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N>1Ko5T103262Q03C)2651A; 1=?>,C60 >J\ J\~^I0}LGT1803CQEO0}2Q51AD2 / C6LG;70K.-O03C 265704HJ:YA=?LG;1AD@X7C60KML?;1HJ:'26:'L?;703C)T7>,2;7LG2 S LGCu2Q570:'2603CsAD26:'L?;18};,030HJ0HF2QLnA?Ko57:<0zEO0 C608P:YHJ>wAD@ !t8>wK.52651A2 S L?C~2Q570 / C6LG;70K.-O03Cc,C6LHJ>1KM2}A77CQLDr:<N%AD26:'L?; AD;1Ht26570# I J 7CQ0KlL?;wHJ:<2Q:<L?;,03Co8 @<0mAmEO0m8 265,08QADNF0y:<NF7CQ0868P:<LG;18l\ }570FT1L?2Q26@'03;70K.-S L?C0A?Ko5_:'2603CsAD2Q:<L?; :<;I7C60KML?;1HJ:'26:'L?;70HJ * V :Y8)8LG@<E:<;7=265704@<:<;,0ADC ( Distribution of eigenvalues of M^(-1)*A for n =250^2 0.3 0.25 percentage 0.2 0.15 0.1 0.05 0 0 c:'=?>7CQ0>\ > :<2Q5 ZADT7@'06>J\ 265,0 0.2 0.4 0.6 0.8 eigenvalue }5704HJ:B826CQ:<T7>,26:<LG;iLDSp2Q57003:<=G03;JEGA@<>708 LS 1 S L?C > F857Lp826570Kl@'>182Q03C6:';7=LS03:'=?03;EGAD@<>,08}AC6L?>,;1H JL?N41ADCQ:Y8LG;FLDSZ:'2603CsAD2Q:<L?;w8 S L?CX26570 CQL?T7@'03N : 0 # J # 2603CsAD2Q:<L?;w8 > > >7 / # 2Q03CoA26:<LG;18 ( G ( > ( 1.4 >7 \ J\ /fCQL?;70mK-O03C 7C6LHJ>wKl2pAD7,C6LrJ:'NnAD2Q:<LG;AD;1H # R J 7C60mKlL?;1H:<26:'L?;70zCo8l\ 1.2 ( PCG_ka = 93 PCG-ic = 66 n = 256 6 4 log(Residual) 2 0 -2 -4 -6 -8 0 10 20 30 40 50 60 70 80 90 100 Iteration c:'=?>7CQ0 >J\ KlLG;1HJ:<2Q:<LG;703Co8M\ 265,06/ KlL?NFwADC6:B8LG;qLDSX26570tCQ0HJ>1KM26:<LG;LS265,0tC608P:YHJ>1A@dS LGC }570IH7AG8570mHg@<:';70:B8 265,C6030HJ: }5,08P: y 0 LDSXh:Y8 03C603;2p7C60l S L?Cn265,0iHJ:BAD=GL?;1A@{,C60KlLG;1HJ:'26:<LG;703C V 2Q570I8LG@<:BH C6L?;,0K-O0zC~7C6LH>1Kl2pAD,7C6Lr:<NnA26:<LG;FA;1HF26570yH7AG85J HJL?2Q260Hn@<:';70 S LGC}2Q570 J57LG@<08-J|O\ 7 :<2Q5 >\ S LGC :<;wKlL?NF,@<032Q0 Comparison of different preconditioners for Laplace stencil matrix 8 Inc−Chol 6 diag KPA 4 B−diag log(Residual) 2 0 −2 −4 −6 −8 0 c:'=?>7CQ0 >\&. HJ:'26:<LG;703Cs8l\ S L?C)2Q570 5 / 10 15 20 25 Iterations 30 35 40 KML?NF1AC6:B8L?;LDS 26570pC60mHJ>1Kl2Q:<L?;LDSZ2Q570pC608P:YHJ>1A@JS L?CS L?>7CXHJ: }570n8PL?@':YHq@<:<;,0FKlLGC6CQ081L?;wH7826L 45 03C603;27C60mKlL?;J # ;wKlL?NF,@<032Q0J57LG@<08-J| V 265704H7AG8570Hi@':<;70t:B8 C6L?;,0K-O0zCu7CQLHJ>1Kl2A77C6Lr:<NnA26:'L?; V 2Q570HJLG26260mH9S LGC{2Q570tT7@'LK-OPH:YAD=GL?;1A@7C60l KlLG;1HJ:<2Q:<LG;703CyAD;1H2Q570tH7AG85J HJLG26260mH @':<;70 S L?Cy26570 }5708: ) 0 LDS 26570t/ A7@YAGKl0t8P2603;1KM:<@[N%AD26CQ:<rh:Y8 HJ:YA=?L?;wAD@~7CQ0KlLG;1HJ:<2Q:<LG;703CpC608P0mKl26:'EO03@<|O\ :<2Q5 > \ > 8LG@<E:<;7= 8|,82Q03N !S L?C J 8P:<;7=I265,0# 7\ !qKML?C6CQ081LG;1H78F26L AG8 V 7CQ0KlL?;wHJ:<2Q:<L?;,03CS L?C26570nNFLH03@7C6LGT7@<0zN 8LG@<E:<;7=I2657098|,82Q03N !q5703CQ0 :B8 2Q570 :<;wKlL?NF,@<032Q0 J57L?@'08-|)HJ0mKlL?NF1L 8:<2Q:<LG;LDS\# ;KlL?;26CsA?8P2 V >18P:<;7=y26570 /fCQL?;70mK-O03C 7C6LH>1Kl2 At7C60mKlL?;1H:<26:'L?;70zC V A/ AD,7C6Lr:<NnA26:<LG;A?8 :<2Q03CoA26:'L?;*\ }51A2:Y8 e y` gS !T|x L?C !tKlLGC6C60m81L?;1H,8 2QL V 570zC60 ( :Y8A !TJ|O NnAD2QC6:'rF8P>1K.5n2Q51AD2EO0K y 7L?Ct265,0FNFLHJ03@7CQL?T7@'03N :'265IL?;7@'| 7LGC 1L?18M\ LDSy8: 0 :B8fTwAD;1HJ0mH 26570 26CQ:YHJ:BAD=GL?;1A@[A;1H ` e 51A?8p26LT08PL?@'EO0HnAD2 NnA26CQ:<r48>1Ko5n2651A2uEO0K !x\ ]A;1H 8P>7TJ HJ:YA=?LG;1AD@B8{;7LG; 0AGK.5 !{LGCu0 O>7:'E?A@<03;26@'|t26L8PL?@<E:<;,= V :<2tKzAD; :Y8pA A;1H T10n857L;R2651A2 03CQLnAD;1HI8LG@<E:<;7=n2Q570F8P|78P2603N C6L?;,0K-O0zC7C6LHJ>wKl2AD77CQLrJ:'NnAD2Q:<L?;n7CQ0KlLG;1HJ:<2Q:<LG;703C V ` / C60 O>7:'C608 A;1H e ADCQ0 1L?18AC60pC60 O>7:<CQ0H V 03r,Kl0372Z2651A2XAu@YAC6=?0103CsKl03;2oA=?0LDS 2657L 80 1LG18 K3A;T1003rJ0mKl>7260mH C6LG;70K.-O03C},C6LHJ>1KM2 :<;n1ACoAD@'@<03@ V T10K3A>180)X08PL?@'EO0uS L?CpN >,@<26:'7@<0CQ:<=G5J251A;1H8P:YHJ0m8l\ 5 "!$# }570 &% JL?;,;70Kl2Q:<L?; qA?Ko57:<;,0 Kl:'03;J2Q: K LDS 2PAC60t/1:<T,CoADCQ| '% J )( /* 26LOLG@ -:<2~S L?CX:<2Q03CoA26:<EO0XNF032Q57LH78~26LG=?032Q5703C:<2Q5nKl>18P26L?N4: 0 H8LG@<EO03Cs8 5,03; 260zNn8~ADCQ0)82QC6>1KM26>7CQ0H[\ }570 ,+ 7CQLHJ>1Kl2X *\ :<;1Kl0v :<;t2Q570uN4LHJ03@7C6LGT7@<03N ,+ 43 43 !@7. !@7. 1+ /% 87 H? GI/ 65 65 65 65 qA?Ko57:<;70 7L?CQ26CsAD; (- 2%"$#4.* 05#-2/#&!HC + *"&*4@ ) ?*)(G:9 -# "5* 5*7216)# ) 2* + + ?*4G6)@>9 ) G*4G6)@>9 + +HG5%7 = " ?*4 * $( ) .* $(39 + +HG5%7 = " G*4 * $( ) .* $3 ( 9 -+ :Y8~A)8P2603;1KM:<@ V }0pH:Y82QC6:'T7>7260mH 2 2Q570p@<:<;,0ADCX8|,8 FC60 O>7:<CQ08~L?;7@'|F;70AC608P2;703:'=?5TLGCpKlL?N4N }57:B8c:Y8*26570vKlL?CQC608P1L?;1HJ:';7="JLG;7;70Kl2Q:<LG; *7C6LE:YHJ0m8A N4LO82XKlLGNFNFLG;L?103CsAD2Q:<L?;FS LGC}265,0"J *:Y8 2Q570pNnAD2QC6:<rO!EO0Kl26LGC v=?C6:BHn8Lt265wAD2p26570)A77@':YK3A26:<LG; K3A@YKl>7@BAD2Q08 * }5,: ( :9 ;9 ;9 ;9 4* 8 % (C$#- * " 45?*) ) ) ,9 * %8(C$#- * " 45?*) ) ) 9 '% 0( J * . LG;FA T| >7;,:<KzAD26:'L?;*\ }57: ( C6L?>,26:<;,02651A2 * %8(C$#- * " 45?*) * %8( C$#- * " 45?*) ) ):9 ; 9,9 ) );9 ;9 9 9 2*7" 2 *780 7L?Cp8PL?@<E:<;,= e 26CQ:YHJ:BAD=GL?;1A@ NnAD2QC6:BKl08 ` S ` LGC AD;1H V }0y2oA -O0y26570/ &)/ 2651A2nADCQ0 e ` HJ0KML?NF1LO8P:<2Q:<L?;LDS 265,08P|JN4NF0326CQ:'K @ @ @ 810KM26:<EO0z@<|O\A?K.-FA;1HS L?CQADCsH8P>7T182Q:<2Q>726:'L?;qAC60)>18P0H%S LGC A;1H % 0 57:<@'0u>7;,:<2@<L}0zC}T7: PHJ:BAD=GL?;1A@ :Y8 2 \ !<( % + 2 :B8~C6037CQ080z;J2Q0H * $ ( #- (- 2%"$#4.* " 2$#0#3+ .(.G37( % + #.78, + # !8#-" %75* #-"5* 5*721 2* + + ! 4G 9 )G 0 4BF 9 + 2* + 2 4G6 * )@> 9 2 \ qA26CQ:<r 62 * 4@6)@0 ) ! ) 2:9 65 5 65 !@7. + H+ G5%7 " 04 * $(D9 + !@7. D+ # ! 4 #D9 $#-"DC<04 # 9 !@7. + +HG57% " * 2 4 * $( ) .* $3 ( 9 2 65 D/ )@ 2 4 * ) 9 /92*4 *780 0% 0.% 0.%<#A/ )@ 2*4 # ) 9 /62 4 # ) 9 *780 0% 0.%<#A/& ) ) 2 4 # ) 9 /92*4 # * *780 0% E ) 9 ! 4 9 2 4 *) 9 04 #D9 ) 9 ! 4 # 9 ]S T|:<2o8 1Cs82p8P>7TJ HJ:YA=?L?;wAD@ #-"5* 5*723# ) 43 43 43 C60l L?C \ }570 :Y8*C603,C608P03;J2Q0H{LG;7@<|)T|:'2o8ZHJ:YA=?L?;wAD@ 2 B B B 8PL?@<E:<;,=t26570)>7;7:'2u2QC6:BAD;7=G>7@YAC T,: HJ:YA=?L?;wAD@WX8P|78P2603N%8l\ p03C60p:Y826570)8P>7T7CQL?>72Q:<;70)265wAD28PL?@'EO08 C6:'=?52c5wAD;1H8:BHJ0 R :Y8*LEO03C6CQ:<262Q03;)TJ| e 2*4 # ) 9 ! \ }570 C65w8 2*7" 2 *780 # NF7@'03N # 0z;J2Q: ;,=F26570 2oAG8- V A?8p:<2 J ,C60KlLG;1HJ:'26:<LG;703C:<; :B8uHJ0zNFL?;18P26CsAD260mH:<; ( */ An1ADCsAD@<@'03@}0z;JE:<CQL?;7NF0z;J2p:Y8tAnEO03C6|IHJ: nKM>7@<2 * S L?C 26570 # I J 7CQ0KlLG;1HJ:<2Q:<LG;703C V }0vKlL?N41ADCQ0H 265,0qHJ:BAD=?LG;1AD@fAD;1H S L?>,;1H 265,0 '% J :'; 2Q570T7@'LK-RHJ:BAD=GL?;1A@ '% ( > !\ :<;wKl02Q5703C60AG8u;7LF=GLOLH_:<NF,@<03N03;2oAD2Q:<LG; 26570 / C6LG;70K.-O03C[,C6LHJ>1KM2}A77CQLDr:<N%AD26:'L?;):<2Q5 7C60mKlL?;1H:<26:'L?;70zCo8>18P:<;7=i2Q570:'NF7@'03N /Z\ }57081030HqLDS9J @AD@'@<L}0H>w8f2QL8L?@'EO0tEO03C6|n@BADC6=G0 NFLHJ03@17CQL?T7@'03Nq\ }5706/ C6LG;70K.-O03C7C6LHJ>wKl2pAD7,C6LrJ:'NnAD2Q:<LG; 0z;O2oAD2Q:<LG;18 :';182sAD;1KM08 LDS 7CQ0KlLG;1HJ:<2Q:<LG;703C};,030HJ0H S 03X03CF:<2Q03CoA26:'L?;18F265wAD;h2Q570_LG26570zC,C60KlLG;1HJ:'26:<LG;703Cs8 V T,>72:'22QLOL-NFL?CQ0q26:'NF0qHJ>,0_2QL KlLGNFN >,;7:YKMAD2Q:<L?;ILEO03C65,0A?H[\ 1=G>7C60 K3A;iT108P0303;i:'; 265wAD;26570NFLGC60XKlLGNF7@<0zr }5708P0CQ08>,@<2o8{S CQL?N "% ;! AG8 265705/ CQL?;70K.-O03C)7CQLHJ>1Kl2A77C6Lr:<NnA26:'L?;iCQ0 x>,:<C60m8yS 0z}03C AF7C60mKlL?;1H:<26:'L?;70zC V S L?C 2Q570KlL?; 6 >7=OA260=?CoAGHJ:<0z;J2p26LKlLG;JEO03CQ=?0 V :<2pC60 x>7:'C60H # 2p:Y8u2Q5703C60MS L?C60LDSX:<;260zC608P2u2QL ;7LG265703C:Y8Q8>70 :Y8n26570 :<NF7@'03N03;O2oA26:<LG;]LDS426570 AD@'=?L?CQ:<2Q57NULG;iAF1ACoAD@'@<03@ KlL?NF,>72603Cz\ K3A@YKl>7@BAD2Q:<;7=n265,05/ 7@'03NF03;2oA26:<LG;_LG; e ` V 2651A2u:B8 2Q570 8PL?@<EO0mHA20A?Ko5n:<2Q03CoA26:<LG;nLDS2Q570KlL?;6>,=OAD2Q0)=?CoAGHJ:<0z;J2z\ @<:';70AC8|,82Q03N 7CQ0KlLG;1HJ:<2Q:<LG;703C:B8fS!AG8260zC ! ) "! 82Q>1HJ|H7A2oAFA@<:<=G;7NF03;2:Y8Q8>70m8 2Q51AD2)AC60)S!AmEOLGCoAT7@<0):<;98LG@<E:<;7= 26:'L?; EO0Kl2QL?C>,;7:<2s8 >,;7:YK3A26: L?;4KlL 82o8LDS 26570KlLGNF7@<0zrt7C60KML?;1HJ:'26:'L?;703Cs8l\ NFLGC60)26:'NF0)2651A;q265708P:<NF7@'03CH:YAD=GL?;1A@7C60KML?;1HJ:'26:'L?;703Cz\ :'265 L?;70m8 V HJ08P7:<2Q026570X=?C60mAD260zC ;>7NT103C*LDS :<2Q03CoA26:'L?;18c;,030HJ0H \ }57:B8 8)}0F51A3EO0n8P0303; V A@<2Q57L?>7=G5 :<2Q03CoA26:'L?;18 '% J >m!;7LHJ0 >J\ J\^I0FLGT1803CQEO0F265wAD2t265,0n8:'NF7@<0m82 :Y8H>70)26L4KlL?NF,>72oA26:<LG;nAD;wHKlLGNFN A 4 C6L?;,0K-O0zC }5,08031ACoAT7@<0@<0AG82)8 x>wADC60m8{A77CQLOA?Ko5_S LGC 7C6LH>1Kl2AD,7C6Lr:<NnA26:<LG;_:B8{08P10Kl:YA@<@'|qAD2Q26CsA?Kl2Q:<EO0tS L?Ct:<N 7C6LKM0868PL?Co84KlL?;7;,0Kl260mH /fCQL?;70mK-O03Cn7CQLHJ>1Kl2IA77CQLDr:<N%Am 4 :';RA NF0m85*\ AD;1H2Q5707CQLKl08Q8L?Cs8)ADC60 wLGCF8P:<NF,@<:YKM:<2P|qAG868>,NF0n2651A2 ADCQCoAD;,=?0HIA?8 A 4 T|x 4 AC6CsAm| 3 Iterations 10 2 10 1 10 1 10 2 3 2 3 10 10 Sqrt(size) of the linear system 4 10 3 10 2 Times in seconds 10 1 10 0 10 -1 10 1 10 c:'=?>7CQ0 >\ 10 10 Sqrt(size) of the linear system KlLGNF1ADCQ:Y8PL?;FLSZ26570p:<2Q03CoA26:<LG;18}A;1H 7CQ0KlL?;wHJ:<2Q:<L?;,03Co8LG;]26570 ;1A@ '% J 8 '_:<2Q5 7C60mKlL?;1H:<26:'L?;70zC V 26570I8PL?@<:BH >q;7LHJ0m8l\ S LGCF265700/ H7AG85J HJL?2Q260H@':<;70pS L?C265,0{T,@<LK.-nHJ:YA=?LG;1AD@!\ 4 10 2Q:<NF02QLtKlLG;JEO03CQ=?0S L?CX2657CQ030 }570qH,A?85,0H H: 03C60z;J2 @<:';70F:Y8S L?C265,0qHJ:BAD=?L C6L?;,0K-O0zCt7C6LHJ>wKl2nAD,7C6Lr:<NnA26:<LG;RAD;1H 2Q570 :<2Q5qKlLG;7;70KM26:<LG;182651A2 NF08Q86A=?08M\ ~C6LKl08Q8LGC A@<@<L :<@<@XK3A@YKl>,@YAD2Q0 K3A@NF03NFLGC6|2Q570nT7@<LK.-,8t e NnA26CQ:YKl08M\ KML?NFN>7;7:BKlA26:<LG;18 2QL nA;1H T10 KlL?NF,@<032Q0H 2Q570F03@'03NF03;2o8 :<;9AD2pNFLO8P2 @<LG= 4 4AD;wH 4AD;1Hi51AG8:<;i:'2o8)@<LD 426LG=?032Q5703C:<265RAIKlL?|ILDS)2Q570Kl>7C6CQ03;2 ;q:<2Q03CoA26:<LG;LS265,08031ACoAT7@<0t@'0A?8P2)8 O>1ADCQ08 ` AD;1H A77C6L A?Ko5iKzAD;qT10HJL?;,0A?8 S L?@'@<Lp8) @'@7C6LKl0m868LGCo8KlL?@'@<0KM26:<EO0z@<|K3AD@BKl>7@BAD26026C e NnA26CQ:<rt:<; @'L?= 4 e )A;1Hi7C6LG1AD= AD2Q0265,0nKl>7CQC603;2 A?Ko5n7C6LKM0868PL?Cp:<;n1ADCsAD@'@<03@ZK3A@YKl>,@YAD2Q08 @'@7C6LKl0m868LGCo8tKlLG@<@'0Kl26:'EO03@<|K3A@YKl>,@YAD2Q0 NnA26CQ:<rt:<; @'L?= 4 26:'NF0y8260z18l\ A?Ko5n7C6LKM0868PL?Cp:<;n1ADCsAD@'@<03@ZK3A@YKl>,@YAD2Q08 :'2o8}L; > }5703CQ0:B8265707C6LGNF:Y8P0FLDSy265,0/ 2QC P` ` :'2o8}L; 4 :'; 2Q:<NF0)8P2603w8l\ )A;1Hi7C6LG1AD= AD2Q02Q570nKl>,C6C60z;J2 4 :'; X26:'NF0 C6LG;70K.-O03C)7C6LHJ>wKl2FA77CQLDr:<N%AD26:'L?; / 26LIT10A 7CQ0KlL?;wHJ:<2Q:<L?;,03C 7CQLDE:YH08)A%EO0zC6|qKML?NF1032Q:<26:'EO0)7:YKl2Q>7C60t:'265IC608P10Kl2)26Ln265,0 J57LG@<08-J:*S!A?Kl2QL?CQ: =GLOLH KzA?80m8l\ c:<=G>7C605>\ 857Lp8265wAD2tL?>,C S L?Ct265,0F8P2603;1KM:<@NnAD2QC6:BKl08 S LGC T1A?8P0HqL?;q265,0t:<;1KlLGNF7@'03260 m & ` 8P2603w8l\ 7CQ0KlL?;wHJ:<2Q:<L?;,03C 26:'L?;70zC e 26:'NF0y8260z18l\ A26:<LG;*\ SvKlLG>7Co8P0 7CQ0KlLG;1HJ: 2Q5703CQ0tADCQ0tNnAD;| :Y8Q8>70m8c2651A2~;7030mH{C60m80ACoKo55703CQ0 V @': -O026570XA3|)26LKo57LOLO8P026570v8: } 0 LDS 2Q570pAD7,C6LrJ:'NnAD2Q:<LG; V 57Lh2QLt51AD;1H@<0)26570)@'LDX03CpHJ:'NF03;18P:<L?;w8 V 26570):B868>,0{LS265,08P26CQ:YHJ0AD;wH8PL LG;*\ Comparison of different preconditioners for Laplace stencil matrix 5 Inc−Chol diag KPA B−diag log(Residual) 0 −5 −10 0 5 10 15 20 25 Iterations c:'=?>7CQ0>\ KlL?N41ADCQ:Y8LG;_LSp26570C60mHJ>1Kl2Q:<L?; LDSp265,0 CQ08:BHJ>1AD@~S LGC{S L?>,CtHJ: [0zC603;2)7C60l KlLG;1HJ:<2Q:<LG;703Co8M\ }570f8L?@':YHt@<:';70pKlL?CQC608P1L?;1H78}2QL$# ;wKlL?NF,@<032Q0 J 57LG@<08-J: V 2Q570yH7A?8P570H S L?C)2Q570 / C6L?;,0K-O0zCu7CQLHJ>1Kl2A77C6Lr:<NnA26:'L?; V 2Q570HJLG26260mH9S LGC{2Q570tT7@'LK-OPH:YAD=GL?;1A@7C60l KlLG;1HJ:<2Q:<LG;703CyAD;1H2Q570tH7AG85J HJLG26260mH , }570y8: 0}LS*265,0 :<2Q5 @':<;70:B8 @':<;70 )HJ:'NF03;18P:<L?;wAD@,8 O>1ADCQ0 , \ S L?Cy26570 HJ:YA=?L?;wAD@~7CQ0KlLG;1HJ:<2Q:<LG;703CpC608P0mKl26:'EO03@<|O\ / AD7@BA?Kl0y82Q03;1Kl:'@7NnAD2QC6:'r :B8 , # ;g2Q57:Y8nKo51AD,2603Cn}0I8P26>1H|g2Q570iKML?;18P26CoA:<;70mH 1;1H:<;7=)2P}L)NnA26C6:BKl08 A;1H ` e / LDS 7C60H032603CQNF:<;,0H)8: 0 8~2651A2 2Q5703:'C+/ :Y8XKl@<L 808P226LA=G:<EO03;FN%AD26CQ:<r\Z^_0)CQ0 x>,:<C60p265,0 82QC6>1KM26>7CQ0 V @<: -O0T103:<;7=8P|JNFN40326CQ:YK AD@B8L7C60m803;28PL?NF0):<;wHJ:YK3A26:'EO0 C6LG;70K.-O03CF7CQLHJ>1Kl29AD7,C6LrJ:'NnAD2Q:<LG;]LDS L?C NnAD2QC6:BKl08 ` *LO037@<:'2 V LGC{T103:';7= AD;1H % CQL?;70K.-O03CZ7C6LH>1Kl2 2QLt51A3EO0 e At8P0mKl:YA@ qADC-OLDEqLGC)L?C62Q57L?=GL?;1A@ \F^I0 A77@<:BK3AD2Q:<LG;18}LDSX8P>1Ko5A77CQLDr:<N%AD26:'L?;18M\ }570):YHJ0mA{LDS AtKlLG;182QCoAD:';70HnAD,7C6Lr:<NnA26:<LG;FLDS ANnA26C6:'rt:<2Q5nAD;7LG26570zCNFL?CQ0)82QC6>1K. 26>,C60H NnAD2QC6:<rI:Y8t;7LG2 ;70z)\ # 2t:B8 A_CQ0Kl03;2:BHJ0Aq26L AD7,C6LrJ:'NnAD2Q0A :<2Q5IAnKl:<CsKl>7@YA;J2pNnA26C6:'rF:<;qLGCoHJ0zC{2QLnC60HJ>wKl0tKlLGNF7>72sAD2Q:<L?;wAD@KlL 82 8|,82Q03N 5703;n26570yNnAD2QC6:<rt:B8 dLO0z7@<:'2 :<N%AD26:'L?;nAG8 T| 26CsAD;7= At7C60KML?;1HJ:'26:'L?;703C}S LGC26570 ( 2QC. * # 2 6* T V AD= ( "! :B8 dLO0z7@<:'2 LSp8PL?@<E:<;,=nAF@<:<;,0ADC A !\ }570)>18P0)LDS 8>1Ko5nAD;A77C6LrO dLO03,@<:<2 7CQ0KlLG;1HJ:<2Q:<LG;703CtLDS N%AD26CQ:<r:<;n265,0 2Q5703C60MS L?C60 V L?;7@'| AD= ( NnAD2QC6:'r KlL?; 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M A8<6 ;@h ADRL?57ADaTADCNG>e\HG[?RL6F= ;>8@g ;08;; ?L5NG>? 6BCH\ATRL;?L5 G[Cb5NG>e OA>O?57;MX6 ;RL; CY? G[ 7RLA>fX6B\HG>?L6BATCb7RLAT`7eB; \H8 MX6F8 eFGcV;>M 8<?+G>RL?L6]C7aOQRLAT\ ?L57;W=@e]AV8<;>8?ADR?L57ATaDATCNG[e \HG[?RL6F= ;>8 ?LA?57; RLADC ;>= cV; R 7RAXMYKN=@? G>77RLA>fX6B\HG>?L6]ADCRL;U8K7e]?L;>M^6]CW=@ADC V;RLaT6BC7a ?LA?L57; aTRLATK7A>Od?57; eBA M ;>8? Q\HGc`'; ?L57;AT7?L6]\HG[e R;>869MYKNG>eF8@g & 6]aDK7R; "$ & `7K76]eFMY6]C7a ?57;H=@ATCN8?R+G>6BCX?\HG[?RL6F= ;>8 e]?L57ATK7aD5 G[C M.698^G[C ;>GT8cJ? GD8)c 6B? 6F8^C7AT? ?LR6 X6FG>e ;>8N;>= 6FG[e]eBc M 57;CH?L5 ;RL;6F8 G^C ;;>MOQADR ;fXN;RL6]\;CX?+G>?L6BATCG>CNM GOQR;VK7;CX? =+5NG[C7aT; "! 86 ;G[CNM= ADCN8<?LR+G>6BCX?2?<cYN;Vg&7ADR ?L57;>8<; RL;>GT8ATCN8 M ;A ;R G A[O 7RAD`7e];\ OQK7CN= ?L6BATC ?L5 G[?)7RLA X6FMY;>8)G=@AD\\_ADC 6]CY?;RO GT=@;2OQADR G[eBeX= ADCN8<?LR+G>6BC7;UM^G[C M^K7CN=@ATCN8?R+G[6]C7;>M G[eBaTADR6B?L5 \H8 \ATRL;A V;R ?L57698 OQK7C =@?L6]ADC aT;C7;R G[?;>84?57; =@ADC 8?LR G[6]CY? \ G[?LR6F=@;U8G[K ?LAD\ G[?L69=G[e]eBcVg 57; aD;CYh ;R G[e)=G>eBe3698 C 7' /H( C $ T C) C &' !HCB !2 TC % !2 TC C C QC)3 * 57;R;SG[eBe `7K7??57; NR 8?W?L57R;;HG[RaDK7\_;CY? 8^G>RL;HAD7?6BATCNG[e gJ698?L5 ;:\ G[?LR6Bfb?Ab`';SG>Yh M 7RA[fX6B\ G[?L;UM C $ G[R; ?57; MX6B\;C 86BATCN8 A>Od?L5 ;\NR 8? RADC7;U= cV;R)O GD=@?ADR 7?L6]ADCNG>e TG>RL6FG>`7eB;U8 G>RL;?L5 ;=@ATC V;RaD;C =@; =@RL6]?L;R6BADC G[C M +!2 cV;c M ATR+M78dG>RL; &'+! g +!2 ?57; \ G[?LR6Bf ?L5 ;8?RL6]C7aG>RLR GUc 8 G[R;W=@ADCX?+G>6BC76]C7a cV; c M ADR+M 8 MY;U8L=@R6B`76]C7a?57;W=@ATCN8?R+G[6]CY? 8@g 57;RL;>= ADaDC 6 ;>M 0 & OQATR ?57;K CN=@ADC 8?LR G[6]C7;>M^= GD8; !#$$&02+% 2+ OQATR eBA M ;RG[CNMHK77N;R?LR6FG[C aDK7e9G[R \HG>?LR6Bf_=@ADCN8<?LR G[6BCX?+8 &'.": .2 0 /0 3 /0 & $+/2 ? !' &.%/ !'&"%/ cV; c M ADR+M 698 KN8;>M OQADR ;6B?57;RG>RLaTK7\;CX?+8 G[C M !+ - %/ 0 / 20- / 0 / 23 0 / g P57; C ?57; 7RA X6FMX;?L57; NCNG>e ?57;WKN8;R 8<57ADK7e9M OQADK R G>RLaDK \;CX?+8 A>O?L5 ;OQK CN=@?L6]ADCS=G>eBe ?L5NG>?G[R;^KN8<;>M?LAW=@R;>G[?;?57; =@ADC 8?LR G[6]CY? 8 2 & !#%$$+&2 % V;>= ?L5NG>? E \KN8<? `N;8<cY\\_;?LR6F= ?LRL69MY6FG>aDATCNG[e V;U= &7ADR2; f7G>\7eB; 6BC ADR+MX;R?LA MY;NC ; G[C M GD8 ?LAW`'; eBA M ; R ?RL6FG>C7aDK eFG[R G>RLaTK7\;CX?+8 0!2 ) -. ) . C "* "!$#%$& =('1_I@9=)'1EF> ;;*'8;+'8J,'17I36EQGX2 > EA36=697T->X=6=?>.--36G G[CNM +!2 G[C M dAV;7e]6B? \WKN8? `N;MX;>=@eFG>RL;>M 7 )20- G ! #0$$ %G )"&' 7 )2 C 7 ' /H( C $.Q 4 C)4 CE0&%UWC D>lU%' 3AD/'LE?> J,' ;0'17TC3AD b AT?L;^?5NG[? G>`7`7RL; Y69G[?L6]ADCN8G[R;KN8;UM:OQADR ?57;W=@ADC 8?LR G[6]CY?CNG[\_;>8 57A M ; V; R G>eBe G[`7`7R; X6FGUh "! ?L6]ADCN8 \WKN8?`N;^A>O2?57; 8LG[\_;e];C7aT?L5S8A?5NG[??L57; 8<?LRL6]C7a_G[RLR Gc6F8 TG[e]6FM6BC "! K7R g OQK7C =@?L6]ADCb6B\_7eB;\W;CV? G[?L6]ADC RLA Y69MY;>8 C AH8;>G>R+=+5:OQATR=@ADC)N6F=@?6BC7a =@ADCYh 8?R+G>6BCX?+8@g ? 698?L57;RL;U8NADCN8<6B`76]eB6]?cWA[O ?L57;KN8<;R ?LAW7RLA X6FMY; \;>G[C 6BC7a>OQK7ed=@ATCN8?R+G>6BCX?+8@g K7; ?A6B?+8 86 ; !S=@AXMY; ?57; 8;5NG V;8;; C8<A^O G[R "$ OQATR OQK7CN= ?L6BATC ' / 6F82e]6F8?;>M6BCW?L57; ' 7N;CNMX6Bf g " !#"' &' G[e]e3eB6]C7;>G>R= ADCN8<?LR+G>6BC7;UMG RLA>fY6]\HG[?6BATCN8 ;f =@;7??57;bG[RPcVA G[R;R G[C"cVh ADC7;SG 7RLA>fX6B\HG>?L6BATCN8 6]?L5eB6BC ;>G[R;VKNG[e]6B?<cb= ADCN8<?LR+G>6BCX?+8 g C?L5 6F88;>= ?L6BATC ; 7RA X6FMX;H8AT\;7RAVA[O 8?L5NG>?RL; V;>G>e?57;HMY6H=@K eB?<cbA[O?57;7RAD`7e];\ G 7RLA GD=+5 ; =G>C 7RA V; ?L57; OQATeBeBA! 6]C7a ,"&; \6BC < % %:9 =! > ?2-2A@ ?B DC FE G4*32 # 324 ; L HIKJML NPOWRQ " ; ! L L L L S 8 5 76 7 P; = L U> V4* L XWY2 [Z Z ; \V]!^+^_a` bc IGpYq t Tn| K7CNMY;ReB6]C7;ADK R ?LAWKN8<;6]?L;R G[?6 V;\;?L5 AXM78?L5NG>? = ADC V;RLaT; ?LAWGW8ATeBK7?6BADC g "#%$&$&')(+*-,+*/.10 324 < G[CNM XbFcDd 5 ; 5 ; I-e3f bhgjikcld cDmnd!o IGp d!b I i J % T L ! ! L = Z & ! ; g [Z ! ! % T % T % T i!gb fqMJsr oKotuvd pYqxw dkt I thi!gb fqxp i J tybcDd IKJ b H d{z 32 u > "#%$&$&')(+*-,+* H = & Id ; ; q d J t I i J 32 u >! ' ! @yW#!@# "C FE G4*32 324 = ) ( &ci H L S < i gb fqVe cDdkm IKq J b T| u> FE d J m q cFi IKJe @ ! % T " +C s@ & G4 $ * u> ; s@ 2& # 32 % = = 4 & = X2 K@y4* F = 3 2' 32 u > 14* XWY2 u > tyi3o Xq b fq cDtyb i gb fqVq-,?r d b I i J t I-J/. s TY 10 gjikc& q b f d b b fq t p d o d c2 It "2 3 J i J z q cFi T gjb q c t rw yt b I b r b IKJe bFi b fq t q!p i J m i!g . % T 40 q-,Xr d b I i J . L &; ; ; e3IXqJ ; i g b fq i wq{p b IXq g rJvp b I i J \V]!^+^_a`+* % T L l 4 DC FE G4*32 L 32 4 * "#%$&$&')(+*-,+*K( > L PWY2 *B u>' \V]!^+^_a` L ! ; L < " L L P; < IKJL ; b fq 0 ! d w i Xq i3o6 J i HI d!oGt&8 . . 3 0 " qnp i J p o r m q 9#9#9 ;: . " 0 b f d b q 0 . f d Xq "5 bi ?> #9#9#9 gjikc=< 0 tuiko Xq " 0 t76tub q H i!g . d i!gVm qe c qq sT J " . H 0 r Ko b I d c I d b q ikcDm q cxbi J m b fq t q b i g p i HH d c I i r t IKJ gji3c H H i J . q b f i%mt d b I i J c q d o 0 q ci ti g d oIc q b fqH t q b i g r oKb I d c I d!b q b fq c q dkm q c I t m I c q!p b q m bi iko 6 J i HI d o t i g c q t r oKb d J b b fqMp ikoKo q!p b I i J d J m Pci d c I d w o q t qp d J wJq c i!gd!cb IGp o q t I-J o-i-6 &ikc H i3c q d3t q t T )( qH T | "! # %$ & ' ( ) * J b fI t paf d b q c q c q t qJ bdVt 6tyb qH +,+,- tMd J m&d bc d J t gji3c H c d Jekq d b I i J t T i gPg d3tybxbcDd J t gjikc H * msi q tyi f d J mso q ikb fq c g dktubbc d J t gji3c H /. Ib f 10 ci Jq!p Xq c ci%m rvp bDt T b fI t[ikc 0 w7q{p d r t qH fq I-J m q# b fq I t c q o d b q m 30 +,+- < IKekf H - 2. 7+,+,- i g q# c q t t I i J t T q cugjikc H d J pYq +,+- t 6 H b fqPIKH i o GI pH w 3 d w tybc d p bxld 6 o qH qJ bDd!b I i J d JI r oGd b I i J tuinb f d bMb fq 30 ci Jq!p Xq c ci%m r p bVoGd#6td bc d J t gji3c H fq t I b r d!b I i J It H 68 +- i gb fq :9 d bFc I pYq t Xikc q c H t b 6s IGp d!oKo 6 p i J bDd IKJ DC i3c q H Xq qJ d w o q d bFc I H Ib f >= ci e cDd H p d J d!bc I g d p bFi3c I !d!b I i J IKH i3cFbDd J bPciko qI-J w d3t q m 4. 30 i J bFid r bFi H ci Jq!p Xq c Xq c 6 p i H d!b I i J :9 z fqJ 6. g d p bikcDt b f d bld c q d bFc IGp q t Xi3cPt H d!bo q 3 d tyb d g q p IKqJ b t76tub q H t p d!oGd c o q#Xq o I b f d H r b d b I i J c 6 i!g q d J m c q t r oKb IKJe d b I i J td c q t q c q m q t p c I-w7q md b b fq bFiMd g d p bi3c I !d!b I i J d bc IGpYq t T d b I-Je tUb f d b p d Jxw qPq# c q tFt q mxdktd[tu d!cDt qH Jq!p Xq ccFi%m r p b ti g m I d e i J d o H bc d J t gji3c H t r t I-Je i tubVi gPi r ccFi e cDd H cDtyb e3qJq cDd!b I i J o IGp d!b q m d J IKJ fq cFi%m r p b I m q J b I b I-q t T - 65 gjikc d r bi H ;+ ci!z d o-o q c i r c IKq c @? BA f d J m?z p c d{gjb q m W a@ 9 dVb q c H I-Jfq c I b q m dnt q ,?rq J pYq r b d b I i J i!gVt H - wr Fb b q c;6 T f dkt J i3b p i H c q t qJ b q mnd w qq J - > i gld d!o e 3 i c Ib f H t H o qHq J bDd b I i J T! rI oGm I-Je b fq f iXt q t H d!oKo d o p i J tubFc r p b I Xq t I q I t p d!oKo q m)b fq d o e 3 i c Ib f H t I tVd J @9 +,+,:5 Jq yt b q m o-iXi3 t[b f d bMm q p d J)w q b fq Jq D9 d!c q q t IKJ p q b fq c q T r b f i3c t b fqH tuo I b -I J bFixbyi i3c m q cVb fq Xq{p d w oq z i g b fq qJ b t b f d b d pDfIKqXq b fq p i rJ bc q-,XrI c q m r t T * i!gdxcDdkm I z 8 p i H X= d cFb d#6)gjikc e3qJq cDd b IKJe td3tPdxt q ,?rq J pYq i!gdktt I-e3JH T fqVIKH +,++,+,- d!oKo bFi i I-J wr bb q c 6 o d c e3q cxi Jq T q J bFcDd o binb fq d i gPb fq t q gji3c H c q i3cFb[b fq vi3 c q t r o-bPd J m 0 r bDd!b I i J fqp i J tybc rvp b I i J b 6s IGp d!oKo 6xo I tybb fq b fI tbDdkt ++,- e c d f thb f d blm q t p c IKw q tb fq d o e i3c I b fH AtVbi wrI oGm d o-o I tPb fq t I-e3J d!o i! gjcFi H m q t Ic q m bFid r bi H t q d c d b q 0 bDdkt d b q t T bikc q o qHqJ bDt d!c q d p z pYq t t q m T T! rI oGm I-Je d)t q-,XrqJ pYq i!gPdktt I-e3JH qJ b tub d b qHqJ bDtb f d bV q cygji3c H i!gd * q bFcDd J t gjikc H ci sI m q ci e c d H ;+,+,- q c d b I i J - t I-Je o q i g t b f d!b[d r bi H o qH I-H i g o qJe b f 8 fIKekf d b IGp d!oKo6 c d3m I ekqJq c d b qMq d pDf d o e 3 i c Ib f H ik p i rJ bld J m w q b b q c HqH ikc M 6 bcDd p d bFb q c J t T tubFc r p b I i J p i H i!g +,+,- 2+ ,- #"%$ &$ d wr b b q c IKq t[b f d bxd!c q r tubMb fq r b q cMd c pDfI b q!p b r c q + ,- wI-J d b I i J T o-b f i re3f 9 bi i3b fq c I b q c d b I Xq cFi e cDd H It H i tub p oKiXt q o 6Mc q o d b q m q HIKJe biMb fq i3c . < C 9 . < @9 < 9 8 . C 67 t f d Xq fq 98 6:3d ; 4 0 fq d r bi H b fq d r bFi H d b I p[e3qJ z wr bb q c I-q t T i g oGd c e3q z cDdkm I c I-e3f bMt I q oGd Je3r d e3q t T " r c t 6tyb q H b I i J t354 - - d cFb T d3m d J b d e3q ti gPc q m r pYq m d!b I i J gjikcd ci%m rvpYq i3c #"%$ &$ '(t rw ci r b I-Jq i3c d*),+-/.0+21 q# b qJ m)i r c[t 6tyb qH p d!b I i J p i H p d!o pYr o d b I i J T q J bDd b I i J t q c HI b t b fqe3q Jq cDd!b I i J fI t I td!bbc d p b I Xqxw q!p d r t q b fq t r i3cFbDtb fqMp i J z ci w o qH e3IXqJ i r bF r bxb f d b I t qI b fq c 9 g rJ p b I i J qxp d J o d Jekr d ekq t[i3c f d c ms d!c q c q t qJ b q m IKJ d J m q dkt I o 6 ty q{pYI 5 dxt q c IKq t i g rw o I p d{z z 6: ; fq c q m I c q!p bPbc d J tyoGd!b I i J i g . 30 ci Jq!p Xq c cFi%m r p b gjikc H r oGd qI t r t q m bFii w bDd IKJ cFi%m r p b I m qJ b I b IKq t bi e3q b fq c[ I b f t T ci e cDd H tubc I m q q c H r bDd b I i J d!bc I pYq tgjci H H :5 - - d!c w qJ pDfH 0 9 b fq c q t r -o b -I JeVp i%m q XtuiVdktUbi m q!pYI m q fI paf g d p bi3c I !d!b I i J t d c q b fqPw q tubgji3c d e3IXqJ bi IKJ 9 - 6 i r c[t 6tyb qH 8 . 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