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OQPS*R &0v M ! kY[rb.N=MY[UcnNIZ KT}eZ n 594! P \cVNIeb KY[eT9[N=Z_TVW[NgY[ZD]_M\cbKb.N=]YVgb.t~TVZib.N`T]_Z.\cjIN=bIv2oM\XjmM \Xb TVNIZleYV].N=oY[Zi].MfQZ.NInrj=].\XY[e6xL#MNdtNIZ.j=NIe2]mT}W[N(YVgyZ.NInrj=].\XY[e6v`T}].Z.\ q2`T}].Z.\^ \Xb N$N/ 224! gY[ZOQPSR v 22"! gY[lZ kY[rb.N=MY[UcnNIZTVeO n 22"! gYVZ;P \cVNIeb &v MA V Q24! v (2" ! TVen 2"!v y V Q:4!v ( ! T}en 4!v # 2( !v 2[" ! TVe n 2( !&v } 92"!@v 9:?"! TVe| n 924!v K 94! v ! TVen L ! L 1 Householder reflections Fill-in in matrix R 3.5E+6 Nonzero elements of R (log) ε=0 ε=0.85 ε=1 4.5E+5 6.0E+4 8.0E+3 1.0E+3 \XW[rZ.Nu^[ SHL400 JPWH991 SHERMAN1 MAHINDAS ORANI678 SHERMAN5 $&'M-FG\ $&% K*G/C %9% T 69+ 69+ )N0:;OCP6('J $&C K:6 G/CDG/+ ; 8&0:%(' GH-,$J Givens rotations Fill-in in matrix R 3.5E+6 ε=0 ε=0.85 ε=1 Nonzero elements of R (log) 9 4.5E+5 6.0E+4 8.0E+3 1.0E+3 SHL400 \XW[rZ.Nu^! 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L 1 O PSR v{]_MN N=Z.ZiY[Zd\XbdUcYoNIZdq2?QYVZ.nN=Z.b(YVg*`TVWVe\c].rnNo\^]_M Z.N=b.tNIj&]].Y A |7 N Q Y[q]_TV\Xe]_MNW[ZiN9T}].NIbl]TVj=jIrZ.TVj=Vv o\^]_M?~UXUk0\ceZ.NInrj=].\XY[e6v gY[Z`T}]_Zi\^ N$NTVen ].MN UXNITVbi]*TVj=jIrZ.TVj=gYVZ#`T}].Z.\ # : e ]_MNoMY[UcNVvygY[Zd]_M\cbsbiN=]YVgw].NIbl]`T}]_Zi\Xj=NIbIvy]_MN!UXNITVbi]TVj=jIrZ.T}]_N!Z.N=b.rU^]_b(o;N=Z.N Y[q]_TV\XeN=n rbi\XeWs]_MNOQPSRTVUcW[Y[Zi\c]_M` *N&[NIZl]_MN=UXN=b.bIv ]_MNxNIZ.ZiY[Z.b*TVZiNSVNIZib.\c`\cUTVZgY[Z ]_MN kY[rb.N=MY[UcnNIZTVen P \c[N=ebTVUXWVY[Z.\^]_M`sbI ? RB 1L * SHL400 - Linear Programming Problems 663x663 10 Numerical error 10 10 1712 els. (0.39%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 0.2 \XW[rZiNSu^F?L 0.4 N$N ε 0.6 0.8 1.0 + '!) G/CP6QE 0:%G/CPCD$&CD- J $&CK:6 G/CHG/+ ;18&0:%(' GH-I$J JPWH991 - Circuit Physics Modelling 991x991 10 10 Numerical error 10 6027 els. (0.61%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 \cW[rZiNu^d( MA 0.2 V 0.4 ε 0.6 0.8 1.0 + '!) G/CP6QE 0:%G/CPCD$&CD- J $&C K:6 G/CHG/+ ; 8&0:%(' GH-I$J ?2 L ! L 1 SHERMAN1 - Oil Reservoir Modelling 1000x1000 10 Numerical error 10 10 3750 els. (0.37%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 \cW[rZiNSucF! 0.2 K 0.4 V ε 0.6 0.8 1.0 + '!) G/CP6QE 0:%G/CPCD$&CD- J $&CK:6 G/CHG/+ ;18&0:%(' GH-I$J MAHINDAS - Economic Modelling 1258x1258 10 Numerical error 10 10 7682 els. (0.49%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 0.2 \XW[rZ.NSu^L # 0.4 ε 0.6 0.8 1.0 + '*) G/CP6QE 0:%G/CPCH$&CD- J $&CK:6 G/CDG/+ ;18&0:%(' GH-I$J ?? RB 1L * ORANI678 - Economic Modelling 2529x2529 10 Numerical error 10 10 90158 els. (1.41%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 0.2 \XW[rZiNSu^5! 0.4 ε 0.6 0.8 1.0 + '!) G/CP6QE 0:%G/CPCD$&CD- J $&CK:6 G/CHG/+ ;18&0:%(' GH-I$J SHERMAN5 - Oil Reservoir Modelling 3312x3312 10 10 Numerical error 10 20793 els. (0.19%) -16 -13 -10 10 10 10 -7 MGS HOUSEHOLDER GIVENS -4 -1 0.0 \cW[rZiNu^9L 0.2 y 0.4 ε 0.6 0.8 1.0 + '!) G/CP6QE 0:%G/CPCD$&CD- J $&C K:6 G/CHG/+ ; 8&0:%(' GH-I$J ?( L ! 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N N ; £/¢ Ô â¦u ¢ ¨PR§ ¥-«¦/ u M ¢ »­33¥ ² À .N ãóô ¥- ¦/ N "!#%$'& ¢±¨ i¢ ¦u / / u© «à ¢ß ª®¥ Û ¨Ñ¡ 7Û ãóô / Ï ¥ ¡ $¢ ¢l)¢ © wª®¥ ¬µàO¶ ¨ ¦u ¥- ¯ ¥ª> ¥- / îñ3ò ¨ /± $ ¡ ¢ *Àu£u ¥- ¨ ¥ M) µ £ ¥ª·/ ª ±¥ à ! § /© " / ¤ ¥- / ¥- ¥-X¦/ / ¥ª M ¨³ P ©-¥ ­½ N ±¥- /e± ) ©-¥ /¥ @ ¥ª N Ù = N /©~ / ¥-¦ ¨/¨ ÛX ¢ ª®¥ ¨ ¥- ¢ ãM ¦ X¥ ¢ § £ ±¥- Z¥O $¥ ¥- ¢¢ N wM ±¥- /w± PÜwÝ îñ3ò ¨ ¨ ­ ¥«± X¢ l ¨ N ¡ × ¦A ¢ £ Ù ­7Ù N=N N .N XN N XN N N N N N N N [N N N .N .N .N N N .N N N .N .N .N N N N N @ N N iN N N ;N uN N N N N .N .N .N N N N @ N N N [N N=N .N .N N N N N =N N ( = N N N [N N N N IN NN uN N iN N VN .N N N N N .N N N N iN N N N=N N N N MS[q p `cq[e h bnXY-S , S h z ^fbM\xS[q 3qY`%XYS Y `cqPXY`ceeZ]bM^aX h \_^a`cb"q h \xS N=N N N 6 N N N 2N 6 VN iN N IN N N N N N ¨ ¢ ¥) u Ö ¥ ¢ N ¦u ò iN N ¢ M N iN ¢ À ¢ À £'¢ ¨ ¥ ã ¢ Ù ¥ ¢ ×* N ¥ª ¥ª1 õ N ¥- Z / æ ¥ Ö ¢ ¡ ϱ 6± N ¢«¬µàO¶ç Oª®¥ ¦u ¨ XÕ ¢W ª®¥ !¦/ óô 1ò N / ¢ ¨J¤ ¥- ä­ M¢ Ö-× ­ Ù N ¢ å Õ ÀA £ £/£ w ¥- ¢$ XN ϼ N N ) @ +* .N ¦u± u© N ¦u±¥- " ¦u ¥- 8 ±¥ £' ª ) ¥ N IN N XN N N VN N [N W ¥-«¦/ u M NN uN À N ­7 ¢ ¥- /Õ ¤ M)¢ /¯Õ® O .N N / R ¢ ± ¥ N ¨ /¥ / N ¢ ¥ u ¨X¡ R ¡ N ¤ ëRì«íPîïRïRð­ ¢ ¦/ / / /©Z ¦A¥ W cN À ­7 £' à ¢ ¢$ =N NN uN N [N N N iN ¢ /¥- /å ¢ ¥ ¯¢ ¦/ / / /© ¥- /©J .N ¢ ©-¦ [N ¤ X¥- u©~¥ N N ß ¥- d­ [N cN N N W) ©-¥ ¨8à Ø' ºK À £/ .N=N ¡ N ¥¦/ ¢ ¦/ / u /©i ¥-¦/ R¥ª ¨ ¢ ± ¥-¦u± ­uª®¥ ¨ã Û N N / ©- N w' N2A ¢ /­3 N NIN N N N Ï N ¥- N £'¢ ±¥- ¢ uX3 ¦u ¨8 N ¦u± u© ( NyN uN ¡ ¢W¬µàO¶ ª®¥ ¯Ø ¢ M ¥- Õ® Rª ¢) ¥- 8¥ªe± ¨ N 6¥ª½ ¢ ¢M¢¢ ±¥ R±W © ¢ å ¢e ©-¥ ©¥ ¥ / {± N N ¢ ±¥X ¦u ¨ /¥ / M ¥- § ¥-¦ N .NIN .N N N iN £/¢ ª ¥ £ ¢ u ¥¦/ ©- u Ø' ¢±£ © ©-¥ N ¦ ¢ ¥- ~¢ X¨ £u¢ ¥ .N N ¨ /¥ / ¨ / ­7 £ M ß¡ ¥- ¥)ª A .N N X¥ ¢ N w¢ ¦/ / / /©Z ¢ N ¨ N ­' ¡ N ¬µàO¶¯ N N ¥-¦/ !¦/ N £'¢ ¥- á¥)ª ¢) ­[ Ì /¥ ¦u ¡ ÏÎ N 576 N Å cN N uN ߣ'¢± ¢ 8 ÏË ) À § :9 Ç $Ì ¦u ¥- "¥ª3 ¯/¥ ¡ ÅwÒ N 6 ©-¥ Ñ Æ Ò ¢ / XN >)/ Ù Ö =<N ±± ­ £'¢ ± § N «Ø' º µ¥ª ´'¦ ¦ N d T <: B!W#%$ ' =)./2 /N1154 , >-0 /= J @ BADC 4 BA @ /JI C I & A = ADC = /J>@ BAD C = 69 A %9 / ¾ 69 ¾ %9 A A ¾ Ú Ú Ù ¹ TÖ ÙR¾ A Ú ¾ A -¾ -Ú Ú ¹A Ù Ú Ö -Ú¹ Ù Ú-Ö Ù ¾ ¾ 7 Ö R¾Ù T¹ ¾ Ù Ú ¾ ¹ Ú ¾ Ú <7 Ú-¹ Ù 87 E: E 9 Ú TÖ 69 Ú ¾ Ù Ù R¾-¾ Ú Ú Ú Ú ¾ ¾ A AT¹u Ú Ú Ö Ú ¾ Ú-Ö Ú Ö R¾ Ù ¾ A A Ú ¾)Ù Ú-¹u A Ú A Ù Ö Ù Úu Ú Ù ÚÚ Ù -¹ ÚÚ Ö Ú Ö ¾ ¹u ÚuM¹-¹u ¾ ¹-¹ Ù Ö ¾ T¹-Ö-¹ )Ú Ö ¾ Ö A Ù ¾-¾-¾ Ú Ö ¾ R¾ ¾ A T¹ Ú Ù ¹ ¾-¾ ¾ Ú Ö-Ú R¾ ¹ Ú ¾ Ú ô ê ¾ Ú Ú-Ú -Ö A Ú Ù A R¾Ù A ¾ ¾ Xì3ñ ¾ T¹-ÚA ¾ Ú Ù ¾ ¹ Ú ¾ Ú >)/ ¾ TÖÚ Ù A Ú Ö Ù ì3ñ ¾ T¹ R¾ Ö ¹ A T¹ ¾ ¾ Ú I[* %9 8 <7 E <7 7 7 : 89 8 7 9E9 7 8< 7 69 7E7 LM/ L I[* 69 %9 <: 9 89 7 7 79 8 8< E: 89 8 S[<S[q h u5egS $ XY`cb t<Z]q h w ¹ Ú ¾ Ö-¹ Ù Pì3ñ = 69 / 69 : 7 6E E7 6E9 E: 7 7 : 8 7 9 8 s7 7 8 7 7 E 8 E9 9 K K I* = / = /= 2' 69 69 69 %9 ), .-0/N1154 7 /= J>@ BADC 4 %9 8 7 %: <7 6:89 7 7 9 7 %9E: 9 E7 BA @ /JI C I& A = E: 89 6 E: 7 9 ADC E 8E %: 8 7 8 9 = /J>@ BADC E9EE 7 6E 8 7% XN Z]eC[S[q` p bn`cb &cS[q D ` S[bM\_q ^aS ^fb e h \dq[^ U Z ^fb]t _\ ^a`cb | ~ 89 I* ðì3ñ ¹ ÚÚ ¾ Ö Ú ¾ Ú Ú ¾ Ö Ù TÖ Ù Ù!¾ ò Ó{Ô "!#%$ /2' ), .= -0/213154 = /?>/J@ I BA&DC 4 HA @ C I A = /= ?>AD @ C BAD C @ ¹ : ;:<: / = @ º º TÚ Ù Ú Ö 7 @ ¹ G7 7 ;9 - 7@ @ ¹ - Ö ¾ Ú-Ö ¾ º TÚ º ¾ ¾ AÚ @ @ - º TÚ Úu Ù 9 º : @ Ö @ ¾ -Ú-Ö 69 - -¹ ¾ @ @ º º º @ ¹ : 7 6 @ ¾ ¹ 9 ;9E9 A F: @ @ % @ @ 7; ¾ Ú @ 7 @ Ö-Ö 7 Ù Ú 7 ;9 : F: 69 @ A º Ù º TÚ @ Ù u ¾ @ @ @ º @ Ù LM/ L Ù ê ô @ : u ¾ @ @ % º - ¾ ¹ º MÚ - ¾ Ö º 9 ¾ º MÚ º M¹ @ ¹ ;9 : ¾ Ù Öu ¹ ¾ º T¹ Ö 8 º Ù A¹ ¾ º TÚ I* Xì3ñ º Úu u º @ @ ¹ @ @ º Ù @ º º : ;E @ ;9 7 @ : - ¾ ¹ ¾ º TÚ º MÚ Öu Ù!¾ ~ @ - Ù Ó1Ô º Ù Öu ¾ º Ù cN ;: Z]egS[q[^aX h uS[q[q `cq Z ^fb]t S[<S[q h unegS $NXY`cb t<ZWq h \d^a`cb | / @ º Ù - ¾ º Ù Ù-Ù @ ¹ 7 ;E9 F: - Ù : º @ ÖuTÚ ¾ = ¹ @ Úu¹ : º I* º Ù ÚuÚ 9 º @ ¹ ; E: º MÚ ¾ - Ú º MÚ @ Pì3ñ º -Ö ¾ º T¹ @ @ ¹ 7E7 Ú 7 º TÚ A ¾ ¾ º Ù A¹-Ö º TÚ ¾ º TÚ - -Ú ¾ º : I[* F: º Ù º Öu º TÚ Ù ¹ º º 7:@ @ -T¹ º @ º @ ¾-¾ / = º Ù Úu¹-Ú ¾ : ì3ñ 6 @ ÖÖ @ º ¾-¾ ¾ K / K = @ @ -Ú º @ - Ù ¾ @ ¹ Ú ;9 º Ù - ¹ º Ù ' =)./2 /213154 , >-0 /= ? @ BADC 4 HA @ /JI C I& A = /= ?>AD @ C BADC Úu Ù º - @ - Ù-Ù 7 E: @ 9 %@ 7 @ : Úu Ú ¹ 6 u I[* ðZì·ñ @ º º TÚ º TÚ ¾ 7 ¹ Fill-in in matrix JPWH991 ε=1 1.0E+6 MGS HOUSEHOLDER GIVENS Nonzero elements of R (log) 5.6E+5 3.2E+5 1.8E+5 1.0E+5 ¼ N N ¨ £ N ¨ ÀA£/ ¡ £ N ¨ ¦ £ ¨ Z ¥ 㥠¢¨ × XN ¢ À [N ¤ ¢§ XN ;: N >)/ ±­ N ¼ .N ¢ £ ;9 Ù u ¢ ¥)ª ¡ § ¥ ¢ VN ¢ ª®¥ ¥- ¡ ­ i ¡ ¢± ©-¦ N ¢ N IN ¤ ­ =N ¥ ¨ Ø/ ºK X N <N .N ¨ .N ¨ ¢ À èÀ ©¦ õ N ¡ N N ­ ±¥- ¤)¢§ ¢ u­uÙ .N %L 7 6Ù ¨ £ N ¢ å ò Ù ¥ ¥ b<MS#" Õ ¨ ¡ N iN N ¢ M ¢ N ¦ ­AÙ ±¥P Ö N = ¨ £ N .N X¢ «¨ à ¢ 1¥ª N ¢ ¥ N / ¡ £ e¥ª ¢ ;º VN ¤ ¡ ­ £'¢ ± § .N N ¢ ! ¤ «§ «¨ ­ [N Ø'M § ±«±u¥ ¦ ¢ N £ XN N ¤)¢ <N .N ¦/ /© .N ¢ ¥)º ¥-± / N e Jóô îñ·ò £ N ª®¥ ¥- d­ w£' × ­ N ¦/ ½ ÑÕ¥ × ¦/ /6 ±¥- © ;9 ( ¤ @ 7 ¨ w ¥ £')¢ «¥)ªe ¥ª ±¥- P / ¢ ¦ ¦ ¥- Ϫ ±¥ 7 ¨' ¢ ¨ N óô 1ò XN ¨8 ¢O¤ ­7Ù/­ ¥- ?¥ª ¢ ¥) ! 3¼ WëRì«íPîïRïRð­ / M1 £ ¢ e¥- ¯± ¢ M × /{¥)ª ¦ ¢± ©-¦ M .N ¢ X¥ Rw Ï Ø Õ /¥A¥± u© ¢ ±±¥ N N iN N XN N J ¥ VN ¤ ¢ ¡ N ¥~Õ¥ ¥- d­ ¥ª £/¢ ¥ ¥ª ¢±£ © Ø ¦u ¢ ¢ ã ¡ /©¥- 8± N iN iN ¢ @ u ­ £ A ¾ '£' ¨ ¢ Àó ¢ M¤ ¥ß 1 N £ !¦/ ­u± ª 6± N Z ¦/ ¨ ¥- ± ¦/ d­ ¨JR§ £ Rw òѪ®¥ ¢ ¥- ¾ XN N ¢ / !¦/ ¨ ­ N p `cq.i<^ S[q S[bM\egS $ '^ &cS ò § ­ ? º Öu Ù ©¥ ¢± ©-¦ 64 Q ufuywx^fb2^fbNe h _\ q ^ U N N ;N=N N cN N ¥ N Ø 1 P ¥- Ï µ± ¨ N / ¥) N /©ß¥- ¯± Û ¦u±¥- Ï ¤ ¥ ± N =N N N N N ¥- /« N N N N £ ­ N TÖR× ­ N 4 16 Number of PEs ëRìZíPîwïRïRð N u /©J¥ µ /w´'¦ ¢ ¥ ¨ N ¢ ±¥- Pª®¥ ¥-¦/ § N N h \dq ^ U ¢ ¢ ¥- "Õg¾ !¤ ¨ = 7 uê Ù /©Ï¥ ç /© iN ¢ ©-¦ 1 N N iN ¢ ¯M NIN ? ¢ ± N R N N ¨ ¨u¨ N !¦/ N N iN ±¥- !¦/ X¢ M ¢O ©¥ § ¤)¢ R¦u± u© ­ £')¢ ¦ ¢ ± Ø/ ºK ¤)¢ ~ ­>ª®¥ ± ¢ ¥ £/¢ N ¡ !­-ª ¢ ¥- ±/¥-¦/© M­ ;N iN N *: 7 , ¡ ¢ ¥ ¨ N N ¢ N ­·± < N iN ¢ N ¢ /i­d N N ÀA£ ¢ N ¦u ¢±¢ ¥ R ¥- ­A /¥)1©- u Ø' ) ) R N ¢ ¥- ç N N ¢ N ¢ Rª®¥ .N N µ± å ¤)¢§ ¦ ¥ N [N ¢±¢ ¥ ¡ ¢ ¨ N ª®¥ N ¢ ± N N ¢±¢ ¥ NG6 e¤)¢ ¥-¦/ ±$¥- J N £ ßØ' º ¢ ­ /¥ª®¥ N XN ¢1£')¢ ¥ª¿¥-¦ N ¢ ¨ /ª ©-R §µ¨ ¤ ¥- N N ¤ ¥- R1¥ª N ¢ ¨8 ¤)¢ R b¢ N §Z£ T¹¹-¹ ¥- N N ¶ ¦ ¨ !N [N ¨ /©Ï¥- 8± ¤ ¥ Ø ¡ ©-¦ ¢ ± /© N ¢ º N N º § ±¥- / u / d T 7 Fill-in in matrix ORANI678 ε=1 1.0E+6 MGS HOUSEHOLDER GIVENS Nonzero elements of R (log) 5.6E+5 3.2E+5 1.8E+5 1.0E+5 ¼ .N ;9 ¢ ©-¦ Ù = ê 1 h \_q ^ U 1ò 4 16 Number of PEs %L Q ufuywx^fbN^fb e h \_q ^ U óô õ 64 p `cq.i<^ S[q S[bM\egS $ ^'&cS ò Fill-in in matrix SHERMAN5 ε=1 1.0E+6 MGS HOUSEHOLDER GIVENS Nonzero elements of R (log) 5.6E+5 3.2E+5 1.8E+5 ¼ 1.0E+5 ©-¦ iN ¢ Ù Ö ê = h \_q ^ U ( 1 îñ3ò = 4 16 Number of PEs Jóô 8Q ufuywx^fbN^fbNe h \dq[^ U ò 64 p `cq i<^ S[q S[bM\egS $ ^'&cS 7 R¾ .0/ . 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M " + 38 K &Y } 9 =3=*= G1 9 x*s 8 I " 3 *8 ) 13 ; /" > 1 "*4<5 '5 636 & .340D,)!*4 >3E 4-, 5 6 M ss@B=C@ ,);<5 2030" + ,; 5 6*6cE 134 > - 6{E 134 M *9 M )9 4 > !H,); M v vv+ *F 5 0 M 6 % *%x8* > ,)2 E 4*!'5NM 3 ,);<5NM > 134 > )9 *9 M M :@'=C@D6FE }M 4 > !L,; 4 > !$,); O M 5 K86 939 5 K86LK 1 3; 4 > !! /G4 > ! =C@ 9- l5 ,);<5 " )9 6 6 G & 9- 5 36i I M 9*9 5 36i 9 M #O M M * M M ! J 9 5 6} $ J K " " > G! - O 1 9 5 6Fv@'=C@ 13 ; /v, > !343 4 > !34 >3EB&Q> 4 5NM & 36 38 K g ! " &Y 13 ; /" > 1 "* 4<5 ' 5 636 & .340D,)!*4 >3E 4-, > ' M 5 6 K&{ 38 K !" ! " =?> ,;<5 " E M 4 > !:! " = 2030" + -I :M ,;<5 M ,)2 E 43! 5 M :9 636 5 3636 M x8 5 6*% 2 4 , E 4 -, E ,)2 E 43!'5NM 636cE 134 > -I *:9 5 36367E 6 5 6J M *9 M ,);<5 20303" + ,;<5 M 5 1*4 > 939 5 36 4 > !$,); 4 > !H,); 1 3; 38 38 4 > !s!" /G4 > ! " > 4 > !! " K K lSVhCmK;=<fIudN d n¥ ` {¦¡%£ O¥: Rn = M¦ ¶n p£ n¦ ¥ _¥ I£ Â¥ ]Q $ Z 8S -Z 1 S q %$ CZ q?~ # Xa S)T -~ R -S IW b 8~ k "! 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