• Aucun résultat trouvé

A new scheme for table-based evaluation of functions

N/A
N/A
Info
Télécharger
Protected

Academic year: 2021

Partager "A new scheme for table-based evaluation of functions"

Copied!
15
0
0

Chargement.... (Voir le texte intégral maintenant)

Texte intégral

(1)A new scheme for table-based evaluation of functions David Defour, Florent de Dinechin, Jean-Michel Muller. To cite this version: David Defour, Florent de Dinechin, Jean-Michel Muller. A new scheme for table-based evaluation of functions. [Research Report] RR-4637, LIP RR-2002-45, INRIA, LIP. 2002. �inria-00071948�. HAL Id: inria-00071948 https://hal.inria.fr/inria-00071948 Submitted on 23 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 new scheme for table-based evaluation of functions David Defour, LIP Florent de Dinechin, LIP Jean-Michel Muller, CNRS. N° 4637 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4637--FR+ENG. THÈME 2. apport de recherche.

(3)

(4)   

(5)  "!#$%&'()&*&+' ,)& -.0/1324- 57698;: <7=?>"@A %B C38;<D57EGFH2I5J- 1KE 57LM 1KE=?>"@A NO50.PEQSRT13LM 5UCVRW: CKC35U<U= XZY[ \ ]^_a`cb&d egf%haijkblnmpo7jnqSjkbal bSrVqSstluqvwlGxzy0` {Omplnju|7vwb }?~mt€9bSr  ~hSiwspjk~b ‚ stƒwƒPm7~zr „ b(~baq†^ba~q†^b‡iOˆ‰7Š7‹UŒeŽ m7bS`‘{~’bdt“7“7de*”p”%ƒwspopbax. •—–I˜a™Sš›œp™p ]^wjnx"ƒPstƒObS~ ƒ~bax’bSiUr†x sZibSžŸxq†^bS`Zb( ¡mp~ r’^b&^Pst~†„ ž¢sp~’b(ba£splkvPsr’jnmpimt   ¡viPq¤r’jnmpiPx jni ¥w¦ bD„0§¨ƒPm7jkiUr ¡mp~`©sraªU ¡m7~"ƒ~’bDqjuxzjnmpiPxvwƒ—rmZ‹7“c{j«r†xS¬?]^wjnx"xq†^bS`Zb%y0jkbaln„x"spisp~q†^wj«rbaq¤rv~b­`©sp„b%mp   ¡m7v~ lnm0mp®U§Kvwƒ—r†st{lnbaxaªwsZ` vwl«rj«§¨mpƒObS~†stiw„sp„w„ bS~DªwstiP„—r9ž¢m©xz`©splklG`‘vlkr’jnƒlkjnbS~†xa¬,]^jux"ibSž4`Zbr^m „jnx bastlnvwsrba„stiw„qm7`cƒPst~ba„ ž"j«r^mpr’^ba~"ƒv{lnjuxz^bD„`cbSr’^m „xa¬ ¯±°U²;³´ µ š¶I˜p¸· jnƒwsp~zrj«rbrst{wlkbDxSªr†st{lnb§¹stiP„0§¨s7„„ jkr’jnmpic`Zbr’^wm0„wxSªU` vlkr’jnƒwsp~zrj«rbpªpº,lkba`cbaiUrst~y& ¡vwiwq¤§ rjkm7iwx. »G¼£½¿¾?ÀKÁ¹ÂSÀ½¿¾ÆĿ¾KÅ&Ã9Ƥƽ¿ÄÇÆȣĿÁ­Ã†¾¢ÃÉÊÁ9¾KÁzÆÉ3ËK¼—ÉÊÁ9ÌtŤÉÊÀ"ņÍGÀ3¼£ÁVΠÆÈtŤÉKÃÀ3Ť½¿É3Á%Ï£Á ÄuÐ ÑÓÒDÍnŤÉÊÔ ÃÀ3½¿ÕaÖ£Á Ï£ÖZ×wÆÉ3ÆĿĿØ9Ä¿½¿¾KÔ"Á ÙڤچÛwܡݤݒޤޤÞwßÓàáDâSã’äåSæ’áwßuç¤èSÝé£ê’ë£ì. Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe, 38330 Montbonnot-St-Martin (France) Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52.

(6)  &' $&"

(7) ¢I ©! !$%&'()&*! &,)&& !  &  ˜

(8)    bSrst~’r’juqlnb±ƒ~hax’bSiUr’b±vwi impvwpbSl­stlnopmp~jkr’^`Zb±ƒOmpv~ l  hSstlnvwstr’jnmpi „ b± ¡m7iwq¤rjkm7iwxHbai 0jn~’o7vlnb ¥¦ b7ªƒOmpv~H„baxZƒ~haqSjnx’jkm7iwxZstlnlnspiUr­€9vwx|Uv   ‹7“ {j«r†xS¬  bSr©splko7mp~j«r^`Zb bDx9r©{ws7xzh±x’v~©vib sp~q†^wj«rbaq¤rv~b&qmp`ZƒOm7x’hSb&„ b ‰HlkbDq¤rv~’bDx „ br†st{lnbaxaªOvisp„„j«rjkm7iibSvw~V` vwl«rj«§¨mpƒOhS~†stiw„baxbr(„ b „ bav ¦ ƒObrj«r†xH`‘vl«rjkƒwlkjnbSv~†xa¬  br’r’b±im7vpbalklnb±`Zhr’^wm0„b±baxzrHhSstlnvhSb±bSr—qm7`cƒPst~hSb±s£pbDq„b±ƒ~’hDqhD„ bSiUrx splko7mp~j«r^`Zbaxa¬ ‡wmpiwµ q™Dr’˜ jn³ mpœ iw x ˜(halk ha`Z]IbSstiU{rsplnbajk~xIba{x jnƒwst~’r’jnbaxaª£`Zhr’^wm0„bax V{Pspx’b„ b¢rsp{lnbaxbr?„  sp„„ jkr’jnmpiPxSª£r†st{lnbax` vlkr’jnƒwsp~zrjkbDxSª.

(9) 

(10)    

(11) 

(12) ! "##%$&(')*+

(13) $, (')  ‹. - .0/21 354760/98 /;:=<?>A@CB?DFEG/IHJDKBL3M/9N /2OPB?EQFBP@5RG<*. <L: :QS.S4T@5RG<*.S3 -.0/1324- 57698;: <7=?B%CK8O<D5UE FH2 5 -13E 5ULMI1KET.OE 2TNO50.OEQSRT13LM 5UCVRT:ICKCK57<. U. V ‘ ©!' ,)&. ]Ist{wlkbS§K{ws7xzbD„`Zbr^m „xI ¡mp~Ir’^wb¢^Pst~†„ ž¢sp~’b,bSstlnvwsrjkm7i mp w ¡viwqr’jnmpiwx ^Ps£pb,{ObSbai r^b"xzvw{ €9baqr mp w` vPq†^ ~baqSbSiUr%strzrbSiUr’jnmpiXWk”7ª;‹wª;‰wªŒ0ªZY[¨¬ ]^baystlnlkmžŸrmqSmp`Zƒv rbcqSmp`Z`Zmpilnyvwx’ba„± ¡vwiwq¤rjkm7iwx(ž"j«r^Jlnmž s7qSqSv~s7q0 y \ÊqSv~~’bai7rlky vƒ r’m d‰ {jkrx

(14) ]cž"j«r^ xzjnopij ¥ qastiUr’lny lkmž¢bS~Z^wsp~„ž¢sp~’bqmUx9rZr’^Psti r’^wstrHmt %s xzr’~†stjnop^Ur’ ¡mp~ž¢sp~„ rsp{lkb%jn`cƒwlkba`cbaiUrsrjkm7iGªUž"^jklnb%{Pbajkio‘ Êspxzr’ba~,r^wsti xz^jk Órz§¹stiP„0§¨s7„„Hsplko7mp~j«r^`©x_^ ` Ma C‚ bCc  mp~ ƒPm7lky0im7`cjustl¢stƒƒw~’m ¦ jn`Zstr’jnmpiwxa¬ ]^wbSy st~b©ƒwst~’r’juqvlust~lny vwx’b ¡vljki „ jko7j«r†stl¢xzjnopiwspl?mp~ jn`©sto7bVƒ~m qbaxx’jkioPª stiw„ stluxzm  ¡mp~ƒ~mUju„ jnio jnijkr’justlGx’bSbD„Hstlnvbaxrm‘jkr’ba~str’jnpb%`cbSr’^m „x"x’vwq†^spxr’^b ŽVbSžrmpi §¹‚ stƒw^wxzm7i splko7mp~j«r^`©x­ ¡mp~&„ jn0jnx’jkm7i stiw„ x’|Uvwsp~’bZ~m0mtr W #Š [,ž"^juq†^ sp~’b©qSmp`Z`Zmpilnyvwx’ba„ jni r^eb dwmUsr’jniop§KƒOmpjniUr¢viwj«r†x"mt  qv~~bSiUr¢ƒw~’m qbDx’x’mp~†xa¬ ]^juxcƒwstƒObS~cjni7r~’m „ vPqbaxcsiwbSž r†st{lnb§¨{wspx’ba„ xq†^ba`cb  ¡mp~Zqm7`Zƒv r’jnioJbSlnbS`ZbSiUrsp~’y  ¡viPq¤r’jnmpiPxS¬ f ^wjklnbr^bJ{wj«§Zmp~ ` vlkr’jk§KƒPst~’r’jkr’b `ZbSr’^m „xst~b{ws7xzbD„ mpi s ¥ ~†x9r’§Km7~„ ba~—stƒwƒ~’m ¦ jn`©sr’jnmpi+mp &r’^b  ¡vwiwq¤rjkm7iGª,r^jux©x’q†^ba`Zb vwx’bax ¥  Ór’^ §¨mp~†„ bS~crbS~`Zxcmp Vr’^wb]Is£y0lkm7~‘b ¦ ƒwstiwx’jnmpiGªž"^jnq†^ xz^wmpvlu„ splklnmž {Obr’r’ba~ s7qSqSv~s7qy(ž"jkr’^&r^b"x’sp`Zbst`Zm7viUrImp w^wst~†„ žst~bp¬]^wb,r~s7„ bSGm gcjuxr’^wstrIr^juxIxq†^ba`cb¢jniU7mplnpbDx x’`©stlnl`‘vl«rjkƒwlkjnbS~†xSh¬ f b b ¦ ƒwlkm7~’b&s©~†stiwopbmt  jn`Zƒlkba`ZbSiUrstr’jnmpir’~†sp„bSm g xVspiw„±qm7`Zƒwst~b(r’^ba`¸ž"jkr’^ ƒw~’baUjnmpvPxzlny©ƒv{lnjuxz^bD„st~†q†^j«rbaqr’v~bax ¡mp~r^bbSsplkvwstr’jnmpimt  ¡viwq¤rjkm7iwxa¬ ]^jux&ƒPstƒObS~ jux&m7~’oUstij iSbD„Js7x( ¡mplnlnmž x jTc¹l i k0baqr’jnmpi d ª ž¢b©b ¦ st`Zjnib—x’mp`ZbZrsp{lnb§¹stiw„0§¹sp„w„ j«rjkm7i `ZbSr’^m „xaC¬ k0baqr’jnmpi ‹Hƒ~’bDxzbaiUrxr^b ƒ~jniwqjnƒlnb‘stiP„opjnpbaxVs©{Pm7viw„±m7ir’^b‘`©sr’^wbS`©srjnqastlspƒƒ~m ¦ § jn`©srjkm7i ba~’~mp~(mp ¢r^b©ƒ~mpƒOm7x’ba„J`Zbr’^wm0„Gm ¬ k0bDq¤rjkm7i ‰„jnxqvwxx’baxjn`Zƒlkba`ZbSiUrstr’jnmpi jnxx’vbax&x’vwq†^ spx ~mpvwiw„ jniowªwspiw„opjnpbDx spqr’vwspl„strsZ ¡m7~Vxzm7`Zb ¡viwqr’jnmpiwxVž"^jnq†^sp~’b&qSmp`Zƒwsp~’bD„HrmHƒ~bS0jkm7vwx’lky ƒv{ § lnjuxz^bD„—~bax’vlkrxa¬ k0baqr’jnmpP i nc„ jnxqvPx’x’bax¢ ¡v rv~b~’bDxzbDst~†q†^—„ jn~’bDq¤r’jnmpiPxS¬. o p I$©) ' q &sr Mt uv y w xCz|{ }~C€,{ }|{,‚Fƒ„‚Z{ 5†M‡ Ijn~xzrƒ~’bDxzbaiUr’ba„—{0yˆ­b s7xM k st~`©s stiP„Š‰str’vlusŠWk”[;jni r’^bx’ƒPbDqj ¥ q(qSspx’b%mt Gr^b(~’bDqjnƒ‰~’m vqastlnlklPba~5 ¡vÇW iwŒ‹¨[q¤ª7rjkstm7iPiG„ ª r^jux?`Zbr^m „‘^Pspx {ObSbai©opbSiwbS~†stlnjnx’ba„‘jkiP„ bSƒObSiw„bSiUr’lny‘{0y2 k q†^0vl«rbVstiP„20k r’jnibW Y#P[ spiw„± ¥ iwstlnlky {UyŒVb jnibaq†^wjki stiP„J]jux’x’bS~†stiP„lW ‹¨[ ¬¨c r qmpiPxzjux9r†x(jki sƒjnbaqbaž"jnx’b ¥ ~†x9r’§Km7~„bS~(stƒƒ~m ¦ jk`©str’jnmpi rm©r’^wb ¡viwqr’jnmpi±ž"^ba~’br^b&`‘vl«rjkƒwlkjuqSstr’jnmpiwx­st~b&s£pmpju„ bD„—{0yx9rmp~jkioHspƒƒ~m ¦ jk`©srba„ ƒ~m0„vwq¤r†x jni. s #Ò Ž  ’‘ “.

(15) #"‹') ( #$, `  =h ' 

(16)

(17) '  # , & P')

(18)  P$, ` ‰. . . . . x’ƒObaqj ¥ q rsp{lkbDxSe¬ c¨r%r9y0ƒjnqastlnlkysplklnmž x" Êspj«r^  ¡vlnlky ~’m7viw„ bD„ ¡viwqr’jnmpiwx­ ¡mp~c”aŠ—{jkrx(mt ?ƒw~’bDqjuxzjnmpijniJs  ¡baž ({0yUr’bDx¢mt Irst{wlkbDx"stiw„sZxz`©splkl `‘vl«rj«§¨mpƒObS~†stiP„ s7„„ bS~D¬. tMu(t. { 5‚ €l{, _z‚ ‹ C‡ ‹ ‡C‡_}|{ } † ƒ ‚ {, C†M‡ ­spxxzlnbS~ZstiP„ ]Isto7sp®0jCW ‰ [ „ baqSmp`ZƒOm7x’b sJƒPm7lky0im7`cjustlstƒwƒ~’m ¦ jn`©sr’jnmpi

(19). .   . . .  r’mJr’^b  ¡viwqr’jnmpi+spxcsJx’v` mp ­ƒ~m „ vwqrx©mp ­{jkrx©mp ­r^b±jniƒv rHž¢mp~†„ ¬ ]^bx’brHmt ­r’^bDxzb±ƒPst~’r’justl"ƒ~m „ vwq¤r†xZjnxH^bSv~jux9rjnqastlnlky „baqm7`ZƒPmUxzbD„jni7rmxzvw{wxzbSrx(ž"^juq†^ st~bcxzr’m7~’bD„ jni rst{wlkbDxSªspiw„ sp„„ bD„ jni s` vlkr’jk§Km7ƒPba~spiw„ sp„„ ba~a¬ Vimtr^bS~Z`Zbr^m „ jnxcopjnpbai {0y f mpio stiP„ f%mpr’m Wk”D“[Kª¢ž"^ba~’bsp„„j«rjkm7iwxZst~b ƒObS~’ ¡mp~`Zba„ {Ob ¡m7~’b spiw„Zs ÓrbS~ r^b"rsp{lnb lkm0m7®7§¨vƒwxa¬Vl«r^mpvwop^Zr’^wbax’b"r9žm(`Zbr^m „x,qSmpiwx’ju„ bS~?^wjko7^bS~’§Km7~„bS~Ir’ba~’`©xaªtr’^wbSy ^Ps£pb?{ObSbai xz^wmž"i(r’mVƒw~’m „ vwqSb,lust~opba~stiw„ xzlnmžba~st~†q†^jkr’bDq¤r’vw~’bDx r’^Pstir^b,o7bSiba~splkjiSbD„(` vlkr’jnƒwst~’r’jkr’b `ZbSr’^m „ ªwspiw„—r’^bay st~bqjkr’bD„H ¡m7~ ~’bS ¡bS~bSiwqSb­m7ilnyp¬. tMu. } 5‚ € †+€&‡C‚ € ƒ„‚Z{ C†h‡ ]^wbb ¦ ƒPm7ibSiUr’justl"x’j`ai bmt  r†st{lnb§¨{wspx’ba„ . . . . . st~†q†^jkr’baqr’v~baxc~bSiw„bS~†x&r^bS` jk`Zƒ~†spqr’juqSstl¢ ¡m7~cƒw~’bDqjuxzjnmpiwx ^wjko7^bS~¢r’^Psti±d‰Z{jkrxa¬ · ~vo7vbS~†sªU}?jbSjn~’m©spiw„Š‰vlklnbS~"^Ps£pb­r’^ba~’bS ¡mp~b­ƒ~bax’bSiUrba„—spisp~q†^wj«rbaq¤rv~b {Pspx’ba„+mpi s x’baqSmpiw„ §Km7~„ ba~Hstƒƒw~’m ¦ jn`Zstr’jnmpi vwx’jkio s ` vlkr’jnƒlnjkba~spiw„ s „ bD„ juqSsrba„ x|Uvwst~bS~©viwj«r W [¨¬ !Gju„„ juqm7strVstiw„ Ilny0ii W n[Ix’jk`Zjnlnsp~’lny—ƒ~mpƒOm7x’b&sti±sp~q†^jkr’bDq¤rv~’b( ¡m7~Vqm7`Zƒv r’jnioZr^b ~baqSjkƒ~m qSspl \ʞ"^m7x’b©]s£y0lnmp~(b ¦ ƒwspiwx’jkm7i jniU7mplnpbDx­m7ilnyvijkr‘qm0#b "©qSjkbaiUr

(20) x ]VvPxzjnio±xzbapba~spl x vijkrxjni ƒwst~†stlnlkbal3ª  ¡m7lklnmž¢ba„H{0ys‘`‘vlkr’jk§Km7ƒPba~spiw„—s7„„ bS~D¬ ]^bJ`Zbr’^wm0„ ƒ~’bDxzbaiUr’ba„+^wbS~bjnx—jkiUrbS~`cbD„ jusr’b {PbSr9žbabSi+rsp{lnb§¹stiw„0§¹sp„w„ j«rjkm7i+`cbSr’^m „xstiP„ `‘vlkr’jnƒlkjuqSstr’jnmpi §¨{ws7xzbD„H`Zbr^m „xS¬ k. IS &"!

(21) ¢ ©!  )† 

(22) &S

(23) ' ¤)&%) . $. Muv w s{ C‚ ƒ„s{ } zT~C~_€ † 5} ƒ J{ } † ]^w~’m7vop^m7v rr’^wjnx ƒwstƒObS~‘žbHžstiUrrmbSstlnvwstr’b ¬Pf b—spxx’v`Zb x ∈ [0, 1[ jnx‘sti ƒOmpjniUr  ¡mp~`ZstrIž"jkr’^ n = 4k +p ¬ f bqasti r’^0vwx ž"~j«frb (x)x = x +x 2 +x 2 +x 2 . ('. *). . ž"^wbS~b&r’^wb x  x(st~b k§¨{jkr%i0v`‘{Pba~x%{ObSlnmpio7jkiwoHr’m x’vwq†^ r’^wstr p < k ¬ f b^ws£pbj i. x=. x0. x1. ª;b ¦ qSbSƒ r% ¡mp~ [0, 1[ 0. x2. 1. −k. x3. 2. x4. ¨§ {jkr ¥¦ ba„ ª r^wsrjux%s §¨{jkr(i0v`‘{Pba~x −2k. n −3k +x4 2−4k 3 p. x4. ]^bV`cbSr’^m „HqSmpiwx’jux9r†x jki—ž"~’jkr’jnio&sti©m7~„bS~’|§ n%]Is£yUlnmp~,b ¦ wƒ stiPxzjnmpiZmt  sr ª0stiP„Z®pbSbaƒjniompilny r^b(r’ba~’`©x~bSlnbSstiUrr’mcr^b(rsp~’o7br spqaqv~†spqSyZr’^wstrjux 2 ¬ fJbo7br j f x 0. −4k+p. Ñ,+JGÑ.-.

(24) 

(25)    

(26) 

(27) ! "##%$&(')*+

(28) $, (')  f (x). n. = f (x0 ) + [x − x0 ]f 0 (x0 ) + 12 [x − x0 ]2 f 00 (x0 ) + 16 [x − x0 ]3 f 000 (x0 ) 1 [x − x0 ]4 f (4) (x0 ) + 24 1 + 120 [x − x0 ]5 f (5) (x0 ) + 1. (T0 ) (T1 ) (T2 ) (T3 ) (T4 ) (T5 ). \9”‹]. ] ^b bS~~’m7~qm7`c`Zjkrzrba„ uj x  = ([x 2 + x 2 + x 2 + x 2 2 c¹iba|Umax vwstr’jnmp|fi \9”|]¤ªž¢b(b ¦ ƒwts iP„Hr^b(r’ba~’` (T ) = [x − x ]f (x ) spx¢ ¡mplnlnmž7j 1 720. 1. 1 −6k 720. (6). −k. 1. 2. 1. −2k. 0. 3. 0. −3k. 4. −4k 6. ] )f (6) (ζ1 ) <. 0. (T1 ) = [x − x0 ]f 0 (x0 ) = x1 2−k f 0 (x0 )+ x2 2−2k f 0 (x0 )+ x3 2−3k f 0 (x0 )+ x4 2−4k f 0 (x0 ). 1 24. ŽVmž lnbr&vwx ¡m0qSvwxm7i rbS~`Zx (T ) = [x − x ] f (x ) ª (T ) = [x − x ] f (x ) ª spiw„ (T ) = [x − x ] f (x )  ¡~mp` Db |7vPsr’jnmpi \9”]¤¬ f b^ws£pbj [x − x ] f (x ) 0. 4 (4). ž"jkr’^ f b%mp{ r†st<jni©r’2^b 2. s #Ò Ž  ’‘ “. 0. 1 −5k 2. 5. 1 2. 2 1 120.  (T2 )                            (T ) 3                 (T4 )            (T5 ). 0. 0 5 (5). 2 00. 3. 0. 1 6. 0. 3 000. 0. (T4 ) =. 0. =. 1 2 −2k 00 f (x0 )+ 2 x1 2 x1 x2 2−3k f 00 (x0 )+ x1 x3 2−4k f 00 (x0 )+ 1 2 −4k 00 f (x0 )+ 2 x2 2 x2 (1/2)2−5k f 00 (x0 )+ 1 −5k 00 f (x0 ) + 2 2 x1 (1/2)2. =. 1 3 −3k 000 f (x0 )+ 6 x1 2 1 2 −4k 000 x x 2 f (x0 )+ 2 2 1 1 2 −5k 000 x x 2 f (x0 )+ 3 2 1 1 2 −5k 000 x x 2 f (x0 ) + 3 1 2 2. =. 1 4 −4k (4) f (x0 )+ 24 x1 2 1 3 −5k (4) x x 2 f (x0 ) + 4 6 1 2. =. 1 5 −5k (5) f (x0 ) 120 x1 2. + 5. ª stiw„ ¬ max |f |  < 2 ʙ3œpª› ™ ° ™a› – °7³ max –I›˜ ° |f¶ |µ š  < ʛ 0{ 2yZ~’baž"max ~’jkr’jni|fo‘º,||Uvwsrjkm7iL<\9”],s7x?2 ¡m7lklnmmax ž xj |f | 00. 3. 1 −6k 3. 000. 4. 1 −6k 24. (4). 5. 1 −6k 120. (5).

(29) #"‹') ( #$, `  =h ' 

(30)

(31) '  # , & P')

(32)  P$, ` Š. . f (x). ž"^wbS~b. . = A(x0 , x1 ) + B(x0 , x2 )+ C(x0 , x3 ) + D(x0 , x4 )+ x2 × E(x0 , x1 )+ x3 2−k × E(x0 , x1 ) + f. . . \3dG]. = f (x0 ) + x1 2−k f 0 (x0 )+ 1 2 −2k 00 f (x0 )+ 2 x1 2 1 3 −3k 000 x 2 f (x0 )+ 6 1 1 4 −4k (4) f (x0 )+ 24 x1 2 1 5 −5k (5) x 2 f (x0 )+ 1 120 1 −5k 00 x (1/2)2 f (x0 )+ 1 2 B(x0 , x2 ) = x2 2−2k f 0 (x0 )+ 1 2 −4k 00 f (x0 )+ 2 x2 2 (1/2)x2 2−5k f 00 (x0 ) C(x0 , x3 ) = x3 2−3k f 0 (x0 ) D(x0 , x4 ) = x4 2−4k f 0 (x0 ) E(x0 , x1 ) = x1 2−3k f 00 (x0 )+ 1 2 −4k 000 f (x0 )+ 2 x1 2 1 3 −5k (4) x 2 f (x0 )+ 6 1 1 −5k 000 x (1/2)2 f (x0 ) 1 2 f ≤ 1 + 2 + 3 + 4 + 5 1 −6k 2 max |f (6) | + 21 2−5k max |f 00 |+ ≤ 720 1 −6k 1 −6k max |f 000 | + 24 2 max |f (4) | 32 1 −6k (5) | 120 2  max |f ≤ 2−5k 21 max |f 00 | + 13 max |f 000 |+ 1 1 (4) | + 120 max |f (5) |+ 24 max |f 1 (6) |+ 720 max |f A(x0 , x1 ). VbSiwqSbpª qSsti {Pb(m7{ rspjkibD„—{0y©ƒObS~’ ¡mp~`Zjkio©d ` vlkr’jnƒlnjnqasr’jnmpiPx"stiw„ sp„„ jnio©x’j ¦ rbS~`Zx¢ž"jkr’^ ps i ba~’~mp~lkbDx’fx(x) r’^Psti  ¬ ]^b¢x’jiSb¢mt wr^brst{wlkbDx ž"^bS~bsplkl0r^bax’b,rbS~`©xIsp~’b¢lnmUm7®pbD„0§Kvwƒ ¡~’m7`ªt„ bSƒObSiw„ ^wbas£0jklny©mt Ir’^wb( ¡viwq¤rjkm7ižbqm7iwx’jn„ ba~a¬f bž"jklnl„ juxqvwxx"st{Ompv r"jkr jkir^b k0baqr’jnmpi‰w¬¿d ¬ c¨r` vwxzr{Pb(impr’juqba„©r’^wstr¢r^b%ƒ~’m7ƒPmUxzbD„Z`Zbr’^wm0„ b ¦ ƒlnmpjkrx¢„ ba~’jnsr’jnpb­ƒ~mpƒObS~’r’jnbaxmt G ¡viPq¤r’jnmpiPxS¬ ]^wb(x’vwqaqbaxx’jk7b(„bS~jksrjk7bax¢` vPx9rV„ bDq~bas7xzbD„H~†stƒju„ lnyHbSim7vop^ r’mZ`©st®7b(~’ba`cm7ba„Hr’bS~`©xjkiPxzjnopij ¥ § qastiUr qm7`cƒPst~ba„©rmZr’^bm7ib(žb(®7bSƒ rD¬ f. V

(33). 

(34) "‘(I)& Vi st{Px9r~s7q¤rVst~†q†^jkr’bDq¤r’vw~’b( ¡m7~ r’^wb ` vwl«rjkƒlnjuqSsrjk7b&`Zbr^m „±jnx­ƒ~bax’bSiUr’bD„mpi Ijnopv~bH”p¬Mc¨r%jnxVpba~’y x’jn`cjnlust~r’mcr^b` vlkr’jnƒwsp~zrj«rb&st~†q†^j«rbaqr’v~b­mp  W ‹#[¨ªž"jkr’^r’^bb ¦ qbSƒr’jnmpimt r^b(r9ž¢mc`‘vlkr’jnƒlkjnbS~†xa¬ %. Ñ,+JGÑ.-.

(35) 

(36)    

(37) 

(38) ! "##%$&(')*+

(39) $, (')  Input word. Tables. Œ. x0. x1. x2. x3. x4. k. k. k. k. p. E A. B. C. D. k+p+g. Small multipliers. 4k+ p+g. 2k+ p+g. k+ p+g. k+ p+g. p+g. p+g. Multi−operand adder Rounding. jnopvw~’bZ”Gj?]^wb(sp{wxzr’~†spq¤r sp~q†^jkr’bDq¤rv~’b­ ¡mp~r^bƒ~mpƒOm7x’ba„ x’q†^ba`Zb 4k+p. Cuv  †x C‡_} „‚ € €&†+€ .  . . c¹i sp„„ jkr’jnmpiHr’m&r^b(stƒƒ~m ¦ jk`©str’jnmpiZbS~~’m7~x,qmpiPxzju„ bS~ba„Zjni©r^b­ƒw~’baUjnmpvPx,x’baq¤rjkm7iGª7r’^juxsp~q†^wj«rbaq¤rv~b jni0pm7lk7bax¢xzbapbS~†stl x’mpv~†qbDx¢mt  ~’m7viw„ jnioZbS~~’m7~ j.  ]^b r†st{lnbax ^Ps£pb r’m"{Ob ¥ lnlnba„(ž"jkr’^ ¥¦ ba„ §KƒOmpjniUrstlnvbax ž"^wjnq†^st~b r’^wb?ƒ~bS0jkm7vwx;`©sr^bS`©srjnqastl. stlnvbax~mpviP„ ba„ r’m©x’mp`Zb(ƒ~baqjux’jkm7iG¬  k0jn`cjnlust~lky7ª0r’^bm7v r’ƒvr mt Ir’^b`‘vl«rjkƒwlkjnbS~†x^ws£pb%rmc{Ob~mpviw„ba„—r’mZr’^wb(rst~opbSr¢ƒw~’bDqjuxzjnmpiG¬  ]^bax’b­spƒƒ~m ¦ jk`©srjkm7i©bS~~’m7~xaªpspiw„Zr’^wb%spƒƒ~m ¦ jn`©srjkm7i©bS~~’m7~aªt~ba|Uvjn~bVvwx,r’mcqmp`Zƒvr’bVž"jkr’^ stijniUr’ba~’iwsplƒ~baqSjnx’jnmpiž"^juq†^jnxV^jnop^ba~ r’^Psti±r^b ¥ iPstlI~’bD|7vwjk~ba„ƒ~baqSjnx’jkm7iG5¬ f b‘qSlns7x’x’jnqastlnlky vwx’b g sp„„ jkr’jnmpiPstl {jkrx"mp ƒ~baqSjnx’jnmpi \¡mp~  $&# ˆ' ) ]¬ i \¡ž"jkr’^  ]^bSir’^bc~’bDxzvwl«rV^PspxVr’m—{Ob ~mpvwiw„ ba„rmHr^b ¥ iPstlIƒ~’bDqjuxzjnmpist ¡bS~Vr’^b ¥ iwsplsp„w„ j«rjkm7l stibS~~mp~sr ž¢mp~†x9rmt  sti^wstlk  !Jk · ]¤¬ ]^ba~’bS ¡mp~bpª0r^b(rsp{lkbx’`j iabaxibaba„ bD„st~b%x’vwq†^r’^wstr j Size(A[x0 , x1 ]) Size(B[x0 , x2 ]) Size(C[x0 , x3 ]) Size(D[x0 , x4 ]) Size(E[x0 , x5 ]). ≤ ≤ ≤ ≤ ≤. 22∗k (4k + p + g) 22∗k (2k + p + g) 22∗k (k + p + g) 2p+k (p + g) 22∗k (k + p + g). ]^wbV` vwl«rjkƒlnjnbS~vwx’ba„©rm `‘vl«rjkƒwlky ž"j«r^ juxs ` v lkr’jnƒlnjkba~spiw„©r^b%mpib­vwxzbD„ rmZ` vlkr’jnƒlny E ž"j«r^ x jux s (k) ×E(p + g)xmpiwbp¬ (k) × (k + p + g) ]^b‘i0v` {ObS~­mt  opvwsp~„{jkrxVjnx%„ bSr’bS~`Zjkiwba„±spx"r’^bcxz`©stlnlnbaxzrVstlnvb ’x vwq†^r’^wstr r’^wb‘x’v`¸mp  r’^b spƒƒ~m ¦ jk`©srjkm7iJba~’~mp~&spiw„ stlnl?r^b—~mpvwiw„ jnio±bS~~’m7~x(juxcxz`©stlnlnbS~r^wsti s !Jk · mt r’^b ¥ iwspl,~bax’vl«r \¡ Êstjkr’^  ¡vwl~mpviw„jkio ]¤¬ 2. 3. s #Ò Ž  ’‘ “.

(40) #"‹') ( #$, `  =h ' 

(41)

(42) '  # , & P')

(43)  P$, ` . . . . ŽVmtr’bcr^wsr(r’^wbc~mpvwiw„ jnio mp ,r^bZ` vlkr’jnƒlnjkba~~’bDxzvlkrxaªGspx%ž¢bSlnl?spx­r’^wb ¥ iwstl ~mpviP„ jkiwowª `©s£y±{Pb ƒObS~’ ¡mp~`Zba„ {0y x’jk`Zƒlnb—r~’vwiwqSstr’jnmpiF\3srZim ^wsp~„ž¢sp~’b—qSm7xzr

(44) ]ª?j« %qmp~~bax’ƒPm7iw„ jnioqm7~’~baqr’jnio±r’ba~’`©x ^Ps£pb%{PbabSi±sp„„ba„ r’mcr’^wb£splkvwb%vPxzbD„—rm ¥ lklGr’^b A r†st{lnbp¬. Cu(t 0‚ C‚ €  zI€ ‚ xCz { *. *. . ]^wb `Zbr^m „ ƒ~bax’bSiUrba„ sp{Pm7bH^ws7x‘{ObSbai jn`Zƒlkba`ZbSiUr’bD„ jni s  ƒ~mpo7~sp`¬ ]^juxcƒ~’m7op~†st` ¥ lnlnx r^bcrsp{lkbZstiP„ „ bSr’bS~`Zjkiwbax­r’^bcrmtrspl bS~~’m7~T\¡{0ybai0v`ZbS~†sr’jnmpi] ¡mp~(jniwq~bas7xzjnio £splkvwbax%mt  g ªGviUrjkl  Êspj«r^  ¡vlG~mpvwiw„ jnioZjnx"s7q†^jkbapbD„ ¬ ]^b,rsp{lkbxzjiSbDx  ¡mp~st~jkm7vwx; ¡viwqr’jnmpiwxopbaibS~†srba„%{0yVr^b,ƒw~’m7op~†st`Tsp~’b ƒw~’bDxzbai7rba„(jnirsp{lnb ”p¬sc¹i s7„„ jkr’jnmpiGª7žb qSmp`Zƒwsp~’bD„ m7v~?~bax’vl«r†x ž"jkr’^Zr’^wb"{PbDx9r,®0imž"icr†st{lnb xzj iSbVmt ;opbSiwbS~†stln`j iaba„‘` vlkr’jnƒwst~’r’jkr’b `ZbSr’^m „7x W ‹ [¨¬ ƒ ]Ix’st`j ia{b lnb qŽ mpvw~~` baq¤{OrbS~ {jkmpr x xz~j biS  b f k ‹ d dUŒŠ ”a‰ ‹0Œ0”Dd sin ‰ ‹ ” ‹ p n t “ U ‰ “ ” ‹ Y d  Yp‰p‰7“ [0, π/4[ n ‹ Œt“ npd dp‹ ”a‹ pŠ7dt‰ ‹ d ‹Udt‹7d ”a‰ Š7dUŒtd exp ‰ ‹ a ” U Š d G p n Š ” ‹ Y t n Š7‹7dt“ [0, 1[ n ‰ Ud‰7‹Ud dt‰ ‹pŠ7Šp“ 7“ ‹ d ‹7G‹ Y7d ”a‰ Œ0”DŠ 2 −1 ‰ ‹ ” 7“ t ‰ ” ‹ Y t n Š7‹7dt“ [0, 1[ n ‰ YpŠp“7“ dt‰ Gd nGYn p‰ ]Ist{lnbZG” j ]sp{lnb&xzj iSbDx"stiw„spqaqv~†spqSyZ ¡mp~‹” n§K{wj«rDªwdt“§¨{jkr stiw„Gd n§K{jkr"jkiwƒv r mpƒObS~†stiP„x x. a i—r’baxzr’bD„Z ¡viwqr’jnmpiwx¢ž¢bV~bS`©st~®‘r’^wstrr^bVr†st{lnb­x’j iSbV~ba|Uvjn~ba„©{0yZ` vlkr’jnƒlnjnqasrjk7b rsp{lnb­{ws7xzbD„ Z` bSr’^m „jnxlnbaxx¢r’^Psti r’^bm7ib(~’bD|7vwjk~ba„ {0yHopbaibS~†stlnj`iaba„—` vlkr’jnƒwst~’r’jkr’b(`Zbr^m „ ¬?]^b(r†st{lnbax A ª B ª ª ^ws£7b jniƒvr¢{jkrxaª0ž"^bS~bas7x r†st{lnb ^ws£pb jniƒv r{jkrxa¬ m7~¢splklOmt  r^bS` r^bVi0v`‘{Pba~ C E mp  mpv rƒv rV{2kjkrxVjnxVst~jnsp{lkb&spiw„±„ baƒPbaiw„x m7Dir’^b&rstp{w+lkb7kªP{vr­stluxzm©m7ir’^b& ¡viwq¤rjkm7iG¬ ]^bDxzb(r†st{lnbax sp~’bHr’^0vwxcxz`©stlnlnbS~&r^wsti r’^mUxzb jki r’^b` vlkr’jnƒwsp~zrj«rb—`Zbr^m F „ \ 2 žm7~„wx&pba~x’vwx 2 ]ª ž"^juq†^ qSmp`ZƒObSiwxsrbax¢ ¡mp~r^b` vwl«rjkƒlnjnbS~†xS¬ 2n/5. Cu. . I€ +‚ { { }  . ( . n/2. . ]^wjnx¢`Zbr^m „H~bSƒ~bax’bSiUrxsti jk`ZƒOmp~’rstiUrjk`Zƒ~mpba`ZbSiUr,mpba~?r^b­`‘vlkr’jnƒwst~’r’jkr’b%`Zbr’^wm0„ ž"^baiHrsp~z§ o7br’r’jnio©lns7x9r’§Ko7bSiba~str’jnmpi }¢f(Vx \ Vjn~zrb ¦ cc"stiw„ ­jk~’r’b ¦ }?~m ]ªž"^juq†^ba` {Oba„±sclust~opb(i0v` {ObS~ mp  x’`©stln+l \ 18 × 18 → 35 {jkrx

(45) ]¢` vlkr’jnƒlnjkba~xa¬?]^bSjn~"ƒ~baqSjnx’jkm7i—jux"`Zmp~bVr^wstibaimpvo7^— ¡m7~r’^jux"xq†^bS`Zbpª spiw„ r’^bay splklnmžTs‘r’~bSbS§3 ¡m7ln„~ba„ vPq¤r’jnmpimpir^brsp{lnb&xzjiSbsr iwmHqm7xzr xzjniwqSb(r’^baysp~’bs£stjnlnsp{lnb­m7i r^b(q†^jkƒ¬ wmp~xz`©splkl k jkr¢`©s£yZbS7bSiH{Ob(ƒPmUx’x’jn{lkbVr’m‘vwxzb%mpiwlkyZmpib%` vwl«rjkƒlnjnbS~r’mZqmp`Zƒvr’b­r’^b­r9žm ƒw~’m „ vwqrxa¬ Ñ,+JGÑ.-.

(46) 

(47)    

(48) 

(49) ! "##%$&(')*+

(50) $, (')  Y. c¨rx’^m7vln„c{Ob"imtrba„‘r’^Psr?r^brsp{lkbDx?qSspiZ{Pb" ¡vw~zr^bS~,qSmp`Zƒ~baxxzbD„&{0y {Pspq†®U§Kbaiw„‘lkm7opjuqmpƒ rjk`Zjias§ rjkm7irmUm7lnxa¬%}?~’ba0jkm7vwxVž¢mp~®*W d#[ ^Pspx%xz^mž"ir’^wstr(sdt“jn`Zƒ~’m7bS`ZbSiUr%mpi±r^b r†st{lnbZxzjiSbcqSstiJ{Pb b ¦ ƒPbDq¤r’bD„ ¬n¬ Ijniwstlnlky7ªjkr&jnx&viwqlnbasp~^mž r^jux`Zbr’^wm0„ ž¢mpvlu„ qm7`cƒPst~b rm*s \Ê`Zmp~bZxzr’~†stjnop^Ur’ ¡mp~ž¢sp~„&] jk`c§ ƒwlkba`cbaiUrsrjkm7i—mp Is  ¡viwq¤rjkm7i vPxzjnioZs‘ƒwjkbDqbSž"jux’b%„ baop~bSb(d m7~¢‹cƒOmplny0imp`Zjustl;bSstlnvwstr’jnmpiHb ¦ ƒwlkm7j«rjkio r^bx’sp`Zb%`‘vlkr’jnƒlkjnbS~†xa¬?]mcmpv~"®0imž"lnba„opbpª x’vwq†^stistƒƒ~m7s7q†^H^ws7ximtr"ypbSr"{PbabSiƒv{lnjnx’^bD„ ªwstiP„ jkr jux¢r’^b&x’v{ €9bDq¤r mt  qv~~bSiUr"žm7~’®;¬ f. Cu. I€ +‚ { { }. (  . . ^wbSi r†st~opbr’r’jnio !Jk c¤ªr’^bqmp`Zƒwsp~’jux’mpi ž"j«r^ ` vlkr’jnƒwsp~zrj«rbHrsp{lnbax&~bax’v`Zbaxr’m±r~s7„ jkiwolkm0m7®7§ vwƒJr†st{lnbHx’jklnjnqSmpi  ¡mp~`‘vlkr’jnƒlkjnbS~‘xzjnlkjuqm7iG¬—]m{Pb© Êspjk~Dªjkr x’^mpvlu„ {Pb©iwmtr’bD„Jr’^wstr‘sti spqr’vwspl !Jk c jn`ZƒlnbS`ZbSiUrstr’jnmpiž"jklnlG{Ob&xzm7`ZbSž"^wstr"„j g;bS~bSiUr ¡~’m7`ž"^wstr stƒƒObasp~xm7i Ijnopv~bZ”Gj  W`‘vlkr’jnƒlkjnbS~qluspxx’jnqastlnlkycqmpiPxzjux9r†x?mp sƒwst~’r’justlPƒ~m0„vwq¤rst~~s£y7ªps qast~~’yU§¹x’s£7b~ba„ vPq¤r’jnmpiZr~’babpª stiw„s ¥ iPstls7„„ bS~D+¬ c¹impvw~Vst~†q†^jkr’baqr’v~b­r’^ba~’bž"jnlkl{Pbm7ilny—mpiwb ¥ iwstlsp„„ ba~ sr"r^b&bSiP„ mp  r’^b©`‘vlkr’jk§Km7ƒPba~spiw„ s7„„ bS~D¬ · bDxzju„ bDxSª  ¡m7~&s k × k mp~s 2k × k ` vlkr’jnƒlnjnqasrjkm7iGªGr^b©„ bSƒ r^ mt "r^b ~ba„vwq¤rjkm7i r’~bSbHž"jnlnl¢{Ob—7bS~y x’`©stlnljni r^b m7bS~†stlnl,q~jkr’juqSstlƒwsr^G¬  lkr’mpo7br^bS~‘r’^b sp„„j«rjkm7iwstl lusrbSiwqSy—„ vb(rmZr’^b`‘vl«rjkƒwlkjnbS~†x"xz^wmpvlu„—{Obpba~’y xz`©stlnlK¬  a v~spƒƒ~m7spq†^%mt 0r~’viPqSsrjkiwor’^b?~bax’vlkrmt s¢ ¡vlnlp`‘vlkr’jnƒlkjnbS~jnxqba~zr†stjnilky,€9vwxzr’j ¥ ba„m7i }¢f(Vx ž"^bS~br^bax’b& ¡vlnlI` vlkr’jnƒlnjkba~x%st~bƒ~bax’bSiUra¬ VmžbapbS~Dªjni !Jk c¤ªOj«r%jnx­ƒ~mp{wsp{lky `Zmp~b st~bast§ stiw„J„ bSlus£yU§K#b "HqjnbSiUrVrm—r~’viPqSsrb r^bcƒwst~’r’justl ƒ~m „ vwq¤rst~~†s£yp¬(]^b‘~mpvwiw„ jnio—ba~’~mp~­ž"jklnl {Pb ^jnop^ba~aª `©s£y0{Pb ~’bD|7vwjk~jkiwo±`cm7~’b—opvwsp~„ {jkrxaª {v rcr’^b mpba~splkl?x’ƒPbaba„ spiw„ sp~’bDs±xz^m7vlu„ {Pb ~’bD„ vwqSba„ ¬?]^bxzr’vw„y—mt Ir’^wjnxr~s7„ b§¨m g±~bS`©stjniwx¢rmc{Ob&„ m7ibp¬  ")& &! '—' I ‡&sr a iwb­qSspi—~bS`©st~®‘r’^wstrr†st{lnbax , x ] = x ∗ 2 ∗ f (x ) spiw„ D[x , x ] = x ∗ 2 ∗ f (x ) xzr’m7~’b‘r’^wbZxst`ZbZstlnvbaxa¬Z]^bS~C[x b ¡m7~’b7ª žbZqSspiJ~ba„ vPqb‘r’^b©m7bS~†stlnlr†st{lnb©xzjiSbc~’bD|Uvjk~ba„ {0y x9rmp~jkiwo m7ilny r^b C rsp{lnbp¬ c¹iZr’^jux¢x’q†^ba`ZbpªU£splkvwbax?ƒw~’baUjnmpvPxzlny‘xzr’m7~’bD„‘jni D ž"jnlklO{ObVlnmUm7®U§Kvƒc ¡~mp` rsp{lkb C s7„„ ~baxx"ž"jkr’^  ¡m7lklnmž¢ba„{0y±s §K{wj«r(xz^jk Óra¬%]^b‘xs£0jkiwo©jki x’ƒws7qb&jux­x’`©stlnl \3st{Ompv rc”0”D‹r^ mp r^b(rsp{lkb&x’j (xiS‹b ]¢, spxiw„) ž"jklnl ƒ~mp{wsp{lky—kimtr"{Ob€9vwx9rj ¥ ba„ ¬ ]^b±`‘vlkr’jnƒlkjuqSstr’jnpb`Zbr^m „xZƒ~bax’bSiUr’bD„ jn0 i k baq¤rjkm7i ‹ juxZ{wspx’ba„ mpi+sti mp~†„ bS~’§ 5 ]Is£yUlnmp~cb ¦ § ƒPstiwx’jkm7iJž"jkr’^ spiJjniƒvr žm7~„Jba|Uvwsplklny x’ƒlnj«r jniUr’L m nx’v{ §¨žm7~„ ¬H]^wjnx`ZbSr’^m „ qSspiJ{Ob©rviba„ {0y jniwqS~’bDspx’jkio r’^b(m7~„ ba~mt r^b]s£y0lnmp~¢b ¦ ƒwspiwxzjnmpiªU{0yH~bSlus ¦ jnio&r^b&xzr’~jnqr 4k + p xzƒwlkjkrzrjkio©ž¢b%^ws£pb vPxzbD„ ª;mp~%{UyqSbSiUr’ba~’jniostƒƒ~m ¦ jk`©str’jnmpiwxaC¬ c¨rž"jnlkl lnbasp„±r’msti jn`Zƒ~mpbS`Zbai7r­jki rst{wlkbZxzj iSbZstiP„ £m7~ jni spqaqv~†spqSyxzjn`Zjklust~%r’mr^wsr&m7{ rspjkiwba„J{UyJopbaibS~†stln`j iaba„ ` vlkr’jnƒwst~’r’jkr’bZ`Zbr^m „x W #‹ [qSmp`Zƒwsp~’bD„±r’m ƒw~’baUjnmpvPx ` vlkr’jnƒwsp~zrj«rbV`Zbr^m „Cx Wn”tªŒ0ª Y[¨¬ ]^wb­„ ~†s£ž"{wspq†® jnx,r’^wstr,r^bVijuqb%]s£y0lnmp~  ¡m7~’`‘vlustb"^ws£7b rmZ{Pb~bSƒluspqSba„ ž"j«r^`cm7~’b(o7bSiba~splOƒOmplny0imp`ZjustlGspƒƒ~m ¦ jn`©srjkm7iw_x \Ê`Zjkijn`©s ¦ mp~"xzjn`Zjklust’~ ]¤¬ c¹iopbaibS~†stlKª7r’^ba~’bst~b%`Zspi0y©r’~†sp„ bam g x¢ž"^jnq†^qSspi {Ob(mp{ r†stjniba„ {0yHjniwq~bas7xzjnio‘r’^b(i0v` {ObS~"mp  x’`©stlnl ` vlkr’jnƒlnjkba~xa¬ c¨r juxƒwst~’r"mt  mpv~¢ ¡v rv~’bjni0pbDx9rjkoUsr’jnmpi¬.

(51). 0. 0. s #Ò Ž  ’‘ “. 4. 3. 3. −3k. 0. 0. 0. 4. 4. −4k. 0. 0.

(52) #"‹') ( #$, `  =h ' 

(53)

(54) '  # , & P')

(55)  P$, `. ”D“. . . . . & ')S& f bc^ws£pb‘ƒ~’bDxzbaiUr’ba„ypbr(stimpr’^ba~­`ZbSr’^m „  ¡mp~(stƒƒ~m ¦ jk`©str’jnio—st~{jkr’~†st~yH ¡viPq¤r’jnmpiPx(vƒrmx’jkiwoplnb ƒw~’bDqjuxzjnmpi \ 24§¨{j«’r ]¢vwx’jkio—`cm „ ba~str’b(r†st{lnb x’j`iabp¬]^brsp{lnb xzjiSbjux ~ba„vwqbD„ vwƒ±r’mZr~’bab§K ¡mplu„±qmp`c§ ƒPst~ba„±rm—r^bZ{PbDx9r&ƒw~’baUjnmpvPxzlnyƒv{lnjnx’^bD„r†st{lnb§¹stiw„ §¨s7„„ jkr’jnmpi `ZbSr’^m „xaªsr%r’^wbcb ¦ ƒPbaiwxzbZmp ,r9ž¢m 7bS~y©xz`©stlnl;` vlkr’jnƒlnjkba~xa¬ ]^jux`cbSr’^m „—^wspx¢{ObSbSijn`cƒwlkba`cbaiUr’ba„ jnis  ƒ~mpo7~sp` ž"^ba~’b­r’^wb­ ¡vwiwq¤§ rjkm7iGª£r’^wb,jniƒv r stiw„&mpv rƒv r spqSqSv~†spqy(qSspi&{Obx’ƒPbDqj ¥ ba„ ¬ ]^jux`ZbSr’^m „&juxIqlnbast~lny­ž¢bSlnlUx’vjkr’bD„&ž"^bai r†st~opbSrzr’jnio I}f(VxVbS` {Oba„w„ jkiwoZx’`©stlnlG` vwl«rjkƒlnjnbS~†xSªspiw„—ƒw~’m7{wst{lnyHGm gOba~xo7mUm „ !sk,cƒOmtrbSiUr’justlK¬ wv r’v~bcžm7~’® jkiwqSlkvP„ b I}f(i  n ¬ stiw„ !sk,c xzy0iUr’^bDxzjux%stiw„±s7q¤r’vPstlI^wst~†„ žst~b&qmp`Zƒwsp~’jux’mpiwxaªwstiP„ r^bž¢mp~®H„ baxq~jk{Oba„ jnix’baqr’jnmpP . . " " ¢" Wk”[7b‘¬ b%spx k sp~’`©sWspiw„„b ¬ f ¬‰srvlus¬ Pstjkr’^  ¡vl©{wjkƒwsp~zrj«rb ‚ a ‰ ~’bDqjnƒ~m0qastl‘rsp{lkbDxS¬ c¹i k ¬ (imž"lnbax­stiwm„ f ¬ &Z¬ ‰qa lnlkjuxzr’bS~DªOba„ jkr’m7~xaª  

(56) #&’' $, #+#$,| 7_’'  , (') ªƒwspopbax(”Œ d ª · sr^Gª ZªG”‹YGY n ¬ c¹º?º?º  m7`cƒwv r’ba~Ck0m qjnbr9yH}?~baxxa¬ W #d [ ¢¬„ hb bVjnibDq†^jkicstiw„ w¬ b­br’~bSy7,¬ ‰±vlkr’jnƒwst~’r’jkr’brsp{lkbDxjni  · j«r†x ¡mp~r’^b¢bSstlnvwstr’jnmpi&mp  ¡viwqr’jnmpiwx m7i  ¡ƒoUspxa? ¬ c¹i   _

(57)   $,  2_ ,' 

(58) ($, I  &! " &| ’ # ')# # #M#  `M ' ’(’')$,| $

(59) # #’' $, ª mp~’r !spvw„ ba~„splkb7ª Ilkm7~’ju„sªP ƒ~jklIdt“p“Ud ¬ VƒO„str’bD„pbS~†x’jkm7i—mt  ! c¹} ~bax’basp~q†^ ~’baƒPm7~zrVdp“p“”S§K‰7‰w¬ W ‹ [ ¢¬;„ b b­jkiwbaq†^jnistiw„± ¬O]juxxzba~spiw„ M¬ k0mp`Zbjn`cƒw~’m7bS`ZbSiUrx mpi±` vwl«rjkƒwsp~zrj«rbrsp{lnb&`Zbr’^wm0„wxS¬ c¹i4Ž bajkl · v~opbaxx—spiw„ !vjnopj  jk`Zjnijkba~swª ba„j«rmp~†xSª &%    '

(60) (#)&’' $, *+#$,| _’'  (') ª7ƒPstopbDx"”Dd +;”a‹ nª ?spjklKª  mplnmp~†sp„ mPª pviwbVdt“p“w”p(¬  ƒ;„srba„Zpba~x’jnmpicmt  ! c¹} ~bax’bast~†q†^ ~bSƒOmp~’rVdt“7“p“§¹‹ w¬ W G‰ [  ¬ Vspxx’lkba~—stiw„WŽ&¬]sp®£spopjK¬ viPq¤r’jnmpiWbSstlnvwsrjkm7i {0y r†st{lnblnm0mp®U§Kvwƒ stiw„ s7„„ jkr’jnmpiG¬ c¹i k ¬ (imž"lnbax­stiwm„ f ¬ &Z¬ ‰qa lnlkjuxzr’bS~DªOba„ jkr’m7~xaª   

(61) #&’' $, #+#$,| 7_’'  , (') ªƒwspopbax(”D+“ ;”aŠª · sr^Gª  ZªG‹” YGY n ¬ c¹º?º?º  m7`cƒwv r’baC~ k0m qjnbr9yH}?~baxxa¬ W n#[ ¬p&¬ !Gju„„ juqm7str?stiwI„ ‰ ,¬ P¬ Ilky0iiG¬  jnop^ §¨ƒObS~’ ¡mp~`Zspiwqb dwmUsr’jnio%ƒPm7jkiUr,„ jn0jn„bp&¬ c¹i $, T')  

(62) (#)&’' $,  '  ' &

(63) (’  ’'   ªDƒPstopbDx ‹ nt-‰ 0‹7Š”7 ª f sp~xs£žªS} m7lnspiw„ #ª k0bSƒ rbS`‘{Pba~Idt“7“”7¬ W Š [.P¬ ‰ C¬ ‰±vlnlnbS~D¬ `2&|#/  $, (') _  ’'  ,  # ’  =(') ¬ · jn~’®0^wspvwx’bS~Dª · mUx9rmpiGª ‹” YGYUŒ ¬ W¿Œ [.P¬ ‰ ,¬ ‰±vlnlkba~a¬I  ¡baž ~’bDxzvwl«r†xm7i©rsp{lnb§¨{wspx’ba„©`ZbSr’^m „xa¬ _ %') 0+#$, '  p&ª n \3‹ ] j dU#Œ Y+d ª ‹” YGYY¬ W [.P¬ &¬G}?j bajk~mwª w¬ b ¬ · ~vo7vbS~†sªGspiw1„ P¬ § ‰ ¬ ‰vlklnbS~D¬ wspj«r^  ¡vl?ƒOmž¢bS~jkioqm7`Zƒv rstr’jnmpi vwx’jkio r†st{lnb±lnmUm7®U§Kvƒ stiw„ sJ ¡vwx’ba„+spqaqv`‘vlnstr’jnmpi r’~bSbp ¬ c¹i Ž bajkl · v~opbDx’xZspiw„ !vjnopj  jn`ZjkijnbS~†sª bD„ jkr’mp~†xaª ,%  2!

(64) # #

(65) ' $, 3+#$,|e_’'    ') ªOƒwspopbax ‰7+“ 0‰0Œ ª ?stjnl3ª  mplnmp~†sp„ mPª 7vib dp“p“w”p¬ Ñ,+JGÑ.-.

(66) 

(67)    

(68) 

(69) ! "##%$&(')*+

(70) $, (') . ”7”. W Y[.P¬ º ¬kUr’jnibstiw„‰ ¬ P¬kq†^Uvwl«rbp¬7]^wb¢x’y0`c`ZbSr’~jnq r†st{lnbs7„„ jkr’jnmpi&`ZbSr’^m „ ¡m7~Is7qSqSv~str’b  ¡viPq¤r’jnmpi spƒƒ~m ¦ jk`©srjkm7iG¬ $,’s 

(71) 

(72) ='  # #  ‹

(73) ’

(74) ’'  pªPd ” \3d]’jn”aŠUŒ O”ŒpŒ0ªw” YGYY¬ Wn”a“[ f ¬ ¢¬#Jf m7io(stiw„ºV¬pf%mtr’mP¬ PspxzrIbastlnvwsrjkm7i&mt r^b¢bSlnbS`ZbSiUrsp~’y% ¡viPq¤r’jnmpiPxjniZxzjnioplnb,ƒw~’bDqjuxzjnmpiG¬ ! ,  G  ')#    +# $,| 

(75)  ª‰7‰=3\ ‹]’j ‰ pn ‹+0‰ 7n Œ ªG‰ sp~q†^ ” YY n ¬ . s #Ò Ž  ’‘ “.

(76) Unité de recherche INRIA Rhône-Alpes 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).   

(77).   . ISSN 0249-6399.

(78)

Références

Télécharger maintenant ( PDF - 15 Page - 366.80 KB )

Documents relatifs

Sonochemical Treatment of Water Polluted by Chlorinated Organocompounds. (Review)

If the temperature of the collapse increases due to an increase in the specific heat ratio (for example by lower initial concentrations of the chlorinated compound) the ratio

Anisothermal thermosetting resin cure monitored by optical fiber refractometer

If a reproducible evolution of the refraction index with temperature and degree of conversion could be available, an absolute value of the degree of conversion would be

La future éruption de la montagne Pelée: Risque et représentations

À partir du moment où la montagne Pelée se réveille, temps au bout duquel une grande éruption peut se reproduire (d’après la population interrogée

Novas cultivares de laranja para o município de Guaçuí, ES.

Contanto, é importante ressaltar que as cultivares ‘Salustiana’, ‘Pera’ IAC, ‘Pera’ “Jetibá”, são mais propícias para suco, podendo a ‘Salustiana’

Langue

Si l’examen du lien qui unit langue et identité trouve dans la première approche une résonnance due au fait que tout sujet parlant construit son identité par le biais de sa

Novos focos de Meloidogyne mayaguensis em goiabeiras no Estado do Espírito Santo.

Ainda segundo os autores, outro ponto importante a ser levado em consideração no combate à disseminação seria o rastreamento, por parte dos órgãos competentes, a partir

ELEMENTS DE REFLEXION POUR UNE DIDACTIQUE DU LEXIQUE DANS LE SECONDAIRE : ENTRE MICRO-SYNTAXE ET MICROSEMANTIQUE

En vertu du principe frégéen de compositionnalité qui règle dans les linguistiques formelles les rapports entre syntaxe et sémantique, la composition syntaxique est

La Confiance, Variable-Clé d'un Contrôle Rénové / "Trust", a Key Variable in a New Approach of Management Control

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des