUeUaiµVUtgQSV;t?aBadgxeUaBact?1^_]hjgVUli¸ , TkSl?gYhjÀ]SVf`pn¬lhd]%$ = ¸,adgQSVUt$¤<^_lV"¤HV"QhiµV;ga eUacT0Sk1gVYgQSVY©cVUt?]SVf`ðµVUefgactad|Thjgt$^¼ ¤Q1^_eNQVUqBkhj`_l<gQSVeUac]SeihjgVU]hjg?^_ac]¬ad.µVUefgact?l hd]S ¸1¤QSVUt?V ^_l<gQSVY©cVUt?]SVf`mµVUefgactad hd]S ^_l

(200) l VUef^p°VULØt?acT 2 q3 ="! W W ="! W . kz . = ! HW - ="! . l^_]SeUV ^_lSVf°]SVUgao%VgQSV0ThB^_Thj`]Sac]Sl?^_]Sbck1` hdtlkSoSThjgt$^¼J¦2P<QSV;eUac]SeihjgVU]hjg?^_ac] + ^_lH^_]SSV1VU¶on ¤Q1^_e8QLhj`p¤HhinBl9^_]Sef`_kSSVUlhd]³hjÀ]SVf`pn^_]SSVU VU]SSVU]glkSoSlVfgGSo%VUeihdkSlV adgQSVUt$/ ¤<^_l8VYhj`p`0%¯±ád`_T;^¼1VU¶µad`_kST0VUlhdt?V»UVUt?ahd]S¶]Sa;t?aradgl<V1^_l?gi¦|P<QSVU]J¸1¤¥Vt?VUeUaµVUthj`p` t?aBadgeUaBact?1^_]hjgVUl

(201) on¶g8hd©r^_]Sbthjg?^_aclad.gQSVµVUefgactVU]gt$^_VUli¦ §¨V²]Sai¤7eUac]SeUVU]gthjgV³ac]¡Thjgt$^¼¡%ad`pnB]SacT;^ hj`_l;hd]SºgQSVf^_t"eUacT0hd]1^_ac]¢Thjgt$^_eUVUl Gáct SVf°]1^pg?^_ac]Sl<hd]Sohdl?^_et?VUlk1`pglHeUac]Slk1`pg 3ìZ } W 5±¦ VU]SadgVgQSV4Q1^_SSVU]²µchdt$^ hdo1`_V

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(203) hd]SQhiµV¥l8eihj` hdt VU]gt$^_VUli¦.§¨V¥t?Vf´VUtðg(a y6s.ðhdl|h & , h

(204) , ¤<^pgQ"SVUbct?VUV hd]S"Thjgt$^¼eUaBVfÀef^_VU]gl y 5 ¦'P<QSV9µchj`_kSVUlad ²gQhjgThd©cV ysml?^_]Sbck1` hdtuhdt?VH^pgl ,

(205) % ! ¦zactVfµVUt$n"Vf^_bcVU]µdhj`_kSV ¸dgQSVUt?V9^_lhohdl?^_l|adgQSV9©cVUt?]SVfàd y6 mSVf°]SVU0on2gQSV , I & ,

(206) ) &(! adgQSV9Thjgt$^¼ %ad`pn1]SacT;^ hj`rhdllaBef^ hjgVUga ¦mP<Q1^_lm^_lJgQSVVf^_bcVU]SSt?aco1`_VUTØactðThjgt$^¼Y%ad`pnB]SacT;^ hj`_l¸jh<ef` hdl8l?^_e St?aco1`_VUT ^_]³`p^_]SVihdt

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(209) áct?T2k1` hjg?^_ac]^_]¶gQSVYSt?VfµB^_ackSl

(210) lVUefg?^_ac]J¦ . y - y6s. z - y - y - z zy - y z,y - y( . Ù'Ù éQPfàSRR$ï.

(211) , , , . 9. ¤QSVUt?V ^_l

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(213) St?VUef^_lVf`pn;gQSV

(214) Vf^_bcVU]µdhj`_kSVUlHad,gQSV

(215) T0ac]1^_e4 ad`pnr¯ ]SacT;^ hj`±¸B¤QSVUt?VihdlH^pgl9t$^_bcQg<Vf^_bcVU]µVUefgact?l<eUac]g8hj^_]³hdlHl8# kSoµVUefgact?l9gQSVxt$^_bcQg9Vf^_bcVU]µVUefgact?lad y- ys8¦¥zact?Thj`p`pn¸%hdll8kST0# V + * / " ; ^_l

(216) h 2 ]Sac]S»UVUt?aZ<3 t$^_bcQgxVf^_bcVU]µVUefgact ad0¸u¤QSVU t?VVihdeNQ D5" ; ¸5 ! ¦¿ P<QSVU] ^_lh 2 ]Sac]S»UVUt?aZ03 t$^_bcQgVf^_bcVU]µVUe¯ gact"ad y - ys hd]S D5 - ¸|áct * ¸¤QSVUt?V ^_lgQSVVf^_bcVU]µdhj`_kSV³ad . eUact?t?VUl8 ac]S1^_]Sb0ga ¦ §¨V]Sa¤hdSSt?VUl8l4gQSV;qBkSVUl?g?^_ac]·ad¥h²l^_]Sbck1` hdt4`_Vihd1^_]Sb¶Thjgt$^¼^_]h]Sac]B¯ T0ac]1^_e0 ad`pnr¯ ]SacT;^ hj`±¦uP<QSV4ád`p`_ai¤<^_]Sbgthd]Sl?Øact?Thjg?^_ac]¶^_lhj`_l8a"kSl8VU^_]gQSV^_T01`_VUT0VU]g8hjg?^_ac]L^_]¶act?SVUt9ga ^_T0St?aµVYgQSVeUac]S1^pg?^_ac]1^_]Sbad'gQSV`_Vihd1^_]SbThjgt$^¼,¦ uÃ ¢ a % M & , h

(217) , y6s. , & , K)& , *

(218) H , % ' f-2 . -& /F& I I , I !& K // , f <

(219) M ) &

(220) d I ->& I /Jh , &*E & cf- & , z z

(221) D f"<-

(222) 1"?$(i & I &A) S% !H & , h

(223) , { gfF & I E" M K // , & I @

(224) & , !JfF& I E" M , M , i % I & I & { AI ? M , % ' ?

(225) , S -, )& & , h . . { gH^_l¥Vihdlngal8VUV

(226) gQhjgHgQSVt?VUlk1`pg?^_]Sb%ad`pn1]SacT;^ hj`¢{ |Qhdl9Thjgt$^¼eUaBVfÀef^_VU]gl

(227) adJgQSV lNhdT0Vthd]S©%¸áctxlk1Àef^_VU]g?`pn·bcVU]SVUt$^_e"l8eihj` hdt?l

(228) ¸%l?^_]SeUV"VfµVUt$nThjgt$^¼^_lxgQSV"lkST ad z 2leihj` hdtSt?aBSkSefgl

(229) ad\ y5$¦uP<QkSl4gQ1^_l<gthd]Sl?áct?Thjg?^_ac]L^_l

(230) adØgVU]Lt?Vf´VUt?t?VU³ga²hdl / ' -, ¦ ÃrÇ Ã Qa f& I < % !2gf& I I ,

(231) 8 , - , 8& I <-

(232) % & , gf ) . ) ="! W i I , I !/! &(> ,

(233) - % f & I & , Jh

(234) , i M " -, &. &( ,

(235) - . !>gf % , & , - , 8 ys , M & , )& , *

(236) > , % ' & I - /% &s , F&( , K)- ?-M$ % & , Agf& I , )% &

(237) ,

(238) & , K &.F S&* ÇJÇ. §¨V2QhiµVl8VUVU]LgQhjgxgQSV2]SVf¤Thjgt$^¼%ad`pn1]SacT;^ hj`\{ H Qhdl4h²]Sac]Sl?^_]Sbck1` hdt4`_Vih¯ 1^_]SbeUarVfÀef^_VU]gi¦7¬act?VUaiµVUtU¸%°]S1^_]Sb^pglxVf^_bcVU]µchj`_kSVUlhd]SVf^_bcVU]µVUefgact?lx^_lxt?VUSkSeUVUgahd] Vf^_bcVU]SSt?aco1`_VUT adgQSV;eUacT0hd]1^_ac]Thjgt$^¼¦BHnQnB%adgQSVUl?^_li¸%gQSV"Vf^_bcVU]SlhdeUVUladhdt?V. ÖØ×.ÙmÖ´Ú.

(239)

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