Additive symmetric: the non-negative case

Texte intégral

(1)Additive symmetric: the non-negative case Marc Daumas, Philippe Langlois. To cite this version: Marc Daumas, Philippe Langlois. Additive symmetric: the non-negative case. [Research Report] LIP RR-2001-06, Laboratoire de l’informatique du parallélisme. 2001, 2+11p. �hal-02101768�. HAL Id: hal-02101768 https://hal-lara.archives-ouvertes.fr/hal-02101768 Submitted on 17 Apr 2019. 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) Laboratoire de l’Informatique du Parallélisme. SPI. École Normale Supérieure de Lyon Unité Mixte de Recherche CNRS-INRIA-ENS LYON no 5668.

(3) .

(4)

(5) . .

(6) Ó . École Normale Supérieure de Lyon 46 Allée d’Italie, 69364 Lyon Cedex 07, France Téléphone : +33(0)4.72.72.80.37 Télécopieur : +33(0)4.72.72.80.80 Adresse électronique :

(7).

(8)

(9) .

(10)

(11) . .

(12)

(13)

(14) .

(15) !

(16)

(17)

(18) " # $ !

(19) %&'( ")

(20) * !

(21)

(22) + .

(23)

(24)

(25)

(26) $.

(27) . " $

(28) +

(29) +

(30)

(31) + %&'( . . , - ./

(32) "/

(33) / " !

(34) 0 ! - $" # "

(35) "

(36) " 1

(37) %&'( .

(38) . "/

(39) + "/

(40) "/$ "

(41) ".

(42) "

(43) 2 3 $. . - $" # + +

(44)

(45) + %&'(.

(46)

(47) £ .

(48)

(49) . 4 " $

(50) !

(51) 5 6 7 89 : # $

(52) " "

(53) 6

(54)

(55) !

(56)

(57)

(58) + # $ ! " " " "$! "") # $

(59) "+ .

(60) ""

(61) +

(62)

(63) "" ! " "$ " 89 6 " " "$!

(64) . ""

(65)

(66)

(67) " 6 . " " # $

(68) 8;9 8(9 "" "" !"

(69)

(70) <

(71)

(72) . # $ !.

(73)

(74) + . . . . . $ # $ ! 6

(75) ".

(76) # $ " ?

(77) + . . . =>. . . =>.

(78) $ 1

(79) " => => . # $ !

(80) %&'(

(81) @

(82) ! # $ ! ": = ! "> ! @ 4 =>5 $ A

(83) B %

(84) B A + B % + C $

(85) $ +

(86) 6 +

(87)

(88) $ . ! # $ ! C

(89) $

(90)

(91) $

(92) $ =;> . !

(93)

(94) $ " + 6

(95)

(96) $ .

(97)

(98) $ + $ 1

(99)

(100)

(101)

(102) $ +

(103)

(104) " =;>+. =(>

(105)

(106) $

(107) + .

(108)

(109)

(110)

(111) . ""$ D; "

(112)

(113)

(114) ! " #

(115)

(116) "!$ $

(117)

(118) .

(119) A

(120)

(121) $ %

(122)

(123) $ B A. + B. B. 6 $ : "" *

(124) "

(125)

(126) " G

(127) * :

(128) G

(129) ; ""$ G

(130) ( @ " $ *

(131)

(132) "

(133) !$ ! *

(134) ".

(135) ""$ * "+ " "

(136) # $ + + " . +

(137) !

(138) * + + "" ": # $ ! * + 6 % &'(

(139) $"

(140)

(141) !"

(142) G

(143)

(144) + +

(145) . +

(146)

(147) .

(148) # $ ! #

(149)

(150) + # $ ! + + " + .

(151) * "" D;

(152) "

(153) . $ " "

(154)

(155) * .

(156) $

(157)

(158) " ! ""

(159)

(160)

(161) ".

(162) * " # $ ! ! 1 . !"

(163)

(164) .

(165) $

(166) *

(167)

(168)

(169) *

(170)

(171) !

(172)

(173) $ " +

(174)

(175) * " #$

(176)

(177) %

(178)

(179) *

(180)

(181) $

(182)

(183) 6

(184)

(185)

(186) $ . !"$ + 6

(187)

(188) $ . . . . "

(189) " * . 6 $ ! . . . . . . + . . .

(190) . 6 +

(191)

(192) "

(193) $

(194)

(195)

(196)

(197) $ . 6

(198) $

(199)

(200) $ G

(201)

(202)

(203) " * )

(204)

(205) *

(206)

(207) $ + " "+

(208)

(209) * " "

(210)

(211) ""$ " " #$

(212)

(213) %

(214)

(215) " =;> $

(216)

(217) * . 6 ! )$ $ 4 =>5 ". + . . . .

(218) * " $ " . + . .

(219)

(220) . 6

(221)

(222) "

(223)

(224)

(225)

(226)

(227) $ . . 6 "

(228) " $ " ! " 6 .

(229)

(230) " #$

(231)

(232) %

(233) 6

(234) ". $ . "" H

(235)

(236) $ . + " +

(237)

(238)

(239) $ . * . . . . . . . . . . . . , $ ! )$ $ 4 =>5 " " +

(240)

(241) " . . . . . . . . 6 $ "

(242) "

(243)

(244) C

(245) ! 8'9+ 89 #$ ! $

(246) 6 C

(247)

(248)

(249) 1

(250) # $ $ "

(251) " "$ 6

(252)

(253) " " "

(254) !

(255)

(256) $ !" . $ A$ "$ + "

(257)

(258)

(259)

(260)

(261)

(262) "

(263)

(264)

(265)

(266) " !" "$ 6

(267) " C ""$ I

(268) + "

(269)

(270) . ='>.

(271) " :

(272) $ . 6

(273)

(274) $

(275)

(276) "+ " ! # $ " $

(277) 6

(278) !" + $

(279) ! " " $ Æ Æ Æ G+ Æ Æ Æ 6

(280) "$

(281) 6 C " 1 $

(282)

(283) $

(284)

(285) " .

(286)

(287)

(288) " 6 "

(289)

(290) "

(291)

(292) .

(293) ! 89 . !

(294)

(295)

(296) ! ! $ " ='> .

(297)

(298) "

(299) 6 C " !

(300) " "

(301) " 1

(302) " $ % + !

(303)

(304)

(305) "

(306)

(307) . !"

(308)

(309) "

(310)

(311)

(312)

(313) ! $ ! % & '

(314) % (

(315) "

(316)

(317) $+. ) " 8&9 C$

(318)

(319) $

(320)

(321) $ .

(322)

(323) /

(324) ".

(325)

(326)

(327)

(328) " " " " . $ % "

(329) $

(330) 1 " 6

(331) $

(332) +

(333)

(334) $ $ : "+

(335)

(336) $ " 6 . ""

(337)

(338)

(339) "

(340)

(341) ""

(342)

(343)

(344)

(345) "

(346) .

(347)

(348) + "

(349) "

(350) $ + D; $ "$ " $ " $ "

(351) . % %

(352) +

(353) . ""

(354)

(355)

(356) " + + " # $

(357) * :

(358) ".

(359)

(360) $

(361)

(362) !" #$

(363)

(364) % % + % 8;9+

(365) . .

(366)

(367) ! + $ " $ <

(368) +

(369)

(370) . + $

(371) 6 " " $ %&'( # $

(372) @ 4 =>5 " $ +

(373) $ 4 :5 4 5 % $ $ ": = ! " > # $ ! $ " # " !" @

(374) %&'(

(375) " 4$ :5

(376) + + ": ! #$ "% " 6 ""

(377)

(378)

(379) " + 6

(380) .

(381)

(382) " $ "

(383) ""$ " %

(384) ")

(385) # # $"

(386) 8(9 ""

(387)

(388)

(389) " 4 5 $ 6 "

(390)

(391) " H # $ !

(392) " = " "> 6

(393) ""

(394)

(395)

(396) " . "$

(397) # $ !

(398)

(399) J

(400) !

(401) # $ ! ""

(402)

(403)

(404) " .

(405)

(406)

(407)

(408) $

(409)

(410) J

(411) 8K9

(412) ". $ " %&'( !"

(413) # $ ! 4 =>5 $ 8L9 $ + ! "" !"

(414) !

(415) G

(416) " ""

(417) " .

(418) # 89+ 89.

(419)

(420) " ! $

(421) .

(422)

(423) * D; $ "

(424) $

(425)

(426) !" =;> *

(427) :

(428) .

(429) " "

(430)

(431) !" => *

(432) $ " + $ $ +

(433)

(434)

(435) $ " => . ! $ + " * $ $ +

(436) + " $ # $ ! 6

(437) .

(438)

(439) $ $ " =>

(440) $ !"$ * : $

(441) .

(442) Ê. Ê. . Ê ´ µ. Ê ¾. Ê ¾. % . Ê. . ´ µ. £. & ' & * &( ) !

(443)

(444) " . 0000000 # " $ 1111111 11111111 +, 00000000 1111111 0000000. ´ µ ¾. . . $ H G " D;

(445)

(446) !" $ " !" ! !

(447) "$ $ !

(448) + !

(449) !

(450) 6 $ ""$

(451) . !

(452) " # $ $ !+. . 6

(453) .

(454) . . * +

(455) % ""

(456) + 6

(457) "" !

(458) "" # $ ! * . . . . * . . . . . . = ! >+ . *

(459) "" " G!: 89 $ M

(460)

(461) 4

(462) !

(463) 5 =

(464) %

(465) >. ) *+,- . . . . . . . . . . . 6 !"

(466)

(467) +

(468) . +

(469)

(470) $

(471) " => * $ ) G

(472) ( 6 "

(473) "

(474) . " . !"

(475)

(476) $

(477) " " * . " . !"

(478) G

(479) (. *

(480) "" # $ ! $ ! % . $

(481)

(482) )

(483) "

(484) +

(485)

(486) !" . $ .

(487) *- . 6

(488)

(489) ": $ " 8;9 "

(490) ". + ""$

(491)

(492) " => . . 6

(493)

(494)

(495) " .. !

(496)

(497) $ ! 6 ! . *- . 6

(498) " . * " ") +

(499) . * $

(500) . $. * + * + % "" + ; * + 4

(501) !

(502) 5 " G!: " 6 6 ! +

(503)

(504) " +

(505) " $ ! 6 # $ !. " =>.

(506)

(507) $ . " => $ . *- . *

(508) $

(509) $ * $ $+ . 6 " => 6 G!: " !

(510) $ $

(511) ! !" $ ! . + . $ *

(512) $

(513) G

(514) (;. *- . G!: "+

(515) . $ $ +

(516) . $ .

(517) " "

(518) G

(519) (;. . *- . "

(520) . $ " "$ " "

(521)

(522)

(523)

(524) *

(525)

(526) "

(527) . $ G

(528) (. *- . *

(529) "" % $ + " => . . * . "

(530) . $ $ ""$ G

(531) (. . . , $

(532) " " # $ ! +

(533)

(534) . . $ " => $ + . . =&>.

(535) ." / $ " / *

(536) Æ. =K>. . 6 ! Æ

(537) .

(538) Æ Æ. . . . . . .

(539) . =L>. *

(540)

(541) # $ ! %

(542) + Æ

(543) Æ ! / $ " / . ! $

(544) =&> * $ => (" +. +

(545)

(546) . . . ": 6

(547) " " " " N " => =>+ . . . . . . . . . . . =>. + . . =>. 6 !" "

(548) " =K>

(549) " 1 ". $ $ " => " =K> *

(550) =;>. "" G

(551) + # $ !

(552) . " ": *

(553) .

(554) " =K> $ ""$ !" " =;>+ 6 1

(555) " $ ! + O " $ ""$

(556) . *

(557)

(558) @ " $ ! "

(559) "

(560) . 6 !" " + " " => C $ " D; *

(561)

(562) . M

(563) = $ > D;. 0 * . ": . + .

(564)

(565) . %

(566) . +

(567) . * % + " . "" " * $ !

(568) $ " =K>

(569) " . 6 . " $

(570)

(571) + . . . . . " !"

(572) $ $ .

(573) $

(574) . Æ. . . . . . . . . . . . . ¼. ¼. ¼. ¼. ¼. ¼. ¼. ¼. . . . . . . . . . . . . . . . . . . . . 6 "

(575)

(576) $ ""$ D;. . . . D

(577) + D

(578) + D; D+

(579)

(580) $ . . %

(581) + 6 . .

(582) " =K> $ !

(583)

(584) * * 6 $ "

(585) $ !. !" Æ $ " =L> *

(586)

(587) !"

(588)

(589) $ $ = > * . . $

(590) . Æ . . . . . . . . . . . . . . . . . . . . . . . . * Æ

(591) . !"$ 6

(592)

(593) " ""$ D D; ""

(594) + D +

(595)

(596) $ +

(597)

(598) $ . .

(599) . =!> 6 " !" $ $

(600) . . . Æ. . . . . . . . . . . . . . ½ . . . . . . . * ""$

(601)

(602) " D D; ""

(603) + D +

(604)

(605) $ "" . . . . . . . . . ¾ . . / $ 1 D; $ . "

(606) . * D+ D D; % $+ $ * +

(607) " !" " " * +

(608) " . +

(609) 6

(610)

(611) " " . !" !" 6

(612) .

(613) $

(614) 6

(615) " " ! * D D; ! )$

(616) . !

(617)

(618) 6 D ) !" !

(619) "

(620) $ "" " % D +

(621)

(622) !"

(623)

(624) $ . . 6 ""$

(625)

(626)

(627) . G

(628) ( * "

(629) $

(630) $ " "

(631)

(632) . D $ + D; + D $ G" "

(633)

(634) "" " * "$

(635) . ! "" $ !$ " 1

(636) . 1 " G

(637) "" * . . D; $ . ½ !

(638) !

(639) " Æ $ % ½ & Æ ' ( Æ

(640)

(641) "

(642) & $ ¾ ) & !

(643) % ! !

(644) &

(645)

(646) ! ½ ¾ ½

(647) ¾ ½¾ ½ * ½ & & (

(648)

(649)

(650) .

(651) 1 " G

(652) "" $ < .

(653) . 1 " ; G

(654) "" $ . D $ D $ . 1 * ""$ " (+ !

(655)

(656)

(657) $ D $ . " 2 *

(658) . 6

(659)

(660)

(661) $ . . . . . 6

(662)

(663)

(664) $ ! ". . . . . . . . . . . . 6

(665)

(666) $ . . !

(667)

(668)

(669) "" ) $ . "

(670) *

(671) # $

(672)

(673) . "

(674) + .

(675)

(676)

(677) $

(678) " ! $

(679) $

(680)

(681)

(682) $ .

(683) "

(684) @ %&'(

(685) 4 =>5 $ E .

(686)

(687)

(688)

(689)

(690) $ C

(691) ".

(692) $ 4 " P ?" " Q C

(693) 5 $

(694)

(695)

(696) $ ! !+ "

(697) " C 6 " = $ >

(698)

(699)

(700) " = ! "> " " $

(701)

(702) " !

(703)

(704)

(705)

(706) *

(707)

(708)

(709) $

(710) +

(711) "

(712)

(713)

(714)

(715) !" ?

(716) $$ $ "

(717)

(718)

(719)

(720)

(721) M

(722) " ) 6

(723)

(724)

(725) $ $ "

(726) ".

(727) $

(728)

(729)

(730) $

(731) $ : <

(732) M

(733) "

(734)

(735) $ " "

(736) . .

(737)

(738) $

(739) 6 " $

(740) . ! # $

(741) * $ " "+

(742) . !"

(743) " "$!

(744) " +

(745) "$

(746)

(747) "" . !"

(748) ) 1 ) +

(749) " . " ! " "

(750) " " " " # $ 8+ ;+ (9 8'9 ?

(751) + " " $ "

(752) *

(753) + " ""$ "

(754) ! .

(755) @ " " !

(756)

(757) $ *

(758) +

(759) .

(760) @

(761)

(762) $ @ = > "= > !" .

(763) " ""$ "$ $ ":

(764)

(765) !" ; . ! # $ ! !

(766) ! % D; "" D B 6 ""$ " $$ !

(767) ""

(768) * + . "" $ + . C $ + .

(769) . . . . . . . . 6

(770) "

(771)

(772)

(773) " 4 " $

(774) !

(775) 5. 89 A 7 + 6 C F$ $+ R" + G

(776) " "$ + ; + * "+ $+ E+ ,G+ LLK 89 F < G!:+ " $F C + F

(777) < ""+ $" C"1 + S+ ,G+ L&( 8;9 , * 7"

(778) + $ :+ H SC T J + " = >+ F

(779) $ C(+ " !. C C

(780) + A $ "+ + ;UL 8(9 SE E""+ G "$!

(781) # $

(782) + H SC T J + " = >+ F

(783) $ C(+ " ! C C

(784) + A $ "+ + U;K 8'9 F . $" + "+

(785)

(786) 1 # $

(787) + H S C T J + = >+ F

(788) $ G

(789) " ! C C

(790) + LL+ LLU(+ =

(791) > 89 F . $" +

(792) "

(793)

(794) $ + T%6 ( =;>+ =6 + " " !" . % %

(795) ('+ ?

(796) ! > 8&9 S <$ +

(797)

(798)

(799) G !"

(800) " "$ + G

(801) % " " E

(802) + F " " + F+ ,G+ LL 8K9 I < + S + I 6! + F V + G # $ ! +. ! =S > 8L9 " + *

(803) =A

(804) LL(>. B+

(805) " !" , . P.

(806) . 89 S E )1+ C

(807) #

(808) F

(809)

(810) "

(811) " + %" + W) 6 ( ! 89 C T E"+ " ! + G%E K =LL'> ++ " * LL'+ U. +. 89 S < + R$

(812)

(813)

(814) " " <?.+ H F

(815) $ % " C

(816) 6 F$ <$ ? .$

(817) + E <""+ S + LL&+ ;&U' 8;9 S G E+ 6 .

(818) + E 7 +

(819)

(820) ""

(821)

(822) ) EA'7K # $ + % 6

(823) C (& =L> =LLK> L;UL 8(9 S < +

(824)

(825)

(826) ) # $

(827) + H W T+ I A)+ <

(828) :+ C F "+ . 6 = >+ % " C