What Every Computer Scientist Should Know Abaout Floating-Point Arithmetic

¢¡¢ £¥¤§¦©¨ !#"¢$&%('0)213 45)6'0718'0@9A'0B&9C)1D!#EF182'GB&45B&'09IHQP5RTSFUV5WYXY`acbTdIe5fgShWYX i§pCq WYr§S qts S i P5bTf§uwvyxr§bTb¢fgSFgutb¢RTS
q r0©b q r§S§X q ShP5dWYS qwp§§ 45§de!gf '09Ag B&$ fh'0d)61©'Gij459k"¢ §l5m5m5l nh$'G!#EFop!#%6B&$&1©)gQqr$&9C§' 5stou!gB 9Yg§1 l5m5m5l5 v hn!#"¢4518!g)6Eh!#9pop!#%6B&$&1©)gQiw45"¢)'09 T§x )"ys 9A'0B&9Y)1©'0 § B&'09C%(nh!g)s 7z45B&B&'0){|y245z1©'GE©!gff!g}¥)2g!g9Agg45)7~54518!g) z7Ck ¢8y¢¢ 7 7©C¢57 ¢8y¢¢ C ¢ g©C ¢8y¢¢ g g#¥57 ¢C ¢8y¢¢ hy7 gz¢k ¢8y¢C g 3¡uC¢ ¢8y¢¢£ 55uuC¤ ¢8y¢¢ z¥ C¦F¢ky 7 ¢8y¢§¢¨ ©© ¢8y¢§¢¨ g C©ª¢§z # C©ª ¢8y¢§¢ ¡u©©© gzu g#¥¢£¢§ u¢¢u C7 ¢8y¢§¢

¢¡¢« ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ±³²´Fµ¶¸·¹°µ º f!#451©)g§»©B&!g)1D459Y1©%6'¼1©"ez"¢!#)2h'09A'0@45)6'0n!#1©'09Y"eh$ ¼½ '0"¢1
g %(45) B&'0!gB&f'§s 7z 9Y451©'09ph$&9CB&9Ch)g '0"¢45$&h'¾&!g451©)gg»hB&!#7)1Dz$ ¿§$&1©!g$&z7)6"T!g%(B&$18'09 h1©'0%(Às v f%6!#n12'0¸'¼9 f745)gg$45gg'Á45u4e¾!g45187)gg»hB&!#)212451©451 B&'0Âr"T!g%(B&$18'09C E©9A!g% Ã opÄt 18!h$B&'09k"¢!#%6B&$&1©'09Y 45§'¾&!g451©)gg»hB&!#7)1D45"T"T'0f'09C4518!g9YnÂ§%(!gh1D"¢!#%(Bf'09Y }¥ff 'G"¢45ff'0@$&B!#)Å18!"T!g%(B&f7'G¾&!g451©)g§»©B&!g)1D45fgg!g9Y718%6zEh9k!g%Æ1©%('G1©!187%('0Â§45) ¸9Y1©$&45ff '0§'09 !#B&'09C4518)2g n18'0%Ç%($&n1D9k'0nB&!#)d18!¾!g4518)g§»©B&!g)1D'0|r"T'0B&1©!#)2 h$"T645 !g§'09C¾&!g}ys0{ B&45B&'09uB&9A'0h'0)218 41©$&18!g9C45f&!g)61©!#n'G45hB&'0"¢18 !#EF¾!#451©)g§» B&!g)1D1824513245§'G479k'0"¢13%6B&45"T1D!g)6'0h7gg)'09Cz!gED"¢!#%6B&$&1©'09u n18'0%65s x 1 '0gg7) }¥1© 45"TÈ§gg9A!g$&)@!g)6¾!g45187)gg»hB&!#)2139A'0B&9A'0h'0)1©45187!#)Å45)@9A!#$))gQ'09Y9A!g9 "T!g)1©)$&'0u}É184In"T$&nh!g)!#E1©' x©Ê{ÊÊ ¾!g45187)gg»hB&!#)21&h1©45)459A 45)6"T!g)"Tf7$&'0 }¥1©6)2$&%('09A!#$z'0|r45%(B&f'0 !#EF!g}Ë"¢!#%6B&$&1©'09 $&f'09Cz"¢45) '01©18'09ph$B&B&!#9C1 ¾!#451©)g§»©B&!g)1s op4518'0g§!g9Y'0 45)@qy$ ¼½ '0"T1 '0h"¢9YB18!g9YÀ0Ì Ã 9Y%(459 §Í oGs Î6Ïop!g%(B&$18'09pq h1©'0%( Ð 9kg§45)~5451©!#)2Ñ0e'0)2'¼9C45f — q r s S©XYf p S q b¢r s WYSFv5W sYq r§Â s Òs Ó6Ï Ã 9k!ggg9C45%(%6)g Ô 45)g§$&45g§'0hÑ Ã 9A!#"¢'0hh!g9Y — p bTdIe q utWYX sAÕ bTe5S q d qtÖ R¢S q bTr§Â§És l s Î6Ï¤×¥$&%('09C"T45f v )245f nhÑ e'0)'09C45f — p bTdIe5f§S©WYXuRTX q S©P5dIWYS qtpÕ WYXYXkb¢XuRTr§RTut` sYqwsYÕ rgf§dIWAX qtp RTu2R¢u b¢X q S©P5d s Ì©qy'¼"¢!#)2459 §Í s Øs l Ïqy!#Eh18}É459k' Ê )g§)'0'09Y)2ggÑh¢Ùc'0¿§$&9A'0%('0)1©5ÚyqrB&'0"¢Û&"¢4518!g) — utRTr0fgRkW s Â s Òs Ó Ã 9A!ggg9C45%(%()2g Ô 45)gg$45gg'0nÑ º !#9C%(45f '0Û&)1©!g) 45)@2'¼!g9 — s WYdIR¢rgS qtps Â s Ós l Ð B&'09Y451©)gQq h1©'0%(hÑh Ã 9A!g"T'0h ij45)45gg'0%6'¼)21 — s `5r p P5XkbTr qtÖ R¢S q bTrDs e'0)'09C45f&r'09C%(5 v fg§!g9Y1©%( § '0ngg) Ô 45)g§$&45gg'0 v 1©!#)245f¸Üz' ÉÝ !g9Au45)2 Ã 29Y45h'0À '0)2!#9C%(45f~5'06)2$&% '¼9 '0|À"¢'0B&187!#) ¾&!g451©)g§» B&!g)1 ¾!#451©)g§»©B&!g)1{n1845)459A gg9C45$&45f2$&)'09C¾&!g} gg$459kÞg§1 ×¥45× !g§'09C¾!#} 9A'0f4518¸'G'09Y9A!g9 9A!#$))gQ'09Y9A!g9 9A!#$))gQ%(!g' $&fB $&)'09C¾!#}ys ßtà µ¶gáFâ ã¥¹°µäá à å $&f7'09Y !#EF"T!g%(B&$&1©'09p h1©'0%(z!gE©1©'0)6)'0'0d)E©!g9Y%64518!g)64 !#$&1D¾!g4518)2gg»©B!#)1 459C18%6'018"ysh{'09A'z459A' !#}É'0§'09 9A'¼%6459YÈ§4 f E©'0}æh!g$&9A"T'0 !#E2'01©45f'0Q)E©!g9C%(45187!#) 4 !g$&1D1s Ð )2'G!#EF182'GE©'0} !#!gÈ§z!g)61©'Gn$ ¼½ '¼"¢1 º f!g451©)gg» Ã !g)1Dop!#%6B&$&1©4518!g) § Ã 451Dqy18'09 '0)~ f!g)gQ!#$13!gEDB9Y)1s02zB&45B'¼9p 41©$&1©!#9C45f&!g)61©!#n'G45hB&'0"¢18 !gE ¾!#451©)g§»©B&!g)1459Y1©%('01©" Ìh¾&!g451©)gg»hB&!#7)1¸'09A'045E©1©'09 Í 1©45145§'z4Á9A'0"T1"T!g))'0"¢18!g) 1©!ç n18'0%6 $f)gDs x 1D"T!g)hn18 !gED1©9A'¼'f!#!gh'0f "T!g))'0"¢18'0dB&459Y1©5s0{'GÛ&9Ch1 qy'¼"¢18!g) §è Ùc!g$&)7)g Ê 9Y9A!#9 é !#)ÅB&45gg' l5ê Ò h"¢$&hn'0z1©'G%(Bf"T451©!#)2z!gED$h)g E8E©'09A'¼)2139A!g$&))2gn189C451©'¼g§'0 E©!g9u1©' 45n"e!#B&'09C4518!g) !#EF451©!#) h$ 189C45"T1©!g) %($f18B&f7"T451©!#)Å45)@¸h7!#)s x 1D45fn!ç"¢!#)21845) 45"¢Ègg§9A!#$)@)Eh!#9C%(451©!#)Å!#)Å182'

ë C¼gì¢wu&Cu¢57r5k® 7¼ C5¢3í C¦ÁzC5¼î5¢CC7
yï7ð#C Cñ u7 ¢¡¢ò 1©}¥!%('01©!# !gED%('045n$&9C)gQ9k!g$&){)gQ'09Y9A!#9 ulps 45) relative error s0{' h'0"¢!#)dB&459Y1Dn"T$&nh'0 182' x©ÊÊ{Ê ¾!#451©)g§»©B&!g)1Dh1©45)459A }É"TÅ '0"T!g%()g 9C45B&f 45"T"T'0B&1©'0 § "¢!#%6%('09k"¢45fg459A}É459k' %645)$&Eh45"T1©$&9A'¼9C5s x )"¢f$&'0Q7)É1©' xhÊÊÊ h1©45)459A@ 1©'G9A!#$&)2)gQ%('01©!#dE©!g9 45h"e!#B&'09C45187!#)Às0'h"¢$&hn!#)Å!#EF182' h1©45)459A69Y45}É !g)182'Á%(451©'09Y45f¸)182'Áqr'0"¢18!g) yè Ù!g$&)2)g Ê 9C9k!g9 é !#)B&45g§' l5ê Òs 'e1©9A(B&459C1&n"T$&nh'0u1©'Á"¢!#)2)'0"T1©!#)2 '018}É'0'0)ç¾!g4518)2gg»hB&!#)21&45)618'e'0hg§) !gED¸459Y!g$& 45hB&'0"T1© !#EF"T!g%(B&$&1©'09p h1©'0%(5s0y!#B&"¢z)2"Tf$&{'G)h1©9Y$&"T1©!g)6h'01D'0ngg) !gB&18%6~5)gQ"T!g%(B&f'09Cz45)2@'0|À"¢'0B&187!#)Å45)f)gDs x 45¸'G189C'0@1©!ç45¸!g@%(45È§)gQh1©451©'¼%6'0)18 4 !g$&1D¾!#451©)g§»©B&!g)1D}¥718!g$&1D45fn! g§§)gQ9A'045h!g) }¥ 1©'Gh1©451©'¼%6'0)18 459A'G189Y$&' '¼nB&'0"T45ff n)"T'18' ½ $&n18Û"T451©!#)2 )¸!#f¸'G)!g182)gQ%(!g9k'G"¢!#%6B&f"T451©'0@1©45)6'0f'0%6'0)18459 "T45f"¢$&f$&Às0!gh' '0|ÀBf45)45187!#)u1©451&459A'e)!#1"T'0)1©9Y45f¸1©!y182'Á%(45)459Ag§$&%('0)145§' '0'0)gg9A!g$&B&'06)1©! 4n'¼"¢187!#)Å"T45ff'0 “ {' '01©45f é h!18245131©' "T45) 'GnÈgB&B&'0dED'0n9A'¼ s x ) B&459C18"¢$&f459 1©'GB&9A!g!#Ehz!gED%645) !#EF18'18'0!g9A'¼%6z45B&B'¼459p)Å187zn'¼"¢187!#)st'G'0) !gE3'045"T(B&9A!g!#EDñ%(459CÈ§'¼@}¥1©(182' % !#fÂg}É'0)(4B&9A!g!#EDñ)!g1{7)"Tf$'0 18' 45B&B&'0459Cz%6%('0451©'¼f Eh!#ff!g}¥7)gQ18'Gn18451©'0%('0)1D!gED1©'G1©'0!#9A'0%s ó áã à â ä àÁôöõ ¶¶#á¶ qy¿§$&'0'0~5)gy)2Û&)1©'0f %(45) 9k'045f§)$% '09C )1©!4Û)1©' )2$&% '¼9c!#E 1© 9A'0¿§$&9A'0 45) 45B&B&9A!#|r%(451©'G9A'0B&9A'¼n'0)1©4518!g)s v f1©!g$&ggÅ182'09k'G459A'G)Û)1©'0f %(45) )1©'0gg'09C ) %(!gh1DB&9A!#g§9C45%(z1©'G9A'0h$f1D!#EF)1©'¼g§'09p"T!g%(B&$&1©451©!#) "T45) 'Gn18!g9A'¼d)6Ò5Ø 18Às x ) "T!g)1©9Y45n1 g§¸'0)645) Û&|r'0@)$% '09p!gE 1© %(!gh1D"T45f"¢$&f451©!#)2z}É18Å9k'045f)$&% '09Y }¥f7f&B&9A!#$"T'G¿§$&45)2181©'0 18451D"¢45))!g1 'G'0|À45"¢18f 9A'0B&9A'¼n'0)1©'¼d$&h7)gQ18451D%(45) 1©5s02'¼9A'0E©!g9A'I1©'G9k'0n$&f13!gED4¾&!g451©)g§»©B&!g)13"¢45f"T$&f451©!g)6%($&n13!gE©18'0) 'I9A!g$&)'0 )6!g9A'09u1©!Û&1 45"¢ÈQ)1©!ç1© Û&)1©'G9A'0B&9A'¼n'0)1©4518!g)s0{z9A!g$&))2g'09C9A!#9p 18' "T2459Y45"T1©'09Y7h1©"ÁE©'0451©$&9A'I!gED¾!g4518)2gg»©B!#)1D"T!g%(B&$18451©!g)s è Ù'0f451©¸' Ê 9Y9A!#9p45)@÷IfB& é !g)6B&45g§' l5ê5ø '0h"¢9Y '0 !g}Ë1D %('045n$&9A'¼us qy)"T'G%6!#n13¾!g4518)2gg»hB&!#)213"¢45f"T$f4518!g) 45§'9k!g$&)7)gQ'¼9C9A!#9p45) }¥4 ¢ !g'¼ 1 %(451©18'09pEF18' 45h"e459Y718%6'018"e!#B'¼9C451©!#) )1©9A!#$"T'G4f7181©f' 1D%(!g9A'G9k!g$&)2)g '09Y9A!g9u1©45)Å)'0"T'0hn459 §ù {451D¿§$&'0h1©!g)6 4%(45)Å182'¼%6'G1829k!g$&g§!g$&1D182zn'0"T1©!#)s è e$459k g§18 é !g)6B&45g§' l5ê5ú h"¢$&hn'0Ff§RTXYvygg1© 4%('045) !#EF9A'¼{$&"T)2g1©' '09Y9A!g9u}É'0)6n$ 1©9Y45"¢18)2g1©}¥!)'0459 g )$&% '09YÀs0e$459kdgg718 }¥'09A'G"T!g)h7'09k'0 h$EûÛ"T'0)1©f %(B&!g9Y1©45)1 gçx å iü18451{) l5m5ø5ú 1{45'0Å4Gg§$&459AÅ{gg1{1©!É18'{!#$ f' B&9A'0"Tn!#)E©!g9Y%(451&)1©'Áq h1©'0%ÚyÒ ø Î459A"T218'0"¢18$9k'ÁÌhh)g§f'ÁB&9A'0"T7h!g)ç45f9A'045 4564 g§$&459A@g§1 Íh 45)@9A'01©9k!gÛ&1©1©'¼d45ff&'0|rh1©)gQ%(45"¢)'0 )61©'GÛ&'0fus0r}¥!'0|r45%(B&f'0 459A'Ggg¸'0)61©!ff$&n189C4518'G1©'G$18f71 !gEDg§$&459A@g§18Às

¢¡ký ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ' x©ÊÊÊ h1©45)459A@g§!#'0 E©$&9C182'09u1©45) ½
$&n139A'0¿§$&9C)gQ1©'G$&n'G!#EF4g§$&459A@g§1Cs x 1 g§§'0 45)645fg§!#9C1©%ÆEh!#9p451©!#) h$ 189C45"T1©!g) %($f18Bf"T451©!#) ¸h!g)645) h¿§$459k'e9A!#!g1 45)69A'¼¿§$9k'0p18451%(B&f7'¼%6'0)18451©!g)uB&9A!g$&"T'e1©'eh45%6'Á9A'0h$f1&45p18451 45fg§!#9C182%s02$& }¥'0)64yB&9k!gg§9Y45%Çz%6!#¸'0@E©9A!g%Æ!g)'G%(45"¢)'G1©!45)!#1©'09 1©' 9A'0h$&f1© !#EF18' 45h"e!gB&'09Y451©!#)2z}Éff 'G1©'Gn45%('G)Å'0§'09 Q 137E !g1©6%(45"¢)'0 h$B&B&!#9C1D182' x©ÊÊ{Ê h1©45)459A s02zg§9A'04518f h7%(B&fÛ'0z1©'GB&!g9Y1©)gQ!gEDB&9A!#g§9C45%(5s Ð 18'09p$&n'0z!gED1© B&9A'¼"¢h'GnB&'0"TÛ"T451©!g)6459A'Ggg¸'0)6) è5Ê |À45"¢18f Ùc!g$&)'0 Ð B&'09Y451©!#)2 é !g)6B&45g§' l5ú5þ s ÿ ¡ £¢¥¤¥¦¨§£©¡£¡¢¥§£©¡¦ñÿ ¡¢¥¤¥¦¨ qy'¼¸'09Y45fEûEh'¼9A'0)1D9A'0B&9A'¼n'0)18451©!g) !#EF9k'045f)$&% '09Y 45§' '0'0)6B&9A!#B&!gh'0 ¸ $&1 g E©459p1©'G%(!gh1D}¥{'¼f $h'0@ 1©'G¾!#451©)g§»©B&!g)1D9A'0B&9k'0n'¼)218451©!g)s º f!g451©)g§»©B&!g)1 9A'0B&9k'0n'0)18451©!g)z245§'G4 45h' β Ìh}¥"¢6 45f}¥4 45hh$%('0@1©! 'G'0§'0) Í 45)@4 B&9A'0"Tn!#)çes x E β l Îe45)2Qe Òe1©'0)É1©' )2$&% '¼9Îs l 9A'¼B9k'0n'¼)218'0(45 l s Î5Î × l Î "!#¢s x E β Ø45)Þe Ø5Ó 182'0)(18'I'0"T7%(45f2)$&% '09 Îs l "T45))2!#1 'I9A'¼B&9A'0h'0)218'0Þ'0|À45"¢18f $13 45B&B&9A!g|À%64518'0f Ql s l Î5Î l¢l Î5Î lTl Î5Î lTl Î5Î l¢l Î5Î l¢l Î l × Ø "!%$ys x )6g§'0)'09Y45f 4¾!#451©)g§» B&!g)12)$&% '09}¥ff ' 9A'0B&9A'0h'0)1©'0Q45 ± v '&v5v )( v × β0 }¥2'09k'Áv '&v5v )( ve "T45ff7'¼(182' sYq r q 1 p RTr§v "2e45)@45 eg§1©5s0ij!#9A'GB&9A'0"Tn'0f ± v 3&§v 4°v "5 … v "687%4 × β0 9A'0B&9A'0h'0)1© 1©'G)$&% '09 Ì l5Í '18'09C% @9pbTRTS q r0©e5b q rgSFrgf§dIWYXu}¥f7f 'G$h'0@1©!%('045)649A'045f&)$% '09p1©451D"T45) ' '0|À45"¢18f 9A'0B&9A'0h'0)1©'0@)61©'GEh!#9C%(451D$&)'09pn"T$&nh!g)s0y}¥!!#1©'09pB&459Y45%6'018'09C 45hn!#"¢4518'0d}¥1©6¾!g4518)g§»©B&!g)1D9A'0B&9A'¼n'0)1©4518!g) 459k'G1©'Gf459Ag§'0h1D45)@n%(45ff'0n1 45ff!g}¥4 f7'z'0|rB&!g)'0)1© W BADC#Ep45)(W FAHGI5snqr)2"T' 182'09k' 459A' β 6 B&!#nh f7'zng§)Û&"¢45) 45)2 W BADC#Ee»FW BAHGIQP l B&!ghn f'G'0|rB&!#)2'0)18 4¾!g4518)g§»©B&!g)1D)$% '09p"T45) 'G'0)"¢!#'0@7) R S T U V W X Y` a b c b d e ` f f ` X f ` a ` g dV d hb g a V f ` i p q r st u v wq v x V g y v su t wp u q sr x V d YV V g y b f g ` f X R V f d YV g y ` f V g y Y` U b X b Yb a R S S e ha d ` f W V a hg df b y d ` y e f g b f a f d e ` V g y b Y` f R h i V g y e V a j ` g ` f V YYf f ` X YV d ` y de ` b Yy ` f d` f W q t r sv v q S d 0 d 1 β 1 – … d p 1 – β p 1 – ( ) – + + +     βe 0 d i β < ≤ ( ) , ± log 2 e max e min – 1 + ( ) log 2 βp ( ) 1 + +

yï7ð#C Cñ u7 ¢¡ "k 1© }É'09k' 1©' Û&)45f lP l °E©!g9F1©' hg§) 1sh{' B&9k'0"¢h' '0)"T!g)gy°)!g1%6B&!#9C1845)21 E©!g9p)!#}¥s '09A' 459A'Á18}É!9A'¼45n!#)2°}É 49A'¼45f§)$&% '09%(g§12)!g1 'Á'0|À45"¢18f 9A'¼B&9A'0h'0)2184 f7' 45 4¾!g4518)g§»©B&!g)1D)$% '09Cs0{'G%(!gh1D"¢!#%(%6!#)Åh1©$&451©!#)Å ff$&n189C4518'0 § 18' '0"T%645f&)$% '09pÎs l s v f182!#$ggÅ13245z4Û)1©'G'0"T%645f&9A'0B&9A'¼n'0)18451©!g) ) )459 1245 45)É)2Û&)1©' 9A'0B&'04518)2g9A'0B&9A'0h'0)1©45187!#)sT$& }É'0) β Ø 1©' )2$&% '09Îs l f7'0 h1©9Y7"T1©f '018}É'0'¼)Å18}É!¾&!g451©)g§»©B&!g)1D)$&% '09Y 45)@ '0|À45"¢18f 9A'0B&9A'0h'0)1©4 f' § )'01©'09 !gE{1©'0%s v f7'¼nñ"T!g%(%(!g)(h1©$&451©!g)Qñ1©451{49A'045f)2$&% '09 7ñ!#$&13!#E39C45)gg' 1©451D 1© 4 h!gf$&1©'G§45f7$&'G f459Agg'09p1845) β × β0 ADC#E !g9un%(45ff'09p1©45) l s Î × β0 AHGI s ij!#n13!gED1© B&45B&'09ph"¢$&hn'0znh$'¼ $&'18!1©'GÛ&9Ch1D9A'045h!g)s 8m¥!g}¥'0¸'09 )$&% '09Y 1©451D459k'G!g$&1D!gED9C45)gg'}¥ff 'Gh"¢$&nh'0@) è5x )2Û&)1 §é !#)ÅB&45gg' l5m5m 45) è5 '0)!g9C%(45f~5'0@×¥$% '09C é !g)6B&45g§' Ø5Î5Øs º f!#451©)g§»©B&!g)1D9A'¼B9k'0n'¼)218451©!g)z459A'G)!g1D)'0"T'0nh459Y7f $)¿§$&'§s º !g9u'0|r45%(B&f' § !#1© Îs Î l × l Î 45) l s Î5Î × l Î "!n9k'0B&9A'0h'0)1{Îs l s x E{182'ef'045)gçgg71p)!g)~5'09A!¥Ì©v 3 ≠ ÎG) '0¿§$&451©!#)ÅÌ l5Í 4 !g§' Í© 1©'0)Å18'G9A'0B&9A'0h'0)1©45187!#)Åzn45@1©! 'Grgb¢XYdIRTu qtÖ WYvs02' ¾!#451©)g§»©B&!g)1D)$&% '09 l s Î5Î × l Î "!n° )!g9Y%645f~5'0 }¥2f'GÎs Î l × l Î ° )!g1Cs Ý '0) β Ø e Ò W BAHGI » l 45)@W BADCoE Øy18'09A'G459A' l5ø )!g9Y%645f~5'0@¾!#451©)g§»©B&!g)1 )$&% '09Y 45 h!g}¥)ç) º g§$&9A' » l sT' !#f(45hç%(459CÈg "T!g9Y9A'0hB&!g)Q1©!)$&% '09Y }¥2!#n'Ghg§)Û"T45)@7 l s Î5Îs0Ùc'0¿§$&9C)gQ1©451D4¾!#451©)g§»©B&!g)1D9A'¼B9k'0n'¼)218451©!g) ' )!g9Y%645f~5'0@%(45È§'0z1©'G9A'0B&9A'0h'0)1©4518!g)6$)¿§$&'§s0÷I)Eh!#9C18$)4518'0f ¢ 1© 9k'0n189C"T1©!g) %(45È§'0z1D%6B&!#nh f'G18!9A'0B&9A'0h'0)1D~5'09k! qp v )451©$&9C45f&}¥4 1©!ç9A'0B&9A'0h'0)1DÎ }¥1© l s Î × β0 AHGI 7r4 h)"¢'G182zB&9A'0h'09C§'0 182'GE©45"T1D1©451D182'G)$&%6'¼9C"T45f!#9A'09C)gQ!#E )!g))'0g§4518¸'G9A'045f&)$&% '09Y "T!g9Y9A'0hB!#) 1©!ç1©'Gf'0|r"T!gg§9Y45B&7"Á!g9k{'¼9C)gQ!gED1©'09 ¾!#451©)g§»©B&!g)1D9A'0B&9k'0n'0)18451©!g)5s Ý '0)Å182'G'0|ÀB&!g)'0)1D h1©!#9A'0@)Å4 ts 1DÛ&'0f 1©4513%('045)2z1824513!g)f Ø 8u» l §45f$'0z459k'G45¸45f4 f'GE©!g9u$&n'I45 '0|ÀB&!g)'0)1© h)2"T'G!#)2' %($h1 'G9A'0h'09C§'0@1©!9k'0B&9A'0h'0)1DÎs ×¥!g18'182451182' × 7)4G¾!#451©)g§»©B&!g)1)2$&% '09°7pB&459Y1!gE{182'e)!g1©4518!g) 45)ÅEûE©'09A'0)1 E©9A!g% 4¾!#451©)g§»©B&!g)13%($&f1©B&f !#B'¼9C451©!#)s¼{'I%('045))2gç!gE31©' × % !#f2n!g$&f 'G"¢f'0459uEh9k!g%Ç18'G"¢!#)218'0|r1Cs º !g9u'0|r45%(B&f' 18'G'0|rB&9A'0hh7!#)ÅÌ©Øs þ × l Î "!rv Í × Ì©Ós Î × l Î "2 Í )¸!#f¸'0z!g)f 4n)g§f'G¾!#451©)g§»©B&!g)1D%($&f718B&f7"T451©!#)s R S e ha V a a W ` a d e ` a V Y V f f V g j ` W ` g d e ` f ` d e ` ` U X b g ` g d ha a d b f ` y d b d e ` Y` c d b c d e ` a hj g hw d V g y S

¢¡ "x ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ î5 y5©¡uï7 zy|{~}#r%"D)}#| β
« " ò " " « ¢¡D £¤¥¦¨§¤£¥¡¦§¨¢¥¤¥©¡©@ª« ¡ qy)"T'G9A!g$&))gQ'09C9k!g9pz)2'09A'¼)213)Å¾!#451©)g§»©B&!g)1D"T!g%(B&$18451©!#) 1Dz7%(B&!g9Y1©45)1 1©!ç245§'G4}É4 18!%('045n$&9A'G187z'09C9k!g9Cs0op!g)h'09p1©'G¾!#451©)g§»©B&!g)1DE©!g9Y%64513}É1© β l Î45)@e Ò }¥"¢6}¥7ff 'G$h'0@1©9A!#$gg!g$&1D1© h'0"T1©!g)s x ED1©'G9A'0h$&f1D!gED4 ¾!#451©)g§»©B&!g)1D"T!g%(B&$18451©!g)6 Òsl Ø × l Î "!%2 45)2@182'G45)h}É'09u}É'0)6"¢!#%6B&$&1©'¼d18! )Û)1©'GB&9A'¼"¢h!g)67s Î5Ò l Ó 1D "Tf'0459p1©451D187z )6'09C9k!g9 g Ø$&)218 )61©'Gf45n1 B&f45"¢'gsTqy%(f459Cf ¢ E&1©'Á9A'045fg)2$&% '¼9Ás Î5Ò l Ó l5þ5m 9A'¼B9k'0n'¼)218'0(45 Òs l Ó × l Î "!%2 182'¼)ç71 )6'09C9k!g9 g s l5þ5m $&)1© )61©'Gf45n13B&f745"T'§s x )6g§'0)'09Y45f ED1©'G¾!#451©)g§»©B&!g)1 )$&% '09pv '&v … v × β0 $&h'0d18!9A'0B&9k'0n'0)1 ÖT 1©'0)61D )6'09C9k!g9 g  v '&v … v» Ì Ö Ú β0 Í β 6¬7r4 $&)218z)61©'If745h13B&f45"¢'§s ¨2Á2'I1©'09Y% ulps }Éff 'G$&h'0@45zh!g9C1845)2 E©!g9 è $&)1©z)Å182'Gf45h1DB&f45"¢'§s éx EF1©'G9A'¼n$&f1D!gED4"¢45f"T$f4518!g)6 1©'G¾!#451©)g§»©B&!g)1 )$&% '09p)'0459A'¼n1D18!1©'G"T!g9Y9A'0"T1D9A'0h$&f71 1Dn18ff%(g§1 'G)Å'09Y9A!#9 g 45 %($&"¢645ys þ ulp s v )!g182'¼9p}É4 1©!%('045h$9k'G1©'G7EûE©'09A'0)"T' '018}É'0'0)64¾!g4518)g§»©B&!g)1D)$% '09 45)@1©'G9A'045f&)$% '09p1D 45B&B&9A!#|r%(451©)gQ XkWAutR¢S q V5WuWYXYXkbTX }¥7"T6 h7%(B&f 182' E8E©'09A'¼)2"T' '01©}¥'0'0)61©'G18}É!ç)2$&% '¼9Cz7§'0 § 1©'G9k'045f&)2$&% '09Cs º !g9u'0|r45%(B&f' 1©'G9A'¼f74518¸'G'09Y9A!g9u"¢!#%6%(1©18'0@}É'0)645B&B&9A!g|À%64518)2gÒs l Ó l5þ5myg Òs l Ó × l Î ®° s Î5Î l5þ5m ÚyÒs l Ó l5þ5m ≈ s Î5Î5Î þ s r!"T!g%(B&$18'G1©'G9A'0f45187§'G'09C9k!g9p18451D"¢!#9C9k'0nB&!#)2z1©!wsþ ulp ! h'09C§'G1©451D}¥2'¼)Å4 9A'045f&)$&% '09pz45B&B&9A!g|À%64518'0 g 18'"Tf!gh'0h1DB&!ghn f'G¾!g4518)2gg»©B!#)1D)$% '09 v '&v5v '&¯&¯&v5v × β° 182'G'09Y9A!#9p"¢45) 'G45 f459Agg'G45 Îs Î5Îsss Î5Î β′ × β° }¥'09A' β Ä2z1©'G7gg1 β ÚrØ 18'09A'G459A'Ge$&)218 )61©'Gngg)2Û&"¢45)@!gED1©'G¾!g4518)g§»©B&!g)1D)$% '09 45)2@e $&)718 !#EFÎ)Å18'Gng§)Û&"¢45)@!gED1©'G'09C9k!g9Cs0{z'09C9A!#9p Ì©Ì β ÚyØ Í β 76 Í × β0 stqr)"¢' R S ± g Y` a a d e ` g W e ` f ² ha YV f j ` f de V g β ³´ µ ¶ · R b f a W V YY` f d e V g β ³´ ¸¹ S º W e ` f a e hd e V f ` b d b c f V g j ` hg d e ha c V a e hb g hYY g b d e ` d b g a hy ` f ` y g d hY c f de ` f g b d hd ` S S » ` d ¼ e ` de ` ½ b V d hg j ¾ X b hg d g W e ` f de V d V X X f b U hW V d ` a S e ` g  ¿ … ¾ À² Á β ³Â β Ã ÄÅ ha ` Æ hÇ V Y` g d db  ¼¾ / YX À ¼Â S À ` ` È É sÊ q w Ë p Ì É r q r sp t Í É s c b f d e ` y ` w g hd hb g b c YX À Â Â S W b f ` V d d f V d ` c b f W YV c b f W ` V a f hg j ` f f b f ha  ¼¾ / YX À Â S ¾ ¾ T y S 0 1 2 3 4 5 6 7

yï7ð#C Cñ u7 ¢¡¢¡ )$&% '09Y !gED1©'GE©!g9Y%Çv '&v5v … v5v × β0 45ff45§'G1©'Gn45%('G4 n!#f7$&18''¼9C9A!#9 ¸ $&1D45¸' ¸45f$&'0z1©451D9Y45)2gg' '018}É'¼'0) β0 45) β × β0 1©'G9A'0f451©§'G'09C9A!#9p9Y45)g§'0 '018}É'0'0) Ì©Ì β ÚyØ Í β 76 Í × β0 Ú β0 45)@Ì©Ì β ÚrØ Í β 76 Í × β0 Ú β0£Î 4 st451D ÌhØ Í x )6B&459C18"¢$&f459 1©'G9A'0f451©§'G'09C9k!g9p"T!g9Y9A'0hB&!g))2g1©!wsþ $&fB@"T45)6¸459 Q§ 4yE©45"T1©!#9p!gE β s02zEh45"T1©!#9p "T45ff'0@1©'GzbTutW . qy'0181©)g ε Ì β ÚyØ Í β 76 1©!182'Gf459Agg'0h1D!gED1©' !g$&) )ÅÌ©Ø Í 4 !#¸' }¥'G"¢45)6n4 1©451D}¥2'¼)Å49A'045f&)$&% '09pz9A!g$&)'0@1©!182' "Tf!gh'0n1¾!#451©)g§»©B&!g)1)$&% '09 1©'e9A'0f451©§''09Y9A!#9°p45f}É4 !g$&)'0 g ε }É"T 9k'0Eh'¼9C9A'0@18!45 dIR p P q rgWuWYe sYq utbTrDs x )18'e'0|À45%6B&f'e4 !#¸' 18'e9A'0f45187§'e'09Y9A!#9°}¥45Is Î5Î l5þ5m ÚyÒs l Ó l5þ5m ≈ s Î5Î5Î þ s x )ç!g9A'09c1©! 45¸!#@n$&"¢6h%645ff&)$% '09C 1©'G9A'0f4518¸'G'09Y9A!g9u7z)!g9Y%645ff }¥9C181©'0)645 4E©45"¢18!g9 1©%('0 ε }¥"¢É)ç18 "T45n' ε Ì β ÚrØ Í β 76 þ Ì l Î Í !rv s Î5Î þ sn{$& 182' 9k'0f451©¸' '09Y9A!#9 }¥!g$&f 'G'0|rB&9k'0nh'0@45 Ìs Î5Î l5þ5m ÚyÒs l Ó l5þ5m5Í Ús Î5Î þ5Í ε ≈ Îs l ε s r!ff$&n189C451©'G18'EûEh'¼9A'0)"¢' '01©}¥'0'0)6$&f7B&z45)d9A'¼f451©¸'G'09Y9A!#9 "T!g)h{'¼9p1©'G9A'045f )$&% '09 Ï l Øs Ò þ s x 13 45B&B&9A!g|À%64518'0 § l s Ø5Ó × l Î ¢s0''¼9C9A!#9p Îs þ ulps 1©'9A'0f451©§'I'09Y9A!g9 ñÎs Ð ε s¼×¥'0|À13"T!g)h7'09 182'"¢!#%6B&$&1©4518!g) ú s¼'I'0|À45"¢1{¸45f$&'I ú Ï m5ú s ú5 }Éf'G1©'G"T!g%(B&$18'0@¸45f$&'G7 ú m s m Ø × l Î Tst'G'09C9k!g9pz)2!#} Ós Î ulps § $1D18'G9A'0f451©§'G'09C9A!#9p h1©ff&Îs Ð ε s02'G'09Y9A!#9p%('045n$&9A'0@) ulps ú 18%6'¼ f459Agg'09 '¼¸'0)61©!g$&g§61©'G9A'¼f74518¸'G'09Y9A!g9u 1©'Gh45%6'gs x )Ågg'0)'09C45f }É'0)61©' 45n'G β 4Û|À'0d9k'0f451©§''¼9C9A!#9p'0|ÀB9k'0nh'0@) ulps "T45)Å}¥! g f' § 4E©45"T1©!g9u!gED$BÞ1©! β s v )("¢!#)2§'09Yn'0f T 45 '0¿§$4518!g)çÌhØ Í 4 !#¸'Áh!g}¥ 4Û|r'¼6'09Y9A!g9c!#E s þ ulps 9k'0n$&f1© ) 49A'0f45187§'G'09C9k!g9p18451D"¢45)6}¥! g f' g β s '%(!#n1D)4518$9Y45f}¥4 18!%('045n$&9A'G9A!#$&)2)gQ'09Y9A!g9u ) ulps s º !g9p'¼|r45%(B&f7' 9A!#$))gQ1©!ç1©'G)'0459A'0h1D¾!g4518)g§»©B&!g)1D)$% '09p"T!g9Y9A'0hB&!g) 1©!ç45)Å'09Y9A!#9p!gEDf'0n 1©45)6!g9u'0¿§$45f&1©!ds þ ulp s 8m¥!g}¥'0§'09 }¥'0)645)245f ~5)2g1©'G9A!g$&))2g'09C9k!g9p"T45$&n'0 g ¸459Y!g$& E©!g9Y%($f45 9A'0f451©§'G'09C9k!g9pz4 '0181©'09p%('045h$9k'§s v g§!g!#@7ff$&h1©9C4518!g)6!gE 1©pp1©'45)45f np!g)QB&45gg' Ø5Ø ø sqr)"¢' ε "T45)Q!g§'09A'0h1©%(451©'e1©''0EûE©'0"¢1!gE{9k!g$&))2g 1©!ç1©'G)'0459A'¼n13¾!g45187)gg»hB&!#)213)$% '09 g 18'G}É! § f'GE©45"¢18!g9u!gE β '09Y9A!#9p'0h1©%(451©'0 !gEDE©!g9Y%6$&f45 }¥ff 'G1©gg218'09p!#)Å%(45"T2)'0z}É1©64n%(45ff β s Ý '0)(!#)f 18'I!#9A'09 !#E3%(45g§)1©$&'I!#E39A!#$&)2)gç'09Y9A!g9 ñ!#E3)1©'09k'0n1 ulps 45) ε %(4 Q 'G$h'0@)218'09A"T45)2gg'04 f T n)"T'G1©' E8E©'09 g 4513%6!#n1D4E©45"T1©!g9u!gE β s º !g9 '0|À45%6B&f' }¥2'0)64¾!g4518)g§»©B&!g)1D)$% '09p )6'09Y9A!g9 § r ulps 18451D%('045)2z1©451 1©'G)$&% '09p!#EF"T!g)1845%6)451©'0@g§18 f!#g β rDs x EF182'G9k'0f451©¸'G'09Y9A!#9p)Å4 "T!g%(B&$18451©!g)6 r ε 1©'0) "T!g)1©45%()451©'0@g§1© ≈ f!gg β r3s ÌhÒ Í 1 2 -- -β p – 1 2 -- -ulp β 2 -- -β p – ≤ ≤ x ˜ x ˜ x ˜

¢¡ "Ñ ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ Ò«Ó ¤¥D©ÕÔ§¡§¤¦¨ Ð )'G%('01©!#d!#EF"T!g%(B&$18)gQ1©'GE8E©'09A'0)"T'
'018}É'¼'0)Å18}É!¾&!g451©)gg»hB&!#7)1D)$&% '09Y 18!"T!g%(B&$18'G1©'G7EûE©'09A'0)"T''¼|r45"T1©f 45)@1©'0)Å9k!g$&)d1D18!1©'G)'0459A'0h1D¾!#451©)g§» B&!g)1D)$&% '09Cs02z §'09 '0|ÀB&'0)n§'G7ED1©'G!gB&'09Y45) E8E©'09pgg9A'04518f )Åh~5'§s v hn$&%()gQe Ò Øs l5þ × l Î 2e» l s Ø þ × l Î "!rÖe}¥!g$&f 'G"T45f"¢$&f451©'0@45 Ï Øs l5þ × l Î 2 ` s Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î5Î l Ø þ × l Î 2 Ï»F` Øs l Ó m5m5m5m5m5m5m5m5m5m5m5m5m5m5ú5ê5þ × l Î 2 }¥2"T69A!g$&) 1©!çØs l5þ × l Î ×2ystÙ451©'09p1845)Å$&n)gQ45ff&1©'0h'Gg§1© ¾!g4518)2gg»hB&!#)21 459A}¥459A'G)!g9Y%645ff !#B&'09C4518'0 !g)64Û|À'0@)2$&% '¼9p!gEDg§1©5s0qy$&B&B&!gh'G1©451D1©' )$&% '09u!gEDg§18 Èg'0B&1Dze 45)@1824513}¥2'0)618'Gh%645ff'09u!gB&'09Y45)@ hEh18'0@9Cgg1 g§18 459A'Gh%6B&f h"¢459k'0dÌ©45 !#B&B&!gh'0d18!9A!#$))g Í s0{'0) Øs l5þ × l Î 2u» l s Ø þ × l Î "!%Ö '0"¢!#%6'¼ Ï Øs l5þ × l Î 2 ` Îs Î5Î × l Î 2 Ï»F` Øs l5þ × l Î 2 'G45)h}É'09uñ'0|r45"T1©f 1©'In45%('I45zED182'IE8E©'09A'¼)2"T'I45 '0'0)6"T!g%(B&$&1©'0Þ'0|r45"T1©f 45)@1©'0)Å9k!g$&)'0 str45Èg'G45)2!#1©'09p'0|À45%(Bf'§ l Îs l » m s m Òst '0"T!g%('0 Ï l s Î l × l Î ` Îs m5m × l Î Ï»F` s Î5Ø × l Î '"T!g9Y9A'0"T1D45)h}É'09u7s l5ê5 n!18'G"¢!#%6B&$&1©'0@E8E©'09A'0)"T'G !#E8E § Ò5Î ulps 45)@7 }¥9A!g)gQ)6'0¸'¼9 g§1 Øp¨m¥!g} 45@"¢45)61©'G'09Y9A!g9 ' ù ÙÚÜÛ¨Ý)Þ8Û¨ßáà â sYq r06R ã9ub¢RTS q r0©e5b q r§S 8äTb¢XYdR¢SD q ShPe5RTXYRTdIWYS©WYX s β R¢rgvye Õ RTr§v p bTdIe5fgS q r¼6v q äåäTWYXkWYr p W s f sAq r06ev q q S sYÕ ShP5WuXkWYuwR¢S q V5WuWYXYXkbTXub æäÁS©P5WuXkW s f§utS p RTrQWuR s utRTXWuR s β çs èÞ8Ý)Ýné v 9A'0f4518¸'G'09Y9A!g9u!gE β » l )Å182'G'0|ÀB&9A'0hn!#) êÏ»D`y!#"¢"T$&9Cz}É'0) ëÏ l s Î5Î … Î45)2 ` .ρρ…ρ }É'09A' ρ β » l s 8mÉ'09k'G`y45zeygg1© Ì©45ff&'0¿§$45f&18! ρ Í st'G'0|r45"T1 E8E©'09A'¼)2"T'I QÏ»D` β ! 6 s ¬m¥!g}¥'0¸'¼9 }É'0)("T!g%(B&$18)2gç1©'I45)h}É'09 $&n)g!#)f e

yï7ð#C Cñ u7 ¢¡ "ì g§18 1©'G9Y7gg1©%(!gh1Dg§1D!#EF`g§'018 hEh18'0d!#E8E 45)dh!182'G"T!g%(B&$&1©'0 E8E©'09A'¼)2"T'G β ! 68í ¢s0{$& 18''¼9C9A!#9p β ! 6 » β ! 68í β ! 6 Ì β » l5Í© 45)2@1©'G9A'¼f74518¸' '09Y9A!g9u β ! 6 Ì β » l5Í Ú β ! 6 β » l s Ý '0) β Ø 1©' 9A'0f4518¸' '09Y9A!g9F"¢45) 'Á45°f459Ag§' 45°1©' 9A'0h$&f1 45)(}¥'0) β l Î 12"¢45) ' m 18%6'0°f459Agg'09Cs Ð 9F18!B&$&12145)!g1©'09F}¥4 ¢ }¥'0) β Ø '0¿§$&451©!g)ÉÌ©Ò Í 4 !g¸'zn!#}É 1©45121©' )2$&% '¼9c!#E&"¢!#)1©45%()24518'0(g§18u f!#g 2 Ì l Ú ε Í f!gg 2 ÌhØ 6 Í esn{4512 45ff§!gE 1©'Geg§1©z)Å18'9k'0n$&f1D459A'G}¥9A!#)g pÀqy$&B&B&!gh'G1©451D!#)2'G'0|À1©9Y4{gg1D 45'0@1©! g§$&459A@45gg457)h1D1©zn1©$&45187!#)ÅÌ©4zf§RTXYvv q q S Í st451D 18'Gn%(45ff'09p)$&% '09p 1©9A$)"T451©'0Þ1©!çe P l gg1© 45)@18'0)6182'I9k'0n$&f13!gE31©'Gh$ 189C45"T1©!#)6z9A!#$&)2'0Þ1©!çe g§18Às Ý 18Å4gg$459A@g§1 1©'GB&9A'0¸!#$z'0|r45%(B&f' '¼"¢!#%6'0 Ï l s Î l 0 × l Î ` Îs m5m Ò × l Î Ï» s Î l5ê × l Î 45)@1©'G45)2h}¥'09p '0|À45"¢1Cs Ý 1©64n)ggf7'Ggg$459kdgg1 1©'G9A'0f45187§'G'09C9k!g9p!#EF18' 9A'0h$&f1D%(4 Q 'Ggg9A'04518'09p1©45) ε 45 ) l¢l Î» ú s þ5m s Ï l s l 0× l Î "2 ` s Î ú5þ × l Î "2 Ï»F` l s Î l5þ × l Î "2 7z9A!#$) 18! l Î5Ø "¢!#%6B&459k'0d}¥1©61©'G"T!g9Y9A'0"T1D45)n}¥'09p!#E l Î l s Ó l5 E©!g9p49A'0f4518¸' '09Y9A!g9 !#Eps Î5Î ø5 }É"T(ñg§9A'04518'09 1©45) ε s Î5Î þ s x )(g§'0)'09Y45f 1©'9A'0f451©§'I'09Y9A!g9 !gE318' 9A'0h$&f1D"¢45) 'G!g)f hf7gg1©f f459Agg'09p1©45) ε s0ij!#9A'GB&9A'0"Tn'0f T ÙÚÜÛ¨Ý)Þ8Û¨ßáî ïä|ÏR¢rgvy`RTXkW D9pbTR¢S q r¼he5b q r§SDr§fgdIWYX suq rQR ðäTb¢XYdR¢SD q ShPe5RTXYRTdIWYS©WYX s β R¢rgvye Õ RTr§v q ä s fg¼S©XYR p S q b¢r qts v5b¢rgWu q ShPe tñ¢çv q q S s óò©q &W ô&§b¢rgW{f§RTXYvv q q S öõ Õ S©P5WYrQS©P5WuXkWYutRTS q V5W XkbTf§r§v q r06WYXYXkb¢X q rQShP5WuXkW s fgutS qts utW sYs S©P5R¢r ø÷ ε s 7z1©'0!#9A'0%Ç}¥ff 'GB&9A!#¸'0)6) è Ù!g$&){)g Ê 9Y9A!g9 é !g)6B&45g§' Ø5Ø þ s v 1©!#)Å )"¢f$&'0@)Å182'G4 !g§'G1©'0!g9k'0%Çh)"¢' ùÏ45)d`"T45) 'GB&!gh1©§'G!g9p)'0gg451©¸'§s ú ¤H©üû¡D £ £¤¥¦"§£¢H© 'f45h1Dh'0"¢18!g)6"¢45) 'Gn$&%(%6459Y~5'0 g h4 )gQ1824513}É182!#$134ygg$&459A@{gg1 1©' 9A'0f4518¸'G'09Y9A!g9u"¢!#%6%(1©18'0@}É'0)6n$ 1©9Y45"¢187)gQ18}É!ç)2'¼459 g ¿§$&45)2181©'0z"¢45) 'G§'09 f459Agg'§s x )¥!g1©'09F}¥!g9A 18' '0§45f$4518!g)É!#E245) '0|ÀB&9A'0hn!#)É"¢!#)1©45))2g4Án$ 1©9Y45"¢187!#) Ì©!g9F45)É451©!g)¥!gE2¿§$&45)2181©'0°}¥1©É!gB&B&!#n18' h7gg) Í "T!g$&fQ9A'0h$&f12)É4Á9A'0f451©§' '09Y9A!g9

"Ñ"ý ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ h!f459Ag§'G18451DR¢uwu21©'G7gg1©z459A'G%6'¼45)2)g§f'0h Ì©'0!g9A'¼% l5Í s Ý
2'0)6h$ 1©9Y45"T1©)g )'0459 g ¿§$&45)1©1©'0 1©'e%6!#n1&ng§)Û&"¢45)1g§18p)Q18'!#B&'09C45)p%(451©"TQ45)6"¢45)"T'0f '045"TÅ!#1©'09Cs0{'09k'G459A'G1©}¥!È§) !#EF"T45)"¢'0ff45187!#)t"T451©45h1©9k!gB&"e45) '0)gg)cs acRTS©R s S©XkbTe5P qtpñp RTr p WAututRTS q bTrQ!#"¢"T$&9C }¥'0)61©'G!gB&'09Y45)2z459A'Gn$ ¼½ '¼"¢1D18!9A!#$))g '09Y9A!g9YÀs º !g9p'0|À45%(Bf'G)61©'G¿§$&45{9Y451©"ÁEh!#9C%($&f4 18''¼|rB&9A'0hn!#)Å F2e»DÓÀR p !#"¢"T$&9C5s '¿§$&45)1©18'0 F2e45)dÓ5R p 459A'Gh$ ¼½ '0"¢131©!9k!g$&)2)gQ'09Y9A!#9C h)"¢'G182' 459A'G1©' 9A'0h$&f1© !#EF¾&!g451©)g§»©B&!g)1D%($&f1©B&f"¢4518!g)Às0qr$B&B&!#n'G1©451D18' 459A'G9A!#$&)2'0@1©!ç1©' )'0459A'0h1&¾!g4518)g§»©B&!g)1&)2$&% '¼9 45)6h!y459A'Á45"¢"T$&9C4518'e18!¥}¥1©) þ&¯ÿ ulp s Ý 2'0)ç1©' 459A'eh$ 189C45"T1©'0 "T45)"¢'0ff4518!g)"¢45)"T45$h'e%(45) !gE182'e45"T"T$9Y451©'egg718p18!¥h45B&B'¼459 f'045§7)g '0)d%(45)f g§1©z"¢!#)1©45%()24518'0 § 9A!g$&))2g'09C9A!#9Cs ~m¥'0)"T'G1©' E8E©'09A'¼)2"T'G%(7gg1D45¸'G45)6'09C9A!#9p!#EF%(45) ulps s º !g9u'0|r45%(B&f' "¢!#)2h'09p Òs Ò5Ó R l s Ø5Ø 45) p Øs Ø ú s0{'G'0|À45"¢13¸45f$&'G!gED B2e»DÓ5R p ys Î5Ø m Øs å $&1D B2Á9A!g$&) 1©! l¢l s Ø 45)@Ó5R p 9k!g$&)2z1©! lTl s l5 2'¼)2"T'G1©'GÛ)45f&45)n}¥'09ps l }¥"¢6 45)6'09C9k!g9 gQê Î ulps '0¸'¼)Å182!#$gg lTl s Ø» lTl s l 7z'0|r45"T1©f '¼¿§$45f&1©!ds l ¢s02'Gh$ 189C45"T1©!#)Å@)2!#1 )1©9A!#$"T'G45) '09Y9A!g9 § $139C451©'09u'0|rB&!gh'0@1©'G'09Y9A!g9u7)189A!g$&"T'0d)61©'G'0459Yf7'09 %($f18B&f7"T451©!#)25s WYr q r p R¢r p WYutuwR¢S q bTr!g"T"¢$&9Cc}É'0)Éh$ 189C45"T1©)g'0|r45"T1©f Èg)2!#}É)¥¿§$45)18718'0Às x E ðÏe45)` 45¸'G)!9A!#$&)2)gQ'09Y9A!#9 182'¼) g '0!g9A'0%ÆØEF182'Gh$ 189C45"T1©!g)6 !#)2'G}¥1©64 g§$&459A@g§1 18'G{EûE©'09A'0)"¢' ùÏ5» 45 4§'09 n%(45ff9k'0f451©§''¼9C9A!#9pÌ©f'0nz1©45)6Ø ε Í s v E©!g9Y%($f4184513'0|À2 1©ñ"T451©45h1©9A!#B&2" "T45)2"T'0ff451©!#)6"T45)(h!g%('01©%('0 'I9k'0459C9Y45)2gg'0 1©!ç'0f%6)451©'G182'GB&9k! f'0% s v gg45)Å"T!g)h{'¼9p1©'G¿§$&459C451©"ÁEh!#9C%($&f4 ÌhÓ Í Ý '0) 18'0) !g'0z)2!#1D)¸!#f¸'G4"¢45)"T'0ff451©!g)645) s å $&1&1©'Á!g1©'09c451©!#)Ì©n$ 1©9Y45"¢18!g) Í )!#)'e!#E1©'ÁE©!g9Y%6$&f45u}¥f7f 45¸'G4"T451©45h1©9A!#B&2"Á"¢45)"T'0ff451©!g)s0r!45§!g@1© %($f18B&f 1©'G)$%('09Y451©!g9u45)2 '0)!g%()451©!g9u!gEDX § R S ¡ ¡ i g b d ¡ S hg d ` SR ¾ S¡ ¢ S¡ ¡ i de ` ` f f b f hg d` f W a b c YX À¡ S¡ Â ha ¡ YX a S¾ ¾ T y S r 1 b – b2 4ac – + 2a -------------------------------------- - r 2 , b – b2 4ac – – 2a ------------------------------------- - = = b2 ac » b2 4ac – b2 4ac – b ≈ b – b2 4ac – –

yï7ð#C Cñ u7 "Ñ¢ Ì©45)dh%6f459Yf E©!g9pX 2 Í 1©!ç! 1©45) Ì þ5Í x E 45) 1©'0)6"¢!#%6B&$&1©)gQX $&h)2gEh!#9C%($&f74Ì©Ó Í }¥ff&7)§!gf§'4 "T45)2"T'0ff451©!#)cs0'09A'0E©!g9k' $h'GÌ þ5Í E©!g9p"T!g%(B&$&1©)gQX 45)@ÌhÓ Í Eh!#9pX 2 s Ð )Å182'G!#1©'09 45) ED ¤£@Î $&h'Ì©Ó Í Eh!#9p"T!g%(B&$18)2gX 45)2@Ì þ5Í E©!#9pX 2 s 'z'0|rB&9A'0hh7!#) Ï¨2 »2` "2uc45)!g1©'09DEh!#9C%($&f4Á1©451'0|À2 1©c"¢451845n189A!#B" "¢45)"T'0ff451©!g)s x 12 %(!g9k'Á45"T"T$9Y451©' 1©!'0§45f$4518'Á1245 Ì öÏe»&` Í Ì nÏüPQ` Í s D÷I)fÈ§' 1©' ¿§$&45{9Y451©"pE©!g9Y%($f4 1©z7%(B&9A!#¸'0@E©!g9Y%Çh1©ff&245z4n$ 1©9Y45"¢187!#) ¸ $&1D1D 4 '0)g§)6"¢45)"T'0ff451©!g)6!gE ¿§$&45)1©1©'0z}É182!#$139A!g$&))2g'09C9k!g9 )2!#1D4"¢451845n189A!#B"Á!g)'§s å {'0!#9A'0%ÆØ 18' 9A'0f4518¸'G'09Y9A!g9u) êÏ»F`z451D%6!#n13Ø ε s0{'Gh45%6'G 189Y$&'G!gE Ï PÞ`¼s0ij$&f1©B&f )gQ1©}¥! ¿§$&45)1©1©'0z}É18Å4h%645ff&9A'0f45187§'G'09C9k!g9p9k'0n$&f1©z)Å4B&9A!#$"T1D}¥1©64yh%(45ff9k'0f451©§' '09Y9A!g9uÌhh'0' è Ùc!#$))g Ê 9C9A!#9 é !#)ÅB&45gg' Ø5Ø þ5Í s x )!#9A'09c1©!y45¸!#6"¢!#)2E©$&n!#) '¼1©}¥'0'0)'0|r45"T145)6"T!g%(B&$&1©'06§45f$'0 182'ÁEh!#ff!g}¥)2g )!g18451©!g)6 $&h'0 s Ý '09A'¼45 Ï»F`'0)!g18'0 182'G'0|À45"¢137EûE©'09A'0)"T'!#E ÜÏ45)@` Ï ` '0)!g18'0°1©'z"T!g%(B&$18'0QEûE©'09A'0)"¢'zÌ©ks'§s }É18É9A!#$&)2)g'09Y9A!g9 Í snqr%6f459Yf ⊕ ⊗ 45) '0)!g1©'G"T!g%(B&$&1©'0@45718!g) %6$&f1©B&f"T451©!g) 45)d§n!#) 9k'0nB&'0"T1©§'0f s v ff "T45B&ñ)"¢4518'182'e"¢!#%6B&$&1©'06¸45f$&'!gE{4GE©$&)"¢18!g) 45p) Ô ×ÉÌ öÏ Í !#9 q ¦¥GÙ3Ì nÏ Í s Ô !#}É'09 "T45n'GE©$&)2"T1©!#)2z45)d189C451©!#)45f%(451©'0%(451©"T45f&)2!#1©45187!#)Å'0)!#1©'G1©'09p'¼|r45"T1D¸45f$&'0 45 )6f)2Ì öÏ Í 45)2 s v f182!#$ggÅÌ öÏ ` Í ⊗ Ì öÏ ⊕ ` Í z45)Å'0|À"¢'0ff'0)1D45B&B&9A!#|r%(451©!#)Å18! þÏ"2e» 2 1©' ¾!#451©)g§»©B&!g)1D)$&% '09Y Ï45)d`%(g§1D1©'0%(h'0f¸'0 'G45B&B&9A!#|r%(451©!g)z1©!h!g%(' 1©9A$'G¿§$&45)1©187'¼ 45) s º !g9u'0|r45%(B&f' 45) %(g§1 'G'0|r45"T1©f Èg)2!#}É) '0"T%645f&)$&% '09Y 1©4513"¢45))!g1 'G'0|ÀB&9A'0hn'0@'0|À45"¢18f ) )459 s x )61© "T45h' '0¸'0) 1©!#$gg øÏ `Ip4Ig§!#!g645B&B&9A!g|À%64518!g)1©! ÏI»{` 1"T45)Q45¸'e4I$gg'e9A'0f451©§'e'09C9A!#9 "T!g%(B&459A'0@1©!18'189Y$&'G'0|rB&9A'0hh!g) 45)dh!182'G45§45)1©45g§'G!#EFÌ öÏtP@` Í Ì nÏ»F` Í !g§'09 ¥Ï"2 » 2 ñ)!g1{45p9C45%(451©"rs¼qr)"¢'"T!g%(B&$18)2gçÌ öÏùPÅ` Í Ì nÏG»3` Í ñ4 !g$&1{1©'h45%6' 45%(!g$&)1D!gED}É!#9CÈQ45z"¢!#%(B$&187)g øÏ"2e» 2 1Dz"¢f'0459Yf 1©'GB&9A'0E©'09Y9A'0@Eh!#9C%Æ)Å187 "T45n'§s x )çgg'0)'09C45f !#}É'0§'09 9k'0B&f45"¢)gy4"T451©45h1©9A!#B&2"p"T45)"¢'0ff4518!g) g 4 '0)g§)ç!#)2' )!#1D}É!#9C182}¥f'G7ED1©'G'0|ÀB'¼)2h'G f459Agg' '0"T45$&n'G182'G)B&$&1D !#Eh18'0)ÅÌ $13)2!#1 R S Yd e b j e de ` ` U X f ` a a hb g À§ ¾ ¨ Â À§ · ¨ Â y b ` a g b d d V a ` V d V d V a df b X e hd d V g d ` YYV dhb g i hd ha a Yhj e dYf Y` a a V d d f V d ` d e V g § © ¾ ¨ © hc b f S g d e ha d V a ` i À§ ¾ ¨ Â À§ · ¨ Â e V a de f ` ` f b g y hg j ` f f b f a i e d U ¾ f e V a b g Yf d b a hg d ` d e ` f b g y hg j ` f f b f d b W W hd d` y e ` g d b W X d hg j de ` a W V YY` f b c § © V g y ¨ © y b ` a g b d V cc ` d d d e ` w g V Y a e df V d d hb g S r 1 2c b – b2 4ac – – ------------------------------------- - r 2 , 2c b – b2 4ac – + -------------------------------------- - = = b2 ac » b 0 > x y » x y « x x ˆ y ˆ x ˆ y ˆ x ˆ y ˆ –

"Ñ¢« ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ 45f}¥4 Í 45)645B&B9k!g|À7%(45187!#)s
å $&1D'0f%()451©)gQ4"¢45)"T'0ff451©!g)6'0)1©9A'0f Ìh45z)Å182' ¿§$&459C45187"ÁE©!g9C%($&f4 Í z}É!#9C1©}¥f7'G'0§'0)6EF1©'G451©4459k')!#1D'0|r45"T1s09A!g$&g§!#$1 1©uB45B&'09 1}¥ff 'e45hn$&%('0Å184511©'e¾!g4518)g§»©B&!g)1)B$&18p1©!y45)Q45fg§!#9C182% 459A' '0|À45"¢1D45)@1©451D182'G9k'0n$&f1©z459A'G"T!g%(B&$18'0d45z45"¢"T$&9C4518'0f 45 B&!#nh f'§s ''0|rB&9A'¼nh!g) øÏ"2 »{` "2zñ%(!g9k'45"T"¢$&9C4518'}¥2'¼)Q9A'0}É9Y1©18'0)Q45pÌ nÏG»{` Í Ì öÏùPÅ` Í{ '0"T45$h' 4e"¢451845n189A!gB&"p"T45)"¢'0ff4518!g)ç°9A'0B&f45"¢'¼(}¥1©É4 '0)g§)É!g)'§s Ý ' )2'0|À12B&9A'0h'0)12%(!g9A' )1©'09A'¼n18)2g'0|r45%(B&f'0 !gEDE©!g9Y%6$&f45 '0|À 1©)gQ"T451©45h1©9k!gB&"e"T45)2"T'0ff451©!#)Å1824513"¢45) 'G9A'0}¥9C1©18'0)61©!'0|À2 1D!#)2f Q '0)g§)6"T45)2"T'0ff451©!#)cs '459k'04!gED4y189C45)g§f'G"T45) 'G'0|rB&9k'0nh'0@9A'0"T1©f )61©'09Y%6z!gED1©'Gf'0)2gg1© !#EF18 h{'¼ R 45) p 45 Ì ø5Í qy$&B&B&!gh' 1©' 1©9Y745)ggf'Á §'09 ¾&451©ÂÀ1©4512 R ≈ P p sn{'0) s ≈ R 45)Q1©'Á18'09C%ËÌ s »2R Í )6'0¿DstÌ ø5Í n$ 1©9Y45"¢18 1©}¥!)'0459 § )$&% '09Y !#)'!#EF}¥"¢6%(4 45¸'G9A!#$))g '09Y9A!g9Cs º !g9°'0|À45%(Bf' E{R m s Î p Ós þ Ò 182'0)1©'"T!g9Y9A'0"T1¸45f$&'!#E s 7 m s Î5ÒI45) Øs Ò5Ó5Øsss§s Ê §'0)61©!g$&g§61©'G"T!g%(B&$18'0@¸45f$&'G!gE s Ì m s Î þ5Í 7z)Å'09Y9A!#9 g !g)f Ø ulps 1©'G"T!g%(B&$18'0d§45f$&'!#E¸ Òs Î5Ó 45)6'09C9A!#9p!#E ê Î ulps s '09A'eu4}É4 18!¥9k'0}É9Y1©'ÁEh!#9C%($&f4IÌ ø5Í n!y182451&1&}Éff§9A'01©$&9Y)45"¢"T$&9C4518'e9A'¼n$&f1©u'0§'0) E©!g9p¾&451D1©9Y45)2ggf'0yÏÜz4545) l5m5ú5ø Ñhs x 1D Ì ê5Í x E{R e45) p !É)!#1{n45187hE R ≥ ≥ p n%(B&f 9A'0)45%('I18'0% '0Eh!#9A'45B&B&f )gçÌ ê5Í s x 1 h1©9Y45g§1©E©!g9Y}¥459A@1©!"T'0"¢ÈQ18451D1©'G9Cgg1©»©245)@h7'0z!gEDÌ ø5Í 45)dÌ ê5Í 459A' 45fg§' 9Y45"¢45ff '0)1©"T45fkst÷n)gQ182'G§45f$'0z!gEDR 45)2 p 4 !g§'Gg§§'0 4"¢!#%(B$&18'0 459A'04!#EFØs Ò þ5 }¥2"TÅ l ulp )6'09C9k!g9p45)@%($"T6%6!#9A'G45"¢"T$&9C4518'G1©45)Å18'Û&9Ch1 E©!g9Y%6$&f4s v f182!#$ggÉE©!g9C%($&f4eÌ ê5Í 7°%($&"¢É%(!g9k' 45"T"¢$&9C4518' 1©45)ÉÌ ø5Í E©!g9F1©°'0|r45%(B&f' 12}¥!g$&f 'G)2"T'G1©!Èg)!g} !#} }¥'0ffÌ ê5Í B'¼9CE©!g9Y%6z)Ågg'0)'09C45fks A s s a – ( ) s b – ( ) s c – ( ) where s , a b c + + ( ) 2 ⁄ = = A a b c + ( ) + ( ) c a b – ( ) – ( ) c a b – ( ) + ( ) a b c – ( ) + ( ) 4 ----------------------------------------------------------------------------------------------------------------------------------------------- - a b c ≥ ≥ , =

yï7ð#C Cñ u7 "Ñ¢ò ÙÚÜÛ¨Ý)Þ8Û¨ß P5WuXkbTf§r§v q r06WYXYXkbTX q r p f§XAXAWAvzP5WArQf sAq r0 ò õDShb p bTdIe5fgShW ShP5WuRTXkWYRÁb åäÁRÁShX q R¢r¼utW qts RTS dIb s S lTl ε Õ e5Xkb¢V q v5WYvyS©P5RTS s fg¼S©XYR p S q bTr qts e5WYX äTbTXYdIWYv q S©PR°f§RTXYvv q q S Õ W ≤ s Î5Î þ5Õ RTr§v S©P5R¢S s f§RTXkWpXkbTbTS s RTXkW p b¢de5f§S©WYvShbÁ q ShP q r l ÚyØ ulp s '"T!g)1©!#)Å18451DW £ s Î5Î þ %('01D)Å§9C18$45ff '¼¸'09 45"¢18$&45f¾!#451©)g§»©B&!g)1 h1©'0%s º !#9p'0|r45%(B&f'G}¥2'0) β Ø e ≥ ú '0)n$&9A'0z1©451DW £ s Î5Î þ5 45)@}¥2'¼) β l Î e ≥ Ò '0)!#$ggcs x )(h1©4518'0%6'¼)218zfÈg'I2'0!#9A'0% Ò182451{n"T$&nñ1©'I9A'¼f451©¸''09C9A!#9u!#E345)6'¼|rB&9A'0hn!#) 1 $&)'09Ch1©!#!g@1©45131©'G'0|rB&9A'0hh!g)6 "T!g%(B&$18'0@$h)gQ¾!#451©)g§»©B&!g)1D459Y718%6'018"ys x )6B&459C18"¢$&f459 1©'G9A'0f451©§'G'09C9k!g9pz45"¢18$45ff !#EF182'G'0|ÀB&9A'0hn!#) q ¥IÙ3ÌhÌ©R ⊕ ( ⊕ pÍ©Í ⊗ ( p ( R ¬õöõ ⊗ Ì p ⊕ Ì©R Í©Í ⊗ ò R ⊕ Ì© pÍ©ÍhÍ ! Ì ú5Í å '0"T45$h'!gE318'I"T$&% '09Yn!#%6')451©$&9A'!gE3Ì ú5Í© )Q1©'n18451©'0%('0)1{!gE318'0!g9A'0%(ñ}¥'I}¥ff $&n$&45ff h4 S©P5W p b¢de5f§S©WYvV5R¢utfgWub æäÁU@9C4518'09p1©45)6}É9Y1©)gQ!#$1DU@}¥1©6"¢9A"Tf' )!g18451©!g)s Ê 9C9k!g9 !#$) 459k'G$h$&45f7f 1©!g!çB'¼nh%6h1©"rs x )Å18'G)2$&%('09C"T45f&'0|r45%(B&f'gg¸'0) 4 !g§' 1©'Á"T!g%(B&$18'0(¸45f$&'Á!gE&Ì ê5Í Øs Ò þ5 "T!g%(B&459A'0(}¥1©ç41©9A$' ¸45f$&'Á!gE&ØsÒ5Ó5Ø l5ø E©!g9p49A'¼f451©¸'G'09Y9A!#9p!gEDÎs ê ε }É"TÅ %($&"¢6f'0h 1©45) l¢l ε s0'G%645)69A'045h!g)6E©!g9 "T!g%(B&$18)2g'09C9k!g9 !g$&) )!#1D1©!gg'01DB&9A'0"Th' !#$) $&1D9C4518'09p1©!ç¸'09YE 1©451 1©'GE©!g9Y%6$&f4!g'0z)2!#1D"T!g)1©45)6)$%('09Y7"T45f&B&9A! f'0%(Às v Û&)245f&'0|À45%6B&f'G!gED45)Å'¼|rB&9A'0hn!#)Å18451D"¢45) 'G9A'0}¥9C1©18'0)Å18!$&n' '0)g§) "T45)2"T'0ff451©!#)ÅzÌ l PþÏ Í I }É'09k' s0 '0|ÀB9k'0nh!g)6459Ch'0 )6Û)45)"T45f "T45f"¢$&f451©!g)5s0op!g)h7'09u'0B&!gh1©)g #" l Î5Î'0¸'09 4 )1©!4 45)ÈQ45"T"¢!#$)1D182451 '0459Y)2z45)Å45))$&45f)18'09A'0h1D9C4518'G!gE ø %$É "T!g%(B&!g$&)'0@{45f s x EFr Ò ø5þ 45)2 q s Î ø5 1©'G45%(!g$&)1D!gED%(!g)' 45"T"¢$&%($f4518'0d45131©'G'0)d!#EF!#)2' '0459p l Î5Î !gff459YÀs x ED1© z"¢!#%6B&$&1©'0@$&n)g β Ø45)@e Ø5Ó 1©'G9A'0h$f1D &"5Ò ê5ø5l5þ s Ó þ "T!g%(B&459A'0@1©!ç1©'G'0|À45"¢1345)n}¥'09u!gE '"5Ò ê5ø5l Ós Î þ5 4n"T9A'0B&45)" !gE '" l s Ó5Îs0'G9A'045h!g) E©!g9p18'GB9k! f'0%Çz'045 1©!h'0'§s0{'G'0|ÀB&9A'0hn!g) l P q ÚrrQ)¸!gf§'0 45)g l 1©! s Î5Î5Î l5ø Ó5Ò ú Ò ø5 n!ç1©'Gf!g} !#9A'09 1©z!gE q ÚyrQ459k'Gf!gh1s02z9A!g$&))2g'09C9k!g9p 45%(B&f7Û&'0@}É'0) l P q Úyr 9C45h'0@1©!18'rg1©6B&!g}¥'09Cs x 1 « 1 i n ⁄ + ( ) n 1 – i n ⁄ ------------------------------- -

"Ñký ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ '189A!#$ f'0h!g%('G'0|rB&9A'0hh7!#)ÅÌ l P
q Úrr Í I "T45) 'G9k'0}É9Y1©1©'¼)Å45 W I)(021 í G38I54 }¥'09A' )!g}ö182' B&9A! f'0% 18!y"T!g%(B&$&1©' f7)Ì l PøÏ Í E©!g9h%645ff Ïs Ð )'Á45B&B&9A!#45"¢É 1©!$h' 1©' 45B&B&9A!#|r%(451©!g)6f)Ì l P¢Ï Í ≈ Ï )6}¥2"TÅ"T45h'G1©'GB&4 %('0)1 '¼"¢!#%6'0 &"5Ò ê5ø5l5ê s Ø ø5 }¥2"T67z!gEûE g "5Òs Ø l 45)d'¼¸'0)6f'0nz45"¢"T$&9C4518'1845)Å182'G! §!g$& E©!g9Y%($f4s å $&1 1©'09k'G7z4}É4 18!"T!g%(B&$18'Gf)2Ì l P¢Ï Í ¸'09 45"¢"T$&9C4518'0f ¢ 45 '0!g9k'0%ÇÓh2!#}É Ï ¯m¥'0}Éf'0181©» Ã 45"TÈ§459k l5m5ú Ø5Ñs02zEh!#9C%($&f4 '0f 6"5Ò ê5ø5l Ós Î ê5 45"¢"T$&9C4518'18!}¥1©)Å18}É! "T'0)1© 'p '0!g9A'0%ÆÓ45nh$&%6'0z1©451 Ô ×ËÌ nÏ Í 45B&B&9A!#|r%(451©'0 f)Ì öÏ Í 18!}¥718) l ÚrØ ulp s0' B&9A! f'0%Æ1Dh!gf¸'0z 1824513}É'0) ëÏyzn%(45ff Ô ×ÉÌ l ⊕ Ï Í z)!g1D"Tf!gh'G1©!f)Ì l P¢Ï Í '0"¢45$&h' l ⊕ Ï45 f!#n131©'G)2E©!g9Y%(451©!g)6)Å18'Gf7!#} !#9A'09 1©z!gE ÜÏs0451D 18' "T!g%(B&$18'0@¸45f$&'G!gEFf)Ì l P¢Ï Í z)2!#1D"Tf!gh'18!1©z45"¢18$45f&§45f$'G}¥'0) s ÙÚÜÛ¨Ý)Þ8Û¨ß7 ïäÁutr ò çtñ¢Ï'õ qwspp b¢de5f§S©WYvf sYq r¼ÅS©P5W DäTbTXYdIfgutR Ï äTbTX l ⊕ Ï = 1 f)Ì l P¢Ï'õ'8 äTb¢X l ⊕ Ï ≠ 1 S©P5WuXAWAutR¢S q V5WuWYXYXkbTX qts RTSFdIb s S þ ε ñP5WYrQÎ ≤ | 9£ Õ e5Xkb¢V q v5WAv s fgShXYR p S q bTr qts e5WYX äTbTXYdIWYv q S©PR°f§RTXYvv q q S Õ W £@Îs l5Õ R¢rgvyuwr qtsup bTdIe5f§S©WYvS©be q S©P q r l ÚrØ ulp s 7zE©!g9C%($&f4}Éff&}É!#9CÈQE©!g9u45) ¸45f$&'G!gE Ï $&1D !#)f )218'09A'0h1©)gQE©!g9 }¥2"T6 }¥'09A'G"T451©45h1©9k!gB&"Á"¢45)"T'0ff74518!g)6!g"T"T$9Yz7)618'G)457§'GE©!g9Y%6$&f4f)Ì l P¢Ï Í s v f182!#$ggÅ18'GEh!#9C%($&f74%(4 h'0'0%Ç% n18'09C!g$& 1©'09A'G 4h%6B&f'G'0|ÀBf45)45187!#)ÅE©!#9 }¥ 1}É!#9CÈ§5s Ý 9C1©'ef)2Ì l PëÏ Í 45 s'f'0E©1245)ÅE©45"T1©!g9 "T45) 'e"T!g%(B&$18'06'0|À45"¢18f ¢r $&1&1©'Á9Cg§1&45)6E©45"¢18!g9 µ Ì öÏ Í f)Ì l PøÏ Í Ú Ï}Éff¸h$&E8E©'09 4f459Ag§'G9k!g$&){)gQ'09Y9A!#9p}¥2'0)645)g l 1©! þÏs 8m¥!g}¥'0¸'09 µ z45f%6!#n1D"T!g)h1©45)1 h)2"T'Gf)Ì l P¢Ï Í ≈ Ïstqr!"T45)2gg)2g þÏyhfg§1©f }Éff&)2!#1D)1©9A!#$"T'G%($"T6'09C9A!#9Cs x ) !g18'09p}¥!g9A E "T!g%(B&$&1©)g }¥f7f 'G4g§!g!#d45B&B&9A!#|r%(451©!#)Å18! Ï µ Ì öÏ Í f)Ì l PþÏ Í s x z1©'09A'G4§45f$'GE©!g9 E©!g9p}¥"¢ 45) "¢45) ' "T!g%(B&$18'0 x 1 « x ln(1+x) 1 x + ( ) 1 – --------------------------- 3 4 -- - x 1 « x ln 1 x + ( ) x ------------------ -     xµ x ( ) = x ˜ x ≈ xµ x ˜ ( ) x ˜ x ˜ x ˜ 1 +

yï7ð#C Cñ u7 "Ñ"k 45"T"¢$&9C4518'0f §ù {'09A'GhÂ§)45%6'0f Ì l ⊕ Ï Í l5§ '0"¢45$&h'18'0) l P '0|À45"¢18f '0¿§$&45f&1©! l ⊕ Ïs '9k'0n$&f1©z!gED1© h'0"T1©!g)6"¢45) 'Gn$&%(%6459Y~5'0 § h4 )gQ18451D4g§$&459A@7gg1 g§$&459Y45)218'0'0 45"T"T$9Y45" }É'0)6)'0459 g B&9A'0"Tn'¼f È§)!g}¥)Å¿§$&45)1©18'0 459A'Gh$ 189C45"T1©'0 Ì '¼)2gg)Q"T45)"¢'0ff4518!g) Í s¢qr!g%('01©%('0u4IEh!#9C%($&f74I18451g§§'0p)45"T"¢$&9C4518'e9A'0h$f18p"T45) ' 9A'0}¥9Y7181©'0)61©!ç245§'G%6$&"TÅg§'09u)2$&%('09C"T45f&45"¢"T$&9C45" Qg $h)g '0)7gg) "T45)2"T'0ff451©!#)2Â§!g}¥'0§'09 18'GB9k!g"T'0$9k'G!g)f }¥!g9CÈg EDn$ 1©9Y45"¢187!#)ÅzB&'09CE©!g9Y%('0 $&n)gQ4g§$&459A@g§1Cst'GB&9C"T'G!gED4ygg$459kdgg1D )!g1Dgg § '0"T45$h'G1D%('09A'0f 9A'0¿§$&9A'0z%(45È§)gQ1©'G45'09p!#)2' 1D}¥{'¼9s º !g9p4 þ Ó 1D!g$ f'GB&9A'¼"¢h!g)645{'09 1©'G451©!#)245f&"T!gh1D f'0h 18245)6Ø $ s º !#9p1© B&9Y"¢' § !g$@gg45)Å18'4 f1 18!9Y$&) %(45) 45fg§!g9Y1©%( h$"T645 1©'GE©!g9Y%6$&f4Ì ø5Í E©!g9p"T!g%(B&$&1©)gQ1©'G459A'¼4y!#EF41©9Y45)g§f' 45)@1©'G'0|rB&9A'0hh!g)6f7)Ì l PøÏ Í s v f1©!#$ggÅ%(!gh1D%(!g'09Y)Å"T!g%(B&$&1©'09Y 45¸'G4g§$&459k g§1 18'09A'G459A'G4E©'0} Ì©n$&"TÅ45zop9C4 %@ h1©'0%( Í 1©451D!)!#1s ¦BA¥¤¥û¦" DC@ ¢ Ó ©¡©¥¡©FE¢¡¡D¤¥¦¨§£¢¥©¡ Ý '0)6¾!g45187)gg»hB&!#)213!gB&'09C4518!g) 459k'G{!#)'G}É1©64g§$&459A@g§1 182' 459A'G)!g1D45 45"T"¢$&9C4518'45pE{18' }¥'09k'"T!g%(B&$18'0Å'0|À45"¢18f 18'0)Q9A!#$)'061©!¥1©')'0459k'0n1¾!#451©)g§» B&!g)1D)$&% '09Cs Ð B&'09Y451©!#)2zB&'09CE©!g9Y%('0d)61© %(45))'09p}¥f7f 'G"¢45ff'0@W ôÏ5R p S©ut` XkbTf§r§v5WYvs °'G'0|r45%(B&f'G%6%('0451©'0f B9k'0"¢'¼{)gQ'0!g9k'0%ÇØh2!#}Éz1©451D4h)2ggf' g§$&459A@gg1D}¥f7f&)!#1D45f}¥4 zg§§'G'0|r45"T1©f 9A!#$)'0@9A'¼n$&f1©5s02'IB9k'0¸!#$zh'0"¢18!g) g§45§' h'0¸'09Y45fg'0|r45%(B&f'0 !#E&45fg§!#9C182%(°1©45129A'¼¿§$9A' 4egg$459kQ{gg12)É!g9A'09F1©!}¥!g9YÈ B&9A!#B&'09Cf st h'0"¢18!g)6g§§'0 '0|À45%6B&f'0z!gEF45fgg!g9Y1©%6z1©451D9k'0¿§$9k''¼|r45"T1 9A!#$))gDs qy!çEh459 182'G'0Û&)718!g)6!gED9A!g$&))2g245z)!g1 '0'0)6g§§'0)cs0Ùc!#$&)2)gQ h1©9Y457gg1©E©!g9Y}É459A }¥1©61©'G'0|r"T'0B&1©!#)Å!#EF!g}Ë1©!9k!g$&)d45fE©}É4 "T45h'0nÂ§E©!g9 '0|À45%6B&f' h2!#$f l Øs þ 9A!#$)@1©! l Ø!#9 l Ò ù Ð )2'Gh"T2!#!gf&!gED1©!g$&g§1D§'0 1©' l Î g§18p)45fE f'0181©)g HG©Î Àl5 Ø Ò Ó %I9A!#$)6!#}É) 45) PG þ5yø5rê5yú5ym I9A!#$)6$&B&Ây18$ l Øs þ }É!#$&f7@9A!#$&)2@1©! l Òst z2!#} 9A!#$&)2)gQ}¥!g9YÈ§ !#) g§1845f Ê ¿§$&B&%6'¼)21 op!#9CB&!g9Y451©!#)2Äw 6Q v ¤RTS "T!g%(B&$&1©'09YÀs v )!g18'09ph"¢!g!#f!#EF18!g$&g§1Dh4 18451Dn)"T' )$&% '09Y '0))2g) þ 459A' 45fEh}¥4 y '01©}¥'0'0)É1©}¥!B&!ghh f' 9A!#$))g§ 18' h2!#$f 9A!#$)@!g}¥)Å45fED1©'G1©%('G45)2@9A!#$&)2@$&B@18'G!g1©'09u245fEs Ð )'G}É4 !gED! 1©45))2g 1© þ Î $ '045¸!#9p1©!ç9A'0¿§$&9A'G1©451D182'G9k!g$&){'¼d9k'0n$&f1D45¸'G18 f'045h1Dngg)2Û&"¢45)1 R S Ya b d b W W b g Yf f ` c ` f f ` y db V a Ê p Ê r w¨ p É t S ¾ ¾ T y S x ˜ x ˜

"Ñ"x ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ g§1 ' '0§'0)sn2$& l
Øs þ 9A!#$)°1©! l Øe9Y451©'09F1©45) l Ò '0"T45$&n' Øe°'0§'0)cs Ý "TÉ!gE 1©'0h'G%('01©!gz7 '0n1 9k!g$&)d$&BÞ!g9p9k!g$&)d18!'0§'0) ù Ùc'0h'09p45)@Üz)$18ªÏ l5m5ê5þ Ñ !gEûE©'09p1©'GE©!gff!g}¥)gQ9A'045h!g)6E©!g9pB&9A'¼Eh'09Y9C)gQ9k!g$&)d18!'0¸'¼)cs ÙÚÜÛ¨Ý)Þ8Û¨ßU V WAS ÜÏR¢rgv`yW D9ub¢RTS q r0©e5b q r§SFrgf§dIWYX sYÕ RTr§vv5W 1 rgW óÏXWY8¢Ï Õ ÏX`Y8 ò ÏXW ` 'õ ⊕ ` Õ … Õ ÏXab8 ò ÏXadc` ` 'õ ⊕ ` ¨& ïä ⊕ RTr§v RTXAWuW £Ï5R p S©ut`XkbTf§r§v5WAvyf sYq r06XkbTf§rgvyS©beWAV5WYr Õ S©P5WYr W q S©P5WYX óÏXae8¢ÏÜäTb¢XuRTutu2rQbTX óÏXaf8 ÏX`~äTbTXpRTutu2r ≥ l s r!"Tf459CE 1© 9k'0n$&f1 "T!g)h{'¼9 β l Î e Ò45)2@f'01 Ï l s Î5Î ` »s þ5þ5þ s Ý 2'0) 9A!#$))gQ$&B 18'Gn'0¿§$&'0)"¢' '0"T!g%('0 ÏXg l s þ5ø5 ÏXh l s þ5ø s þ5þ5þ l s Î l5 ÏXh l s Î l ⊕ s þ5þ5þ l s þ5ê5 45)2@'045"TÅh$&"¢"T'0hn§'§45f$&'!#E ÜÏXa°)"¢9k'045n'0 § s Î l5 $&)1©f ðÏXa m s Ó þ Ì©) ≤ ú Ó þ5Í ¢s÷I)'09 9A!#$)Å1©!É'0§'0) ÏXañ45f}É4 l s Î5Îs{ñ'0|À45%6B&f' h$ggg§'0h1©z1©451D}¥'0)6$h)gQ182'G9k!g$&)@$&B@9A$&f7' "¢!#%6B&$&1©4518!g) "T45)6g§9Y45$45ff 9YEh1 $&B&}É459k }¥'09A'045z}É'0)6$h)gQ9A!#$)@1©!'0§'0)61©'G1©'0!g9k'0%Çh4 z1© "T45))2!#1 45B&B&'0)st9A!#$gg2!#$&1D1©'G9A'0h1D!gED1© B&45B&'09 9A!#$)@1©!ç'0¸'0)6}Éff 'G$h'0 s Ð )'G45B&B&f"¢4518!g)6!gED'0|r45"T1D9A!#$))gQ!g"T"¢$&9Y )6%6$&f1©B&f'GB&9A'0"Tn!#)Å459Y1©%6'¼1©"ys '09A'G459A'G1©}¥! 45h7"Á45B&B&9A!g45"T'0 18!g§'09pB&9A'0"Th7!#)s Ð )'G45BB&9k!g45"TÅ9k'0B&9A'0h'0)218 ¾!#451©)g§»©B&!g)1D)$&% '09Y $&n)gQ4¸'¼9 f459Agg'Gngg)2Û&"¢45) }¥2"TÅzn18!g9A'¼d)645) 459C9Y4 !#EF}¥!g9A 45)d"T!g'0z1©'G9A!g$&187)'0zEh!#9p%(45)B$&f451©)gQ18'0n'G)$&% '09Y ) 45hn'0% f f45)g§$&45g§'§s02'Gh'0"T!g)@45BB&9k!g45"TÅ9A'¼B&9A'0h'0)218 g§'09uB9k'0"¢h!g)6¾!g4518)g§» B&!g)1D)$&% '09Y 45z45)Å459Y9C4 !#EF!#9A)459 ¾!#451©)g§»©B&!g)1D)$&% '09Y }¥'09A'G45)2g 1©'G'0f'0%('0)1©z!gED1©'G459C9Y4 )Å)Û&)218'GB9k'0"¢h!g)69A'0"T!g§'09Cz1©'Gg§6B9k'0"¢h!g) ¾!#451©)g§»©B&!g)1D)$&% '09Cs x 1Dz1© h'0"¢!#)d45B&B&9A!#45"¢61©451D}¥ff 'Gn"T$&nh'0@'09A'§s '45§45)1©45g§'G!#EF$&h7)gQ45)6459Y9C4 !#EF¾!#451©)g§»©B&!g)1D)$&% '09Y 18451D1D"¢45) ' "T!g'0@B&!g9Y1©4 f )Å4g§6f'0¸'0f&f45)g§$&45g§' § $&1D1D9A'0¿§$&9A'0z'0|r45"T1©f 9A!#$&)2'0 459C18%6'018"ys 'IÈ§' 18!É%6$&f1©B&f"¢4518!g)()(18ñ n18'0% ñ9A'0B&9A'0h'0)1©)gç4B&9A!#{$&"T1 qÏ 45ñ4Gn$&% }¥2'¼9A'G'045"¢6h$%(%(45)d45 18'Gn45%('GB&9A'0"Tn!g)645 Ï45)@`¼s0{ "T45) 'G!#)2' g hB&f7181©)g øÏ45)d`¼s Ý 9C1©)g øÏ ÏXiP@| %pÀ45)@` ` XiÜP p 182'G'0|À45"¢13B9k!g$&"¢137 Ï5` ÏXi&` XiPêÏXiD` Xp'P ÏXp5` XiqP ÏXp5` Xpws x E Ï45)@`45¸'Ie 13hg§)Û&"¢45) 182'Ih$%(%(45)z}Éff 45fh!45¸'Ge 13ng§)Û&"¢45) B&9k!g¸'0@1©451 qÏqp ÏXi ` Xi ` Xp5"¢45) '9k'0B&9A'0h'0)1©'0@$&n)g  eÚyØ  1©5s Ý '0)Åe '¼¸'0) 1D '045 1©!çÛ)@4yhB&f1©187)g3st'G)$% '09 ÏXghs ÏXh … ÏXrtsuhF"T45) 'G}É9Y1©18'0)Å45z1©'Gn$&%Æ!gE ÏXghs ÏXh … ÏXrwv swh45) Îs Î … Î 'ÏXrv … ÏXrXsxhs Ý 2'¼)Åe !# 182zn%(B&f'GnB&f1©18)2g%6'018!g@}É!#)Ät1D}¥!g9CÈ3s R S y e ` g g ¢ i U ¢ S i U · ¡ S ¢ R ¡ S¡ i V g y R ¡ S¡ ¾ ¡ S ¢ S S e ` f ` c b f ` i U ¢ U c b f t S

yï7ð#C Cñ u7 "Ñ¢¡ v )Q'¼|r1©9Y4 1"¢45) !#}É'0§'09 # 'gg45)2'¼ g $&h)2gÉ)'0g§4518¸'e)$% '09C5s º !g9°'0|r45%(B&f' E β Ø e þ5 45)2 êÏ s l Î lTl¢l5 ÏG"T45) 'hBf145 QÏXi s lTl 45) ÏXp »s Î5Î5Î5Î l s2'09k' %(!g9A'G1845)Å!#)2'G}¥4 1©!çnB&f1D4)$% '09Cs v nB&f1©18)2g%6'018!g@1©451D '¼45 18! "T!g%(B&$18'Á°{$&' 1©! '0È§Èg'09 Ï l5m5ê5l Ñ À $12129A'0¿§$&9A'0°%6!#9A' 1©45)ç4eh7)ggf'Ágg$459AQg§1Cs ÙÚÜÛ¨Ý)Þ8Û¨ß V WAS3eGW°ShP5W 9ubTR¢S q r0©e5b q r§S{e5XkW pCqtsYq bTr Õ q S©PGShP5W°XkW s S©X qtp S q b¢rÉShP5RTS3e qws WYV5WYrçñP5WYr β Ø Õ RTr§vGR sYs fgdIW°S©P5RTS ¨9pbTR¢S q r¼he5b q rgS{b¢e5WYXAR¢S q bTr s R¢XkWcW ôÏ5R p Shut`IXkb¢fgr§v5WYv '& P5WYr qä s8jÏB°ÚyØ5Ñ qws P5RTu äzShP5W°e5XkW pqtsAq b¢r ò XkbTf§r§v5WYvGf§e 'õ{R¢rgvd 8 β uðñ ltÕ Ï p RTrç¼W s e5u q S3R s Ï8êÏXiqñêÏqpÕ zP5WAXAW ÏXi8 ò d ⊗ Ï'õ ò d ⊗ Ï Ï'õ Õ ÏXp8 Ï ÏXi Õ R¢rgvIWYR p P ¡ÏX qts XkWYe5XkW s WYr§S©RT¼uwWf sYq r¼wÏB°ÚyØ5Ñ q S s b æäÁe5XkW pqwsYq bTr & r!h'0'G!g} 187z1©'0!#9A'0%Ç}¥!g9YÈ§z)Å45)6'0|r45%(B&f' f'01 β l Î e Ó Òs Ó ê5ø5 R Òs Ó ø Ò 45) p Òs Ó ê5m s'0)Q »ÞR p 9A!g$&)'0Å1©!¥1©')'0459k'0n1¾!#451©)g§»©B&!g)1{)$&% '09 ys Î5Ò5Ó ú Î }¥7f'G ⊗ l Øs Î ú5 R ⊗ p l Øs Î þ5 45)dh!1©'G"T!g%(B&$18'0@¸45f$&'G!gED » R p ys Î5Òs02z 45)6'09C9A!#9p!#EFÓ ú Î ulps s0÷In)gQ'0!g9k'0% ø 1©!ç}É9Y1©'G Òs þ »ñs Î5Ø5Ó R Òs þ »cs Î5Ò ê5 45)2 p Òs þ »ñs Î5Ø l5 '0"T!g%('0 Òs þ 2e»DØ × Òs þ × s Î5Ø5Ó tP s Î5Ø5Ó s Ê 45"T h$%(%(45)dz'0|r45"T1 n!ç l Øs Ø þ »zs l5ø5ú P s Î5Î5Î þ5ê5ø5 }É'09k'G1©'Gn$&%Çzf'0Eh1 $&)'0¸45f$&451©'0@45131© B&!#7)1Cs0qy%(f459Cf T R p Òs þ 2e»DÌ©Òs þ × s Î5Ò ê P@Òs þ × s Î5Ø l5Í P s Î5Ò ê × s Î5Ø l l Øs Ø þ »us Ø5Î5Ò5Î ùP°s Î5Î5Î ê5ê5ê s º )45ff T h$ 189C45"T1©)gç182'0h'18}É!¥n'09Y'0ñ18'09C% g 18'09C% g§§'0z45)6'0h1©%(451©'IEh!#9u »DR p !#EDÎ ⊕ s Î5Ò þ Î s Î5Î5Î5Ø5Î l s Î5Ò5Ó ú Î }¥"¢6z'0)1©"T45f 1©!ç1©'G'0|r45"T1©f 9k!g$&){'¼d9k'0n$&f1s0r!h!g} 18451D{'0!#9A'0% ø 9k'045ff 9A'0¿§$&9A'0z'0|r45"T1 9A!#$))g "¢!#)2h'09ce Ò β Ø 45) Ï ê s¢'0)d þ5 d Ï Ò þ5 45)6d ⊗ Ï Ò5Øs x EDn$ 1©9Y45"¢18!g)6 B&'09CE©!#9C%('0@}É1©64n)ggf7'Ggg$459kdgg1 1©'0)6Ìhd ⊗ Ï Í Ï Ø ú s '09A'0E©!g9A' ÏXi Óy45) ¢ÏXp Ò '0)"¢' ùÏXp5 )!g1D9k'0B&9A'0h'0)1©4 f'G}É1© [ e £÷5Ñ l 1Cs v 4Û&)45f§'0|r45%(B&f'Á!gE&'0|À45"¢129A!#$&)2)g "¢!#)2h'09{§)2gyd gyl ÎsT2' 9A'0h$&f712 4 ¾!#451©)g§»©B&!g)1D)$&% '09p18451D}Éff&)Ågg'0)'09C45f&)!g1 'G'0¿§$&45f18!dÚ l Îs Ý '0) β Ø %($f18B&f )gQdÚ l Î gQl Î}Éff&%(9C45"T$f!#$hf 9A'0h1©!#9A'Gd B&9A!#¸'0@'0|r45"T1D9A!#$))g '0)gÉ$&n'0 s v "T1©$&45ff ¢ 4G%(!#9A'eg§'0)'09Y45fE©45"T1Ìh$&'e1©!¥Üñ45245) Í p1©9A$'§s¢'B&9k!g!gE )g§'0)!g$& y $&19k'045'09Cu)!g1)1©'09k'0h1©'06)Qh$&"¢'01©45fu"¢45)hÈ§B(45'045Å18!¥qr'0"¢18!g) è ' x©ÊÊÊ qr1©45)459A é !g)6B&45g§' l5ú5m s ÙÚÜÛ¨Ý)Þ8Û¨ß HP5WYr β 8ÅØ Õ#q äzd R¢rgvGrçR¢XkW q rgShWûWAX s q ShP Fkd ÅØ r%cw` RTr§vGrçP5R s S©P5W s e5W pCq R¢u Fänb¢XYd r 8@Ø ñ@Ø wd Õ S©P5WYr ò d r õ ⊗ r e8@d Õ e5Xkb¢V q v5WYv Ü9pbTR¢S q r¼he5b q r§SDbTe5WYXYR¢S q bTr s RTXkWuW ôÏ5R p Shut` XkbTf§r§v5WYv '&

"Ñ"Ñ ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ èÞ8Ý)Ýné qy"T45f)g g 4B&!g}¥'09p!gED1©}¥! 459Y%6f'0h
h)"¢'G1D"T45)2gg'0 !#)f 1©'G'0|rB&!#)2'0)1 )!g1{1©'Ih7gg)Û"T45)2 s x E dÚrr 18'0)6h"T45f'Irh!ç1©451{Ø rtsuh ≤ r e£ÅØ r 45)Þn"T45f'Id h!ç1©451 £ £ l s¼{$& Ø rts £Þd f£ÅØ r s¼qy)"T'Id 45zehg§)Û&"¢45)1 1© 1345ñ451 %(!gh1D!g)' 1D1©!18'G9Cg§1D!#EF182' )459 B&!#)21Cs0op45)2gg)2g1©'Gngg)Å!#EFdæ 459C%(f'0h h!45nh$&%6'G18451 Îs x E d r 1©!B&9k!g§'18'G1©'0!g9A'¼%Ç9A'¼¿§$9A'¼ h2!#}É)gQ182451 Ì m5Í 451D '0"¢45$&h'Gdæ245z451D%(!gh1 l 139Cg§1D!#EF182' )459 B&!#)21 n!r }Éff 9A!#$)@1©!çds0y!ç'045f}¥1©61©'G45fEh}¥4 "T45n'G}¥'0) hgr »Dd ig )!#1©'G1©451 h)2"T'G1©'G)1©45f&$)h"¢45f'0@dæ45 jgd ige£@Ø rtsuh 1©zf!g}¥»h!#9A'09 1D}¥45 Î h!1©' f!g}¥»©!g9A'09 13!#ED1©'Ih"¢45f'0Þd 7ñ45fn!çÎs¼$ 245fE©}¥4 "¢45h'0z}¥ff&9A!#$)Þ1©!çds qy$&B&B&!gh'e1©451 s h … 45)6f7'¼1 s h … r l sr!¥'0n18%64518' gr »{d ig Û9Yn1 "T!g%(B&$18' Fg » g glkÚyØ rtmnh »FdÚrr og }¥'09A' kæ 45)6!g@)1©'0gg'09s0qr)"¢'Gr Ø PQØ d 45)@Ø r¦suh ≤ r e£@Ø r 1D%($h1 'G1©451Dr Ø r¦suh PÞØ u E©!g9un!#%6'GÈ ≤ eDØ 45) 1©$& s 'G)2$&%('09C4518!g9u7z45)6)218'0g§'09 45)@n)"T' ok !g 1Dz)ÅE©45"T1D45)6!g@)218'0g§'09Cs $ g » g ≥ l ÚrÌhrgØ rqmphXs u Í s v hn$&%(' £ Ì©1©'Á"T45n' h%6f459 Í s D'0) r £@d 45) gkdhr qg dI©r r§Ì Í r§Ì »©Ìü»©Ø 76¬7r4 ÍDÍ ≤ Ì©Ø 6 !n¤P&Ø )r Í Ø 76¬7r4ts Ø 7687%4 Î r 7z'0h1©4 f7h'0 Ì m5Í 45)2@B&9A!#¸'0z1©'G1©'0!g9k'0%s 2 R S º b dhd ` d e V d hg e hg V f f i u d V g g b d ` Æ V Y S ¾ ¾ T y S 1 2 -- - q nq m – 1 4 -- - ≤ q q 1 4 -- - q ˆ q q ˆ q ˆ q – nN 2p 1 + m – n2p 1 + ----------------------------- - 2p 1 – k – 1 + ( ) N 2p 1 k – + m – n2p 1 k – + -------------------------------------------------------------------- - = = q ˆ q ˆ q ˆ q ˆ q q q q q ˆ n 2 p – 1 – 1 n2p 1 k – + --------------------- - –     1 4 -- -

yï7ð#C Cñ u7 "Ñ"ì '18'0!g9A'¼%Ç!gf 189Y$&'GEh!#9p45) Q 45n' β 45zf!g)gQ45 Ø P@Ø wd2z9A'0B&f45"¢'¼ g β P β ds v β g§'01©zf459Ag§'¼9 !g}¥'0§'09 '0)!#%6)451©!#9Cz!gED1©'GEh!#9C% β P β dÅ459A'GE©459C182'¼9p45) E©459C182'¼9p45B&459C1Cs Ý 'G459A'G)!#} )Å4B&!#n1©!#)Å18!45)n}¥'09p18'¿§$&'0h1©!#) § !g'0z1D%645181©'09uEF1©' 45n" 459C18%6'018"e!#B'¼9C451©!#) )1©9A!#$"T'G4f7181©f'G%(!g9A'G9k!g$&){)gQ'09Y9A!#9p1©45)6)'0"¢'0hh459 §ù '45)h}É'¼9p 1824513713!g'0 %(45181©'09 § '¼"¢45$&n'G45"T"T$9Y451©' 45n"Á!gB&'09Y451©!g) '¼)24 f'G$& 1©!çB&9A!g§'G1©451DE©!g9Y%6$&f45 459k' è "¢!#9C9A'0"T1 é )61©'Gn'¼)2h'G1©' 45§'G4yh%(45ff9k'0f451©¸' '09Y9A!g9Cs è op45)"T'0ff74518!g) é !g)6B&45g§' l5ê5m n"T$&nh'0@n'0§'09C45f&45fg§!#9C182%(z1©451D9A'0¿§$&9A' g§$&459A@g§18 1©!çB&9A!g$&"¢'G"T!g9Y9A'0"T1D9A'0h$&f1© )61© h'0)h'§s x EF182'G)B&$&1D1©!18!gh' E©!g9Y%6$&f45 459k'G)2$&% '¼9C 9k'0B&9A'0h'0)1©)gQ%(B9k'0"¢h'G%('045n$&9A'0%('0)1© !g}¥'0¸'09 1©' !g$&) !gED2'0!#9A'0%( Ò45)@Ó '0"T!g%('Gf'0nz)1©'09A'0h1©)g3st'G9A'045h!g)6 1©451D18' '0)g§)6"¢45)"T'0ff74518!g) ëÏ»F`"¢45) '0"¢!#%('"T451©45h1©9k!gB&"eE ÜÏ45)@`459A'G!g)f 45B&B&9A!#|r%(451©!g)p1©!¥n!#%6'e%6'045h$&9A'0Å¿§$&45)2181 s å $145"¢"T$&9C4518'!gB&'09Y451©!g)p459A'$&h'0Eh$&f '0§'0)Å)61©'GE©45"¢'G!#EF)'0|r45"T1D451©4 § '0"T45$h'G1©' '0)4 f'G$z1©!'0h1©4 fhÅ'0|À45"¢1 9A'0f4518!g)nB& fÈ§' 1©!gh'Áh"¢$&nh'0()ç'0!g9k'0%( ø 45) ê sT'0n' 459A'Á$&h'0Eh$&f§'0§'0)çE '0§'09 ¾&!g451©)gg»hB&!#7)1D§459C4 f'G !#)2f 45)645B&B&9A!g|À%(451©!g)61©!h!g%('G45"T1©$&45f§45f$&'§s v9w6xçßtõÞõÅõ y µ· à â ·¶¸â '09A'G459A'G1©}¥!E8E©'09k'0)1 xhÊÊÊ n1845)2459AzE©!g9p¾!#451©)g§»©B&!g)1D"T!g%(B&$&1©451©!#)cs x©Ê{ÊÊ ê5þ Óz4 )459 h1©45)459AÞ1©45139A'0¿§$&9A'0 β Ø e Ø5ÓEh!#9uh7)ggf'GB&9A'0"Th7!#)645)@e þ ÒE©!g9!#$ f'ÁB&9k'0"¢h!g)dÏ xhÊÊÊ(l5m5ú5ê Ñs x 1245fh!yhB&'0"T7Û&'0 1©' B9k'0"Tn' f74 !g$&12!gE 1© ) 4h)2ggf'Á45)(!g$ f' B9k'0"Tn!g)s x©ÊÊ{Ê(ú5þ Óe45f7f!#}É '01©'09 β Ø!g9 β l Î45)($&)2fÈg' ê5þ Ó !g'0z)2!#1DhB&'0"¢E !g}Ë¾!g4518)2gg»hB&!#)213)2$&% '09Cz459A'G'0)"¢!#'0d)1©! 18yÏop!g '01D45fks l5m5ú Ó5Ñs x 1D!g'0z)!g1D9k'0¿§$9A'G4B&459Y1©"T$f459p§45f$&'E©!#9pe § $&1D)h1©'045@1DnB&'0"TÛ'¼ "T!g)n189C45)1©z!g)61©'G45ff!g}¥4 f'G§45f$'0z!gEDeEh!#9ph7)ggf'45)@!g$ f'B&9k'0"¢h!g)s0{' 1©'¼9C% ï UUU i S©R¢rgv5R¢XAvy}¥ff 'G$&n'0@}¥'0)Åh"¢$&hn)gQB&9A!#B&'09C187'¼ "T!g%(%6!#)Å18! !g1© h1©45)459A5s 7zn'¼"¢187!#)ÅB&9k!g§7'0z41©!g$&9p!#EF18' x©ÊÊÊ h1©45)459A s Ê 45"¢6h$ h'0"¢18!g)67h"T$hh'0 !g)'G45hB&'0"¢13!gEF18'Gn1845)2459A@45)@}É 1D}¥45 )"Tf$'0 s x 137z)!g1D182'GB&$&9CB&!#n'G!#E 1©zB45B&'09u1©!459kg§$&'G1©451D1©' xhÊÊÊ h1©45)459A@ 1©' '0n13B&!ghn f'G¾!g45187)gg»hB&!#)21 h1©45)459A $&1D9Y451©'09p18!45"T"¢'0B&1D18'Gn1845)2459A@45zg§¸'¼)Å45)@B&9A!g§'45) )1©9A!#$"T1©!#)Å18!1©z$h'§s º !#9pE©$ff&'01©45fz"¢!#)2h$&f1D1©'Gn1845)459A 182'0%(h'0f¸'0Ï x©ÊÊÊ l5m5ú5ê Â§op!g '¼1D45fks l5m5ú Ó5Ñs S » ` c d V a V g ` U ` f d ha ` db de ` f ` V y ` f z ` U d` g y d e ` X f b b c d b e V a ` a b de ` f d e V g S ¾ ¾ T y S

"ì"ý ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ÿ ¡¢¥¤H¦"¤¥©¡©FE¢¡¡D¤¥¦¨§£¢¥©¡ {o|n}¨Û x 137z"¢f'0459u}É
Qx©ÊÊ{Ê@ú5þ Ó45ff!g}¥ β l Îs å 45h'18'0)67z!g} $&%(45)2z'0|r"T45)g§'G45) 1©)ÈQ4 !g$&1D)$&% '09YÀs0÷Ih)g β l Î '0hB&'0"¢45ff 45B&B&9A!#B&9C451©'GE©!g9u"¢45f"T$f4518!g9Y }¥2'¼9A'G1©'G9A'0h$&f713!gED'045"¢6!gB&'09Y451©!g)6 hB&f4 '0 § 1©'G"¢45f"T$&f74518!g9u7)6'0"T%645fks '09A'G459A'Gh'0¸'09Y45f&9A'045h!g) }¥ QxhÊÊÊ@ú5þ Óy9k'0¿§$9k'0 18245137ED1©' 45n'Gz)2!#1 l Î 1 %($h1 'GØs è Ùc'0f451©§' Ê 9Y9A!#9p45)d÷fB é !g)6B&45g§' l5ê5ø %('0)1©!g)'0@!g)'G9A'045h!g)01©' 9A'0h$&f1©u!gE'09C9A!#9°45)45f h'0u459A'e%($"T1©g§18'09°}¥2'¼) β uØ '0"T45$&n'e4I9k!g$&)2)gÉ'09Y9A!#9 !gE s þ ulp }¥! g f'0 g 4IE©45"T1©!g9c!gE β }¥'0)"¢!#%6B&$&1©'0645u4I9A'0f4518¸'e'09Y9A!#9 45)6'09Y9A!g9 45)45f h'0 459k'45f%(!gh1D45f}¥4 zn%(B&f'09p}¥2'0) 45n'0@!#)Å9A'¼f74518¸'G'09Y9A!g9Cs v 9A'¼f74518'0 9A'045h!g)645 18!!}¥718Å18'G'0E8E©'0"T1©¸'GB&9A'¼"¢h!g)6Eh!#9pf459Agg' 45h'0Às0op!#)n'09 β l5ø5 e l "T!g%(B&459A'0@1©! β Ø e Ós å !#1©6 n18'0%6z45¸'GÓ 18 !#EFhg§)Û"T45)us op!#)2h'09u1©'G"T!g%(B&$&1©451©!#)6!gE l5þ Ú ú s Ý '0) β Ø §l5þ z9A'0B&9A'0h'0)1©'0Þ45 l s l¢l¢l × Ø X~ 45) l5þ Ú ú 45 l s lTl¢l × Ø g s0qr! l5þ Ú ú '0|À45"¢1Cs 8mÉ!#}É'0§'09 }É'0) β l5ø5§l5þ 7 9A'0B&9k'0n'0)18'0d45z × l5ø g }¥'09A'GQ 18''0|À45'0"¢%(45f&7gg1DE©!g9 l5þ s å $&1 l5þ Ú ú 9A'0B&9k'0n'0)18'0d45 l × l5ø g }É"TÅ45 !#)f !g)' 713"¢!#9C9A'¼"¢1Cs x )Ågg'0)'09C45f § 45n' l5ø "¢45) f!gh'G$&B@18!Ò 18 h!1©451D4B&9A'0"Tn!#)Å!#EFe'0|r'0"T%645f&g§18 "T45)Å45§'45) '0EûEh'0"T1©§'eB&9A'0"Th7!#)45uf!g} 45uÓ5eI»ÒI9C4518'09°1845)ÓÀe )459 É 1©5s¢qr7)"T'ef459Agg'e§45f7$&'0 !gE β 45¸'Á1©'0h'eB&9A! f7'¼%6 }¥ x å i "¢!g!#n' β l5ø Eh!#9c1©u h1©'0%ÚyÒ ê Î ù Ð )2f x å i Èg)!g}¥ E©!g9ph$&9A' § $131©'09A'G459A'G18}É!B&!#nh f7'G9k'045n!#)Às02'GÛ&9Ch1D )"T9A'045h'0 '0|ÀB!#)'0)1D9C45)g§'gstqr)g§f'GB&9A'0"Tn!g)6!g)61©'G h1©'0%ÚyÒ ê Î45 β l5ø5 e ø s 8m¥'0)"¢' 1©'Ángg)2Û&"¢45)69k'0¿§$&79k'0uØ5Ó 1©5s¢qy)"T'e182u%($&n1&Û1&)1©!yÒ5Ø 1© 1©uf'045§'0 êI 718 E©!g9°1©''0|rB&!#)2'0)145)2Å!#)2'E©!g9°1©'hg§) 1Cs{$&ñ182'%(45gg)218$'!#E39A'0B&9A'¼n'0)1©4 f' )$&% '09Y 9C45)gg'0 E©9A!g%Æ4 !g$&1 1©!ç4 !#$1 s0r!g§'01D4h%6f459 '0|ÀB!#)'0)1D9C45)g§'G}¥'0) β Øy}¥!g$&f@9A'0¿§$&9A' m 18 !gED'0|rB&!#)'0)21 f'045¸)gQ!g)f Ø5Ø 1©ñEh!#9u18'Ingg)7Û&"T45)2 s ¬m¥!#}É'0§'09 13}¥45 ½ $&h13B&!g)18'0Þ!g$&13182451{}¥2'0) β l5ø5 18' '0EûEh'0"T1©§'ÁB&9A'0"Tn!#)"T45) 'Á45uf!#} 45 Ó5e»2Ò Ø l 1©5s Ê ¸'0)ç}¥!g9Ch' }¥'0) β Ø1& B&!ghh f'G1©!çg§45)Å45)6'0|À1©9C4 1D!gEDB&9A'0"Tn!#)ÅÌ©45 '0|ÀB&f45)2'0@f451©'¼9p)Å187zn'¼"¢187!#) Íh h! 1©' β Ø%(45"¢)'G45 Ø5Ò 18 !#EFB&9A'0"Tn!#)Å18!"T!g%(B&459A'G}É18Å49Y45)2gg'G!gEDØ l s Ø5Ó 1© E©!#9p1©' β l5ø %(45"¢)'§s v )!g18'09FB!#nh f' '0|rB&f45)451©!#)ÉE©!g9F"¢!#!gh7)g β l5ø 45°18!!}É18Éh2E©1©)gDs Ý '0) 45)gQ1©}¥!ç¾!g45187)gg»hB&!#)213)$% '09C ED1©'09u'0|rB&!g)'0)1©z459k'GE8E©'09A'0)1 !g)'G!#EF18' hg§)Û"T45){z}Éff&45¸'G18! 'GhEh18'0@1©!%(45È§'G182'G9Y45|¥B&!g)1©zf7)'G$&B hf!g}¥)g !g}¥)182'Á!gB&'09Y451©!#)cs x )ç1©' β l5ø5 e l n18'0% 45ff¸182'Á)$&% '09Y '018}É'¼'0) l 45) l5þ 245§'1©'n45%(''0|rB&!g)'0)1 45)Þh!É)!ÉnE©1©)gçñ9A'¼¿§$9A'0Å}¥2'¼)(45)2gç45) !#E 16 26 – 1626 228

yï7ð#C Cñ u7 "ì¢ 1©' B h l Î þ B&!ghh f'eB&459Cu!gE{h1©)"T1)$% '09CuEh9k!g% 1©ph'01s "m¥!g}¥'0§'09 )1©' β Ø e Ó n18'0% 18'0n'G)$&% '09Y 45¸'G'0|ÀB&!g)'0)1©z9C45)g§)gQE©9A!#%ÇÎ1©!Ò 45) h2E©1©)gQz9A'0¿§$&9A'0@Eh!#9 ê Î!gED1©' l Î þ B&459C5s x )6%(!gh1D%(!g'09C)6459A}¥459A' 1©'GB&'09CE©!#9C%(45)"¢'Gg§45)'0 g 45¸!#7)gQ4hEh1DE©!#9p4 h$ h'013!gE3!gB&'09Y45){ñz)'0g§fgg f' 45)Þn!ç18'Gh%(45ff&}¥! g f'I!gE β Ø%(45È§'¼z131©' B&9A'0E©'09Y4 f' 45h'§s v )!g1©'09c45¸45)1©45gg'e!gE$h)g β ØIu1©451182'¼9A'ep4I}¥4 18!¥g§45) 45)6'0|r189C4 713!gEDng§)Û&"¢45)"T'§s °qy)"T'¾&!g451©)gg»hB&!#7)1D)$&% '09Y 459k'45f}¥4 )!g9Y%645f~5'0 1©'G%(!gh13ngg)7Û&"T45)21 13!gED1©'Ingg)2Û&"¢45)Þ7z45f}¥4 l5 45)@1©'09k'G )!9A'045h!g)61©!ç}É45h1©'G4 1D!gEDn18!g9Y45g§'G9A'0B&9A'¼n'0)1©)gQ1Cs º !g9Y%64518 1824513$h'G1© 189C"TÈ 459A'Gh45d18!45¸'G4P q v5v5WYr 1Cs x 1D}É45z45f9A'045 B&!#7)18'0d!#$&1D) è º f!g451©)g§»©B&!g)1 º !#9C%(451© é !g)B&45g§' l5ê ÓI184511©p9k'0¿§$&9A'0p4IhB&'0"T745f¸"T!g)§'0)218!g)E©!g9°Îs¢'e%6'¼1©!g g§§'0)61©'09A'G}¥45 1©451D45)6'0|ÀB!#)'0)1D!gEDW tp » l 45)2@4ngg)7Û&"T45)2@!gED45ff&~5'09A!g 9A'0B&9k'0n'0)18 )!g1 § $&1D9C4518'09pÎs xhÊÊÊ(ê5þ Óen)g§f' B&9A'0"¢h!g)ç '0)"T!g'0()çÒ5Ø 18u$&h)2g l 12Eh!#91©'Áhg§) rúe 18uE©!#9 1©'I'0|rB&!g)'0)1 45)@Ø5Ò 1©zE©!g9u1©'Ing§)Û&"¢45) s 8m¥!g}¥'0§'09 1D$&h'0z4{'¼) 1 h! 1©'Ihg§)Û&"¢45)ÞñØ5Ó 18zÌ©e Ø5Ó Í© '0¸'0)(182!#$&g§(13ñ'0)"¢!#'0Þ$h)g!#)2f Ø5Ò 1©5s èÞ8Ût t}toÝn ' x©ÊÊÊ h1©45)459A@'0Û)'0zEh!#$&9pE8E©'09A'0)1DB&9A'¼"¢h!g)À0h)g§f' !g$ f' h)g§f'0» '0|À1©'0)'0 45)2@!g$ f'0»h'¼|r1©'0)'0 s x ) ê5þ Ó n)g§f'G45)@!g$ f'GB&9A'0"Th7!#) "T!g9Y9A'0hB!#)d9k!g$&g§f 1©!ç}É451D%(!gh1D¾!#451©)g§»©B&!g)1D459A}¥459A'GB&9A!#¸'05stqr)g§f' B&9A'0"Tn!#)É!g"T"¢$&B&'0°4en)g§f' Ò5Ø 1}É!#9A !g$ f' B&9A'0"Tn!#)ç18}É!"T!g)n'0"T$&1©§' Ò5Ø 71 }¥!g9A5s Ê |À1©'0)'0ÅB&9k'0"¢h!g)Qp4IE©!g9Y%6451182451!#E8E©'09Yp451f'045n14If7181©f'e'0|r189C4IB&9A'0"Tn!#) 45)@'0|rB&!g)'0)1D9Y45)g§'GÌ©y4 f' » l5Í s R#SreØhaV#XBXF`nVnfåadbeØV#Çô`¬wBfradoe£`ö`ngDXFBeôYhaæeF`öye£f b£YyBeô`öf%j¥R%#h#ni×V#YdeFb£Bj£eD¨gØBdeHÀ RrnRriXBV#jô`¨R×R Â Vnd dfåhe£Bd `öa d eØhahy`öV8d b¨¨b£gØfrV#yp¬Fa`FS C¢2¡uï D ¡ "kký {~}#5nD"}#"æ}# Parameter Format Single Single- Extended Double Double- Extended B ØtÓ ÒtØ þ Ò ø Ó W t P l Ø ê l ÎtØtÒ P l ÎtØtÒ > ltø Ò ú Ò 1.0 2e min 1 – ×

"ì¢« ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ' x©ÊÊÊ h1©45)459A@!g)f hB&'0"¢Û&'0
4f!g}¥'09 !#$)@!g)6!g} %(45) '0|À1©9C4 1© '0|À1©'0)'06B&9A'0"Tn!#)B&9A!g§'0ÀsT{'e%()%6$&% 45ff!g}¥4 f'Á!g$ f'0»©'0|À1©'0)'06E©!g9Y%6451 h!g%('0187%('0z9A'0E©'09C9k'0d18!45 Ð%5© q S 8äTb¢XYdR¢S '0¸'0)61©!#$ggÅ18'184 f7'Gh!g}¥ 1D$&h7)g ê5m 718Às0'G9A'045h!g)6 1©451D459A}¥459A'G%(B&f'0%6'0)18451©!g)z!gEF'¼|r1©'0)'0@B&9A'0"Tn!#) )!g9Y%645ff !#)Ät1D$&n'G4{'¼) 1 45)dh!}¥!g$&f@$h' ú Î9C451©'09u1©45) ê5m 1©5s 'h1©45)459A@B&$&1©z1©'G%6!#n13'0%6B&45h !g)6'0|r18'0)'0dB&9k'0"¢h!g) %645Èg)gQ)! 9A'0"T!g%(%('0)451©!g)y"¢!#)"¢'09Y))2g!#$ f'ñB&9A'0"Tn!#) t $1§n189A!g)g§f 9k'0"¢!#%6%('0))g1©451 ï dIe5utWYdIWAr§S©R¢S q bTr ss P5b¢fgutv s f§e5e5bTXYS&S©P5WW ôÏ5S©WYr§v5WYv ãäTb¢XAdIR¢S p b¢XYXkW s e5bTr§v q r0çS©bpShP5WF q v5W s S&R sYqtp äTbTXYdIRTS s fge5e5b¢XYS©WYv Õ … Ð )'G%(!g18¸451©!#)ÅE©!g9u'0|r18'0){'¼dB&9A'¼"¢h!g)6"¢!#%6'0zEh9k!g%Æ"¢45f"T$f4518!g9Y }¥2"T6}Éff !gE©18'0)çnB&f4 yl Îg§18 À $&12$&n' l Òg§18 )218'09C)45ff s å {hB&f4 )gy!g)f yl Î!gE&18' l Òg§1© 1©'G"¢45f"T$&f451©!g9u45B&B'¼459C 18!1©'G$&n'¼9p45 4 è5 f45"¢È !#| é 1©451D"T!g%(B&$18'0 '0|ÀB!#)'0)1©45f "T!gh)2'¼ '01©"rs01©! l Îg§18 !#EF45"T"¢$&9C45" s º !g9p182'G"T45f"¢$&f451©!#9p18! "T!g%(B&$18'GEh$&)"¢18!g) fÈg''¼|rB f!ggQ45)@"¢!# 1©!ç}É182) l Î7gg1©z}É1©69A'045h!g)4 f' '0EûÛ"T'0)" T 1&)'0'0u4IE©'0} '0|À1©9Y4Ig§18u1©!y}¥!g9YÈ¥}É1©s x 1&n)Ät1&459A(1©!yÛ&)64Ih%(B&f7' 9C4518!g)45f&'0|ÀB&9A'0hn!#)61©451345BB&9k!g|À7%(4518'0zf7!#g}É1©(45)6'09C9A!#9u!gE þ Î5Î$&)1©z)6182'If45n1 B&f45"¢'gst$& "T!g%(B&$18)2g}É1© l Òg§1©zg§§'0 45)645)2h}¥'09p"T!g9Y9A'0"T1D1©! l Îg§18Às å È§'¼'0B&7)gQ18'0n'G'0|À1©9Y4yÒg§18 '0) 18'"T45f"T$f4518!g9pB&9A'¼n'0)1©z4n%(B&f'G%6!#'0f18! 1©'G!#B'¼9C451©!#9Cs R#SrDd×d×b£f%yBhgFjdb'V#eFVngFi×`öUod`ögFyB`öy XFfr`öd×haæhb£gDeFVnahd8eôhdab£c¤aæhjôgØhwBd×V#gØyeô`öd×VnBa`¬deFVnd V#ad eØ` hy`öaædoXBfr`öd×haæhb£g V#dfrb£aæa eØhd×eDd×V#frfrfXFfrb£XFV#jôVndhb£gDd×b£BYye£`¬yBb£gF`¬b£gDdeF` gØd `öYn¡## hd eØb£BdohgØd×fr`öV#aæhgFjdeF`¬dfôd×Y`¬dhWH`l'V#eØV#g Rrn##×S W t » l Ø ø ≤ » l ÎtØtØ » l ÎtØtØ ≤ » ltø Ò ú Ø Ê |ÀB2!y)')1§}¥t&1û¥t) t18 ú ≤ lTl lnl ltþ º !y9Y%Q4t1g}¥t&1ût) t18 ÒtØ ÓtÒ ø Ó êtm C¢2¡uï D ¡ "kký {~}#5nD"}#"æ}# Parameter Format Single Single- Extended Double Double- Extended

yï7ð#C Cñ u7 "ì¢ò Ê |r18'0)'0dB&9A'¼"¢h!g)6)Å182' x©Ê{ÊÊ h1©45)459A@n'09Y¸'¼ 4n%(f459pE©$)"T1©!#)cs x 13'0)4 f'0 f 9C459Y7'¼ 1©!ç'0E8Û&"¢'0)1©f "¢!#%6B&$&1©'G¿§$&45)1©1©'0z1©!}¥1©)64 !#$1ps þ ulp )6n)g§f'GÌ©!g9 !g$ f' Í B&9A'¼"¢h!g) g§¸)gQ182'G$&h'09p!gED1©!#n'Gf 9C459Y7'¼ 4n%(B&f'G%6!#'0f )45%('0f 1©451D'¼45"¢6B&9C%(1©¸'G!#B&'09C4518!g) § 'G1D4h%6B&f'G%($f18Bf !g9p45)6)¸!g"T451©!#)Å!#EFf!#g 9A'018$&9C) 4§45f7$&'G45"T"¢$&9C4518'G1©!}¥1©)64 !#$1ps þ ulp s 8m¥!g}¥'0§'09 }¥'0)Å$&h)2g '0|À1©'0)'0@B9k'0"¢h!g) 1D %(B&!g9Y1©45)1D1©!ç%645Èg'Gn$&9A'G1824513718 $&h'z1©9Y45)2hB&459A'0)1D18! 1©'G$&n'¼9s º !g9p'0|À45%(Bf' !#)Å4"T45f"¢$&f451©!#9 ED1©'G)218'09C)45f&9A'0B&9k'0n'0)18451©!g)6!gED4 nB&f4 '0d§45f$&'z)!g1D9A!#$&)2'0@1©!ç1©'Gn45%('GB&9A'0"Tn!#)Å45z1©'GnB&f4 ¢ 182'¼)Å182' 9A'0h$&f1D!gEDEh$&9Y1©'09p!#B&'09C45187!#) }¥f7f&'0B&'0)@!g)61©'G7'0)6g§1©z45)d45B&B&'0459 $&)B9k'0"¢184 f'G18!1©'G$&n'09Cs r!ff$&n189C451©'G'0|À1©'0)'0@B&9A'0"Tn!#)ÅE©$&9C1©'09 "T!g)n'09u1©'GB&9A! f'0%Æ!gEF"T!g)§'09C18)2g '01©}¥'0'0) xhÊÊÊdê5þ Óh)g§f'GB&9A'0"Tn!g)645)@{'¼"¢%(45fks x '045ff ¢ n)ggf7'GB&9A'¼"¢h!g) )$&% '09Y }¥ff 'GB&9Y7)18'0@}¥7186'0)!g$&g§6gg1©zn!ç1824513}¥2'¼)61©'G'0"T%645f2)2$&% '09 z9A'045 45"¢È) 182'Ih)2ggf'GB&9A'¼"¢h!g)6)$&% '09u"T45) 'G9k'0"¢!#¸'09k'0 s x 1D18$&9C)z!g$&131©451 m '0"¢%(45f&g§1©z459A'G'0)!g$&g§61©!9k'0"¢!#¸'09u4n)g§f'GB&9A'0"Tn!#) )459 )$&% '09uÌhh'0' è å )459 18! '0"¢%(45f&op!g)¸'09Yh7!#) é !#)ÅB&45gg' Ø5Ò ø5Í s Ý '0)6"¢!#)¸'09Y1©)gQ4'0"¢%(45f )$&% '09 45"¢Èy18!1© $&)¿§$' )459 9k'0B&9A'0h'0)1©451©!#) 4e9A!#$))gy'09C9A!#945 h%(45f7fg45 l ulp E©451©45f § '0"¢45$&h'G713}Éff&g§§'18'G}É9A!#)gQ45)n}¥'09Cs 8mÉ'¼9A'G 4n18$4518!g)6}É'09A' '0|À1©'0)'0@B9k'0"¢h!g)6 §1©45f&Eh!#9p45)6'0E8Û&"¢'0)1D45fgg!g9Y718% s Ý '0)6n)g§f'0»©'0|À1©'0)'0@7 45¸45f4 f' 4¸'09 h1©9C45gg1©E©!g9Y}É459A@%('01©!#d'0|Àn18 E©!g9p"T!g)§'09C18)2g4y'0"T%(45f )$&% '09c18!y4n)g§f'ÁB&9A'0"Th7!#) )459 !g)'§s º 9Ch1&9A'045()ç1©' m {'¼"¢%(45f¸gg1©u45 45) )1©'0gg'09 k g§)!#9C)gQ1©'G'0"T%645f&B&!g)1s º 9A!g%Ær4 f' » l5 e ≥ Ò5Ø 45)@n)"T' l Î X£Ø X~ ≈ Ós Ò × l Î X k "¢45) '9A'0B&9A'0h'0)1©'0Å'0|À45"¢18f )(h)g§f'0»©'0|r18'0){'¼us×¥'0|À1{Û&)2 1©'G45B&B&9A!#B&9C451©'GB&!#}É'09 l Î Xñ)2'¼"¢'0hn459 18!n"T45f' ks0{z}Éff 'G4"T!g% )45187!#)Å!#E 1©'Á'0|ÀB&!g)'0)1&!gE&1©'Á'0"T%645f§)$&% '09 1©!#g§'018'09c}¥1©ç1©'ÁB&!gh1©!g)ç!gE&1©'ÁÌ©$&B($&)1©f )!g} Í gg)2!#9A'0@'0"T7%(45f&B&!g)1s0op!#%6B&$&1©' l Î '8s x E Pgk Bg ≤ l Ò 1©'0)61© 45fh! 9A'0B&9k'0n'0)18'0d'¼|r45"T1©f ¢§ '0"T45$h' l Î h ~ Ø h ~ þ h ~ 45) þ h ~£QØ X~ s º )45ff %($f18Bf Ìh!#9 ¸'GEFe £@Î Í kæ45) l Î '8s x EF18 f45n13!gB&'09Y451©!g)6 !g)'G'0|À45"¢18f ¢ 1©'0)61©' "Tf!gh'0n1 )459 )2$&% '¼9p 9A'¼"¢!#¸'09A'¼us è å )459 1©! '0"¢%(45f&op!g)§'09Ch!g) é !g) B&45g§' Ø5Ò ø n!g}¥p2!#} 1©!É!¥1©'f45h1{%($&f1©B&f Ì©!g9°¸' Í '0|r45"T1©f s$pE©!g9 gk Bg ≤ l Ò 18'Á$&h'Á!#E&1©' n)g§f'0»©'0|À1©'0)'0(E©!g9Y%(451&'¼)24 f'0 m 7gg12'0"¢%(45f§)$&% '09Y 1©! ' "T!g)¸'09Y1©'0618!¥18'e"¢f!#n'0h1 )459 )$% '09°Ì©ks '§s¢'0|À45"¢18f 9A!g$&)'0 Í s x E Bgk Bg l Ò 1©'0) h)2ggf'0»h'¼|r1©'0)'0@ )!g13'0)!g$&g§6Eh!#9p1©'G4 !g§'G45fg§!g9Y1©%Ç18!45f}¥4 "T!g%(B&$&1©'G1©' '0|À45"¢18f 9A!g$&)'0 )459 '0¿§$§45f'0)1 § $1Dou!g!g)'0) Ï l5m5ú Ó5ÑFh2!#}Éz1©451D1Dz'0)!g$&g§ 1©!g§$&459C45)1©'0'Á18245121©'Á"¢!#)2§'09Ch!g)ç!gE )459 1©!y'0"T%645f§45) 45"TÈ¥}Éff§9A'0"T!g§'09c1©' !g9Yg§)45f )459 )$% '09Cs

"ìký ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ x ED!g$ f7'GB&9A'¼"¢h!g)6 h$B&B&!g9Y1©'0
182'0)61©'G45fg§!#9C182%Æ4 !g§'}¥!g$&f 'G9A$)6) !g$ f'GB9k'0"¢h!g)69C4518'09p1©45)6n)g§f'0»©'0|À1©'0)'0 § $&1D18!"¢!#)¸'09Y1D!g$ f'B&9k'0"¢h!g) 1©!ç4 l5ê g§13{'¼"¢%(45f)$&% '09u45) 45"TÈQ}É!#$f@9A'0¿§$&9A'G182'G!#$ f'0»©'0|r18'0){'¼ E©!g9Y%6451Cs o¡ Ýn Ût£¢ qy)"T'G1©'G'0|rB&!#)2'¼)213"¢45) 'GB&!#n187§'G!g9u)2'0gg451©§' n!#%6'G%('01©!#d%($&n1 'G"T2!#n'¼) 1©!ç9A'0B&9A'0h'0)1D1©zngg)cs0r}¥!"¢!#%6%(!#)Å%('01©!#{z!gED9A'0B&9A'0h'0)1©)gQhg§)'0@)$% '09C 459A'Ghg§)Úy%(45g§)18$'G45)@1©}¥!gÄtz"¢!#%6B&f'0%('0)1s0qygg)Úy%(45g§)1©$&'G 1©'G h1©'0% $&n'0ÅE©!g9°1©'ngg)(!#E31©'hg§)Û"T45)Þ)Q1©' xhÊÊÊ Eh!#9C%(451©5!g)' 1{ñ$&n'0Å18!É!gf 1©'Ghg§) 1©'G9k'0n13!gED1©' 1©z9A'0B&9A'¼n'0)131©'G%(45g§)1©$&'G!gED1©'G)$&% '09Cs0'G1©}¥!gÄt "T!g%(B&f'0%6'0)1D9k'0B&9A'0h'0)1©451©!#)Åz!gE©1©'0)6$&n'0@)Å)18'0g§'09p459Y1©%('01©"rs x )Å18 h"¢'0%(' 4I)$&% '¼9c7)ç1©'Á9C45)g§'¥Ï»hØ X¤ sh Ø X¤ sh » l Ñu9A'¼B9k'0n'¼)218'0 g 1©'Án%(45ff'0n1&)!#)2)'0gg451©¸' )$&% '09p18451D "T!g)g§9A$'0)1D18!1D%(!g$&f!Ø X¤0s ' xhÊÊÊ( )459 h1©45)459A(!#'0 )!g12$&n' '01©'09!gE&18'0n' %6'¼1©!g 1©!9k'0B&9A'0h'0)1&18' '0|ÀB!#)'0)1 § $&1D)h1©'045@$&n'0z4y q R s WYv9A'0B&9A'¼n'0)18451©!g)s x )61©'G"¢45h'G!gEDn)ggf7' B&9A'0"Tn!#) }¥'09A'G1©'G'0|ÀB&!g)'0)1D h1©!#9A'0@) ú 1© 18' 45 l Ø ê ÌhE©!g9u!g$ f' B&9A'0"Tn!#)Å1D l Î5Ø5Ò Í s Ý 451D182z%6'¼45)2z 1©451DE z1©'G¸45f$&'G!gED1©'G'0|rB&!#)2'¼)21 1©ñ)1©'09YB&9A'018'0Þ45ñ45)Q$&)2hg§)'0Å)218'0g§'09 1©'0)(18'I'0|ÀB&!g)'0)1{!gE{1©'I¾&!g451©)gg»hB&!#7)1 )$&% '09c » l Ø ê s¢2uu!gE©1©'0)"T45ff'061©'ef§rg q R s WAvIW ôÏ5e5b¢rgWYr§S1©!yn18)2gg$hE©9A!g% 1©' 45n'0@'0|ÀB&!g)'0)1 s Ùc'0E©'09Y9C)gQ1©!çr4 f' » l !#)ÅB&45gg' l5m5l5 h)g§f'GB&9A'0"Tn!#)Å45 W tp l Ø ê 45) W tp » l Ø ø s0{'G9A'045h!#)ÅE©!g9u245§)g ¥gkW tp ge£@W tp h!1©451D18'9k'0"¢B&9A!#"T45f!#EF182' h%645ff'0h1D)$&% '09 }¥f7f&)!g13!g¸'¼9C¾!#}¥s v f18!g$&g§61D 1©9A$'G18451D1©' 9A'0"TB&9A!#"¢45f&!gED1©'Gf459Ag§'¼n1D)$&% '09u}Éff&$)'09Y¾!g} $&)2'09Y¾!#} $&h$45ff f'0h h'09C!#$z1©45)Å!#¸'09Y¾!#}¥s è å 45h' é !g)6B45gg' l5m Î'0|rB&f45)'0@1©451DW tp » l $&n'¼dE©!g9 9A'0B&9k'0n'0)18)2gÎ 45) è qyB&'0"T45f ¥G$&45)2181©'0 é !g)6B45gg' l5m5ê }Éff&)1©9A!#$"T'G4$h'GEh!#9 W tp¦ P l s x ) xhÊÊÊ h7)ggf'B&9k'0"¢h!g) 1© %('045) 18451D1©' 745h'0@'0|rB&!#)2'¼)218 9Y45)2gg' '01©}¥'0'0)6W tp » l » l Ø ê 45)@W tp¦ P l l Ø ú5 }¥'09A'045z1©'G$) 45n'0@'0|ÀB&!g)'0)1© 9C45)gg' '018}É'¼'0)6Î45)ÞØ þ5þ5 }¥"¢(459A'I'0|r45"T1©f 1©'I)2!#))2'¼g§451©§'I)2$&% '09Cñ1©4513"T45) 'G9A'0B&9A'0h'0)1©'0@$&n)g ú 1©5s k k k 1 2e min ⁄ ( )

yï7ð#C Cñ u7 "ì"k § Û¨Þ%|n¢5¤Ýn£} ' x©ÊÊÊ h1©45)459A@9A'0¿§$&9A'0z1©451D182'G9A'¼n$&f1D!gED451©!g) h$ 1©9Y45"¢18!g) %($f18B&f7"T451©!#)Å45)@¸h7!#) 'G'0|À45"¢18f 9A!g$&)'0 s0{451D 1©'G9A'0h$f1D%($&n1 ' "T!g%(B&$18'0@'0|r45"T1©f 45)@1©'0)69A!g$&)'0d18!1©'G)'0459A'0h1D¾!#451©)g§»©B&!g)1D)$&% '09 Ì©$h)g¥9A!#$)(1©!y'0§'0) Í s è e$459k g§18 é !g)çB&45g§' l5ê5ú B&!g)1©'0(!g$&1&1824512"¢!#%6B&$&1©)g 1©'G'0|À45"¢1DEûEh'¼9A'0)"¢'G!#9ph$%Æ!gED1©}¥!¾!g4518)g§»©B&!g)1D)$% '09Cz"¢45) 'G¸'09 '0|ÀB'¼)2h¸'G}¥'0)Å18'09p'0|rB&!#)2'¼)218 459A'Gh$ h1©45)18745ff EûEh'¼9A'0)1s024513n'0"T1©!#) )1©9A!#$"T'0@g§$&459A@g§1© }É"TÅB&9k!g¸'G4B&9C45"T1©"T45f&}É4 !#EF"T!g%(B&$&1©)g E8E©'09A'¼)2"T'0 }¥f'Gg§$&459C45)1©'0'¼7)gQ18451D1©'G9A'0f4518¸'G'09Y9A!g9u7zn%(45ffks 8m¥!g}¥'0¸'09 "T!g%(B&$18)2g}É18Å4h7)ggf'gg$&459A@{gg1D}Éff&)!g1D45f}¥4 zg§¸'G18'h45%('G45)2h}¥'09p45 "T!g%(B&$18)2g1©'G'0|À45"¢139A'0h$&f1D45)@1©'0)69A!#$&)2)gDs å )1©9A!#$"T)gQ4h'0"¢!#)@g§$&459A g§1D45)@41©9A s S qtp sT` 1 E8E©'09k'0)"¢'0z"¢45) '"T!g%(B&$&1©'0@451D!#)2f 4f1©18f'G%6!#9A' "T!gh1218245)É}¥1©É4Án)g§f' gg$&459AQg§1 À $11©' 9k'0n$&f1 18' n45%(' 45°E21©'z{EûE©'09A'0)"T' }¥'09A'G"T!g%(B&$18'0d'¼|r45"T1©f 45)@1©'0)Å9k!g$&)'0 Ïe!gf '09Ag l5m5m Î5Ñs0{$& 18'h1©45)459k "T45) 'G%6B&f'0%('0)1©'0@'0EûÛ"T'0)1©f s Ð )'G9A'¼45n!#)ÅE©!g9u"¢!#%6B&f'018'0f nB&'0"TE )gQ1©'G9A'0h$&f1© !#EF459Y1©%('01©"e!#B&'09C4518!g) 18! %(B&9A!g§'G1©'GB&!g9Y1©4 f71 !gEDn!#E©1©}¥459A'§s Ý '0)Å4B&9A!#g§9Y45%Ç %(!g§'0 '¼1©}¥'0'0)Å18}É! %(45"¢)'0 45) !g1©6h$B&B&!g9Y1 xhÊÊÊ 459C18%6'018" 1©'0)6EF45) )1©'¼9C%('0451©'G9A'0h$&f1 E8E©'09Y 1{%($&n1 ' '0"T45$h'e!gE{h!gE©1©}¥459A' $&gg )2!#1Eh9k!g% E8E©'09k'0)2"T'0p)Q459Y1©%('01©"rs v )!g18'09p45¸45)1845g§'G!gEDB&9A'0"Tn'GhB&'0"¢Û&"¢4518!g)6 1©451D1D%(45Èg'0 1D'045h'09p18!9A'045h!g) 4 !g$&1D¾!#451©)g§»©B&!g)1s Ã 9A!g!#Ehz4 !g$&1D¾!g4518)g§»©B&!g)1D459A'G459A@'0)!g$&g§ }É18!g$&1 45¸)gQ18!'045f&}É1©6%($f18B&f7'G"T45h'0 459Y7h)gQE©9A!g%Æ%($f187B&f'GÈg7)z!gEF459Y1©%('01©"rs ¨ $h1D45z)218'0g§'09uB&9A!ggg9C45%( "T45) 'GB&9A!#¸'0)61©! 'G"T!g9Y9A'0"T1 n!ç"¢45)6¾!g4518)2gg»©B!#)1 B&9A!#g§9Y45%6 45f1©!g$&g§6}É451DzB&9A!g§'0)67)61©451D"T45h'G7z1©451D182'G9k!g$&){)gQ'09Y9A!#9p!gE 1©'G9A'¼n$&f1Dh451©nÛ&'0 "T'09Y1©45) !#$)5st'0!g9k'0%ÇÓ 45)6'0|À45%6B&f'G!gEDn$&"TÅ4B&9A!#!gEs '0n'B&9A!#!gE©ñ459A'%(45'%6$&"T('045h'09 }¥2'0)Q1©'!gB&'09Y451©!#)2 '0)gç9A'¼45n!#)2'¼Þ4 !g$&1 459A'GB&9A'¼"¢h'0f nB&'0"TÛ'¼us Ð )2"T'G45)645f7gg!g9Y1©%ÇzB&9A!g§'0)Å18! 'G"¢!#9C9A'¼"¢1DE©!g9 xhÊÊÊ 459C18%6'018" 1D}¥ff}¥!g9YÈQ"T!g9Y9A'0"T1©f !g)645) %(45"¢)'Gn$&B&B&!g9Y1©)gQ1©' xhÊÊÊ h1©45)459A s å 9A!#}É) Ï l5m5ú5l ÑF45 B&9A!#B&!gh'0@45|r!g%( E©!g9u¾!g4518)2gg»©B!#)1D1©451D)"¢f$&'G%(!gh1D!gED1©' '0|À7h1©)gç¾!#451©)g§»©B&!g)1{459A}¥459A'§s ¨m¥!g}¥'0¸'09 B&9A!#!gE©ñ)(18ñ n18'0% "T45))!g1{§'09CE 1©'G45fg§!#9C182%(z!gEFh'0"T1©!g) §è op45)"¢'0ff451©!#) é !#)ÅB&45gg' l5ê5m 45)2 §è5Ê |r45"T1©f Ù!g$&){'¼ Ð B&'09Y451©!#)2 é !g)6B&45g§' l5ú5þ5 }¥2"T69A'0¿§$&9A'GEh'¼451©$&9A'0 )!#1DB&9A'0h'0)1D!g)645ff459A}¥459A'gs º $&9Y1©'09C%(!#9A' å 9A!g}¥)Ät 45|À!g%( 459A'G%(!g9k'G"¢!#%6B&f'0|¥1845)Åh%6B&f '0Û&))2g !gB&'09Y451©!#)2z18! 'IB'¼9CE©!g9Y%6'0Þ'0|r45"T1©f 45)@1©'0)69A!#$&)2'0 s¼{$&zB&9A!g§)gQ18'0!g9A'0%( E©9A!g% å 9A!#}É)Ätz45|r!g%( z$h$&45ff %6!#9A'GE8Û&"¢$&f1D18245)6B&9A!#¸)gQ1©'0%Æ45nh$&%6)g !gB&'09Y451©!#)2z459A'G'0|À45"¢18f 9A!g$&)'0 s

"ì"x ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ '09A'I7ñ)2!#13"¢!#%6B&f'018'G45gg9A'0'0%('0)13!g)6}¥4513!gB&'09Y451©!g)z4¾!g4518)2gg»©B!#)1Dh1©45)459k h2!#$&f7@"T!g§'09Cs x )Å451©!#)Å18!1©'
45n"Á!gB&'09Y451©!g) P » × 45) Ú 1©' xhÊÊÊ h1©45)459A@45fn!hB&'0"TÛ'0z1©451Dh¿§$459A'G9k!g!#1 9A'0%(45){'¼9 45)@"¢!#)¸'09Yn!#) '01©}¥'0'0) )1©'0gg'0945)Q¾!#451©)g§»©B&!g)1 ' "T!g9Y9A'0"T1©f 9A!g$&)'0 s x 145fn!9A'0¿§$&9A'0°1©4512"T!g)§'09Ch!g) '01©}¥'0'0)6)218'09C)45f&E©!g9C%(4518 45)d'0"T%(45f 'G"T!g9C9k'0"¢18f 9A!#$)'0@Ì©'0|r"T'0B&1DE©!g9p§'09 f459Agg'G)2$&% '¼9C Í s0Üñ$fh"¢645)diw9C45)È§'09IÏ l5m5ú5ø ÑD45¸'GB&9A!#B&!gh'0d45)gQ))'09 B&9A!#$"T1D1©!ç1©'Gfn1D!#EF!#B&'09C45187!#) 1©451D459k'GB&9A'0"T7h'0f hB&'0"¢Û&'0 s0{' )!#1©'G1©451 }¥2'¼)())2'¼9 B&9A!g$&"¢18ñ459A'"T!g%(B&$18'0Þ) xhÊÊÊ 459Y1©%6'¼1©" 1©'Û)45f45)n}¥'09 "T45) ' ¿§$&1©'G}¥9A!g)gDs º !g9p'0|À45%(Bf'Gh$&%6z459A'G4nB&'0"T45f"T45h'G!gEF))'09pB&9A!#$&"¢18 45)d18' h$%ÆÌ©ÌhØ × l Î s~ g P l Î X~ g Í » l Î X~ g Í » l Î s~ g '0|À45"¢18f '0¿§$45f&18! l Î s~ g ¸ $&1D!g)64%645"T)' }¥1© xhÊÊÊ 459Y1©%('01©"e18'G"¢!#%6B&$&1©'0@9A'0h$&f1D}Éff 'G» l Î s~ g s x 1DzB&!ghn f'G1©! "T!g%(B&$18'G)2)'09uB9k!g$&"¢18 18!}É182) l ulp }É1©6f'0h 459A}¥459A'G1©45)61D1©45Èg'0 1©! %(B&f7'¼%6'0)1D4E©45n13%6$&f1©B&f'09GÏÜz9k"¢)'09p45)@Üz$&fn l5m5ú5ê Ñs 2 v ff&1©'G!gB&'09Y451©!#)2z%6'¼)218!g)'0@)Å182'Gh1©45)459A@459A'G9A'¼¿§$9A'¼d18! ''¼|r45"T1©f 9A!#$)'0@'0|r"T'0B&1D"T!g)§'09Ch!g) '018}É'¼'0)Å'0"T%645f&45) )459 s02'G9k'045n!#)Åz1©451 '0EûÛ"T'0)1D45fg§!g9Y1©%( E©!g9u'0|r45"T1©f 9A!#$&)2)gQ45ff&1©'G!gB&'09Y451©!g)z459A'GÈ§)!g}¥) '0|r"T'0B&1 "T!g)¸'09Yh7!#)s º !g9u"¢!#)2§'09Ch!g) 1©' '0n13È§)!g}¥)Å'¼E8Û&"¢'0)1D45fg§!#9C182%(zB9k!g$&"¢' 9A'0h$&f1© 18451D459A'Ghf7gg1©f }¥!g9Yn'G18245)6'0|r45"T1©f 9A!g$&)'0d!#)'0yÏop!g!#)'0) l5m5ú Ó5Ñs ' x©ÊÊÊ h1©45)459A@!g'0z)!g1D9A'¼¿§$9A'G189C45)h"¢'0)'0)1©45f&E©$)"T1©!#)2z1©! 'G'0|r45"T1©f 9A!#$)'0 '0"¢45$&h'G!gEF18'S©RTutWudIR "sTWYX t©s v q uwWYdIdIRrstr!ç7ff$&h1©9C4518' n$&B&B&!gh' !#$@459A' %(45È§)gQ41©4 f'G!gED1©'G'0|rB&!#)2'0)1845fE©$&)2"T1©!#)Å18!ÓB&f45"¢'05s0{'0) '0|ÀBÌ l s ø Ø ø5Í þ s Î ú Ò þ s0qr2!#$f@1© 'G9A!#$&)2'0@1©! þ s Î ú Òy!#9 þ s Î ú Ó ùÁx EF'0|ÀB&Ì l s ø Ø ø5Í "T!g%(B&$18'0@%6!#9A'G"¢459k'0E©$ff ¢ 1 '0"T!g%('0 þ s Î ú Ò þ Îs v )@1©'0) þ s Î ú Ò þ Î5Îs v )d18'0) þ s Î ú Ò þ Î5Î5Îs0qy)"T'G'0|rBÞ 1©9Y45)n"T'0)'0)1©45f 1© "T!g$&f@g§!!#)Å459 189C459Yf f!g)g '0Eh!#9A' n18)g§$&n)gQ}É'018'09p'0|rB&Ì l s ø Ø ø5Í 7 þ s Î ú Ò þ Î5Î … Î5v5v5v!g9 þ s Î ú Ò5Ó m5m5m … m v5v5vs0$ 17°)!g1B&9C45"T1©"T45fg1©!hB&'0"¢E 184512182'zB&9A'0"Tn!#)É!gE21©9Y45)n"T'0)'0)1©45fgE©$&)2"T1©!#)2 ' 182' h45%6' 45 E&182' }¥'09A' "T!g%(B&$18'0(18!)2Û&)1©' B9k'0"Tn!g)É45)(1©'0)É9A!g$&)'0 s v )!#1©'09 45B&B&9A!#45"¢6}É!#$f 'G1©!hB&'0"¢E 189C45)h"¢'0)'0)1©45f&E©$)"T1©!g)z45fg§!g9Y1©%("¢45ff s å $&1 1©'09k'G{!#'0 )!#1D45B&B&'0459p1©! 'G4h7)ggf'45fgg!g9Y1©%Ç1824513}É!#9CÈg }¥'0ff45"T9A!#nz45f7f 459A}¥459A'G459A"T718'0"T1©$&9A'05stÙ451©!g)45f&45B&B&9A!g|À%64518!g) o Ð Ù x o ve45)@f459Ag§'G184 f'0 R S b W ` V f j W ` g d a V j V hg a d hg d Y y hg j hg g ` f X f b y d d V a b g ` b c d e ` e V a hd b X ` f V dhb g a V f ` X f ` a ` g d` y e f V e V g V g y » ` ª YV g d R S S hf d e g ` f f hd ` a z d ha X b a a he Y` d b d b W X d ` hg g ` f X f b y d da d b hde hg R YX hg e V f y V f ` hg b g ` X V f d hV Y X f b y d d X ` f d Yb d « d f d Y` S e ` V y y hd hb g V YYf g ` ` y ` y e V f y V f ` d b W X V f ` a db d e ` W Yd hX Yh` f V f f V f g ` ` y ` y V g f V f c b f d e V d a X ` ` y S ¬ S ® ¯ ° ha V g V d f b g f W c b f b b f y hg V d ` ¯ b dV d hb g ° hj hd V Y b W X d` f V g y ha V W ` de b y b c d b W X dhg j d f V g a d ` g y ` g dV Y c g d d hb g a de V d a ` a W b a d Yf a e hc da V g y V y y a Àh S` Si Ç ` f f c ` W Yd hX Yhd V d hb g a V g y y hÇ ha hb g a Â y V Yd e ` f R R S d ha d e ` W ` de b y V y y hd hb g V YYf g ` ` y ` y e V f y V f ` d b W X V f ` a db de ` W YdhX Yh` f V f f V f g ` ` y ` y V g f V f c b f de V d a X ` ` y S y a ` y b g e b de d e ` g d` Y ¡ V g y de ` b db f b YV h R S

yï7ð#C Cñ u7 "ì¢¡ 459A'G1829k'0'GE8E©'09A'0)1D18'0"¢)¿§$&'0 1©451D459k'G$h'0@Eh!#9p"T!g%(B&$18)gQ1©9Y45)2h"T'0){'¼)21845f !g) "T!g)1©'0%(B&!g9Y459 %645"T)2'05s Ê 45"¢6 45B&B&9A!#B&9C451©'GE©!g9u4{EûE©'09A'0)1D"Tf45nz!gEF459k}É459A' 45)@451DB&9A'0h'0)1D)!n)g§f'G45fgg!g9C18%Ç}¥!g9YÈ§ 45"T"T'0B&1©4 f !g§'09p1©'G}¥'9Y45)g§'G!gE "T$9Y9A'0)1D459A}¥459A'§s ± ¡¡û§£¤¥ £² Ó ¤¥©¡¦¨§£¦"§£¡D Ð )6h!g%('G¾!g4518)2gg»hB&!#)2132459k}É459k''¼¸'09 Q 713B&451©1©'09Y)69A'0B&9A'0h'0)1©z4y§45fd¾&!g451©)g§» B&!g)1D)$&% '09Cs02' x å i q n18'0% ÚrÒ ê Î 45)6'0|r45%(B&f'!#EF1875s Ð )61©'G!g18'09p45) 1©' Q v ¤RS 9k'0n'¼9C§'0 h!g%(' 1DB&45181©'09Y) 1©!ç9A'0B&9A'0h'0)1DhB'¼"¢45f&)$% '09Cz"¢45ff'0 XkW s WYXYV5WYvbTe5WYXYRTr§v s s02z'04ygg!g'0 45"TÈQ18!1©'Go o ø5ø Î5Î }¥"¢645 1DB&451©18'09C) E©!g9p18'GnB&'0"T45f¿§$&45)1©187'¼ x × Ê º x × x Ê 45) x × º x × x ´³3s ' x©Ê{ÊÊ n1845)459A6"T!g)187)$&'0u)ç1©u1©9Y451©!g)ç45)2(45 NaN uÌ kb¢S&R £kIf§d¼WYX Í 45) )Û)1©'05s Ý 182!#$1345) nB&'0"T45f¿§$&45)1©18'0 1©'09k'G7z)!g§!#!g@}É4 1©!45)f' '0|À"¢'0B&187!#)45fh1©$&451©!#)2zfÈ§'G1©45Èg)2g1©'Gn¿§$&459A'G9A!#!g13!gEF4)'0gg451©¸'G)$&% '09 !g18'09 1©45)64 !g9C18)gQ"¢!#%6B&$&1©4518!g)s0÷I)'09 x å i q h1©'0%ÚrÒ ê Î º Ð Ù3Ù v × 1©'G'0E©45$f1 45"T1©!g)6)Å9k'0nB&!#)n'G1©!ç"¢!#%6B&$&1©)gQ182'Gh¿§$&459A'G9A!g!#1D!#EF4)'0g§45187§'G)$% '09pfÈ§'G»©Ó 9A'0h$&f1©ñ)(182'B&9C)1©)gç!gE345)Q'09C9k!g9 %('0nh45g§'§sqy)"T'I'0§'09 ç 13B&45181©'09Y)(9A'0B&9k'0n'0)18ñ4 ¸45f@)$&% '09 1©'G9A'018$9Y)Å§45f$&'!#EFh¿§$&459A'G9A!#!g1D%($&n1 ' h!g%('G¾!g4518)2gg»hB&!#)213)2$&% '09s x )Å182'G"T45h'!#EFq h1©'0%ÚyÒ ê Î º Ð Ù3Ù v × 9k'01©$&9C)'0 s x ) xhÊÊÊ 459C1©%('01©" 4 NaN z9A'01©$&9Y)2'¼d)61© h1©$&451©!#)cs ' x©ÊÊÊ h1©45)459A@hB'¼"¢Û&'0 1©'GE©!gff!g}¥)gQhB'¼"¢45f&¸45f$&'0 Ì©h'0'Gy4 f' »hØ Í ± Î '0)!g9Y%645f~5'0Q)$% '09C ±∞ 45) NaN Ì©1©'09A' %(!g9k' 1©45)ç!#)2' NaN 45 '0|ÀB&f45)2'0 )61©'G)2'¼|r1Dh'0"¢18!g) Í st'0h'GnB&'0"T45f§45f$'¼ 459A'G45ff&'0)"¢!#'0d}¥1©6'0|rB&!#)2'¼)218 !#E '01©'09uW µ¤¶w·P l !g9 W t¡ » l Ìh13}¥45z45f9A'045 B&!g)1©'¼@!g$&131©4513Î245z45)('0|rB&!#)2'0)13!#E W tp » l5Í s C¢2¡uï D ¡ "kký ¸¹ ºØ%"2» %l Exponent Fraction Represents W W lp » l ä Î ± Î W W lp » l ä ≠ Î W l ≤ W ≤ W l ¼ l s ä × 2° W W dpðP l ä Î ∞ W W lpðP l ä ≠ Î ×y4t× 4 – 2 = 0.f 2e min ×

"ì"Ñ ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ NaN r9C451©!#)45f7f T 1©'G"¢!#%6B&$&1©4518!g)6!gEDÎÚyÎ!g9
245 '0'0)Å189A'04518'0d45z45) $&)9A'0"T!g¸'¼9C4 f'G'09C9A!#9p}¥7"T6"¢45$&h'0 4"¢!#%(B$&18451©!g)61©!45f1Cs ~m¥!#}É'0§'09 18'09A'G459A' '0|À45%6B&f'0z}É'09A'G1D%(45È§'¼ h'0)2h'GE©!g9p4"T!g%(B&$18451©!#)Å18!"¢!#)1©)$'G)6n$&"TÅ4 h1©$&451©!g)s0op!g)h'09p4n$ 9A!#$18)2'G1824513Û) 182'G~5'09k!gz!gEF4E©$&)2"T1©!#) ä h4 zero(f) s0r9C451©!#)45f7f T ~5'09A!çÛ)'09Cz9A'¼¿§$9A'G18'G$&n'09u1©!ç7)B&$&1D45)6)1©'09Y¸45f°ÏR ¼Ñ !g)6}¥2"T61©'GEh$&)"¢18!g)6 '0Û&)2'0@45)@!g§'09p}¥2"TÅ18'G~5'09A!Û&)2'09u}Éff&n'¼459A"¢s 451D 1©'Gh$ 9A!#$&1©)'G7z"¢45ff'0@45 zero(f a b) s v %(!g9A'G$&n'¼Eh$&f&~5'09A!Û&){'¼9 }¥!g$&fd)!#1D9A'0¿§$&9A'G1©'G$&n'09u1©!)B&$&1D1© '0|À1©9Y47)E©!g9Y%(451©!g)s02z%6!#9A'Gg§'0)'09Y45f ~5'09A!çÛ)'09pz'0nB&'0"T45ff 45B&B&9A!gB&9Y451©'GE©!g9p"T45f"¢$&f451©!#9C }É'09A'G1Dz)451©$&9C45f&1©! h%6B&f Èg' )Å4E©$&)2"T1©!#) 45)@45}ÉÈg}É459kd18!1©'0)645¸'G18!nB&'0"TE 1©'G!g%(45)s m¥!g}¥'0¸'¼9 1Dz'045 1©!h'0'G}É %(!gh1D~5'09k!Û)'09Y 9A'0¿§$&9A'G4!g%(45)s0{'G~5'09A! Û)'09u!g'0 18 }¥!g9YÈ g B&9A! )gQ1©'GE©$)"T1©!#) f 451D§459C!g$&z¸45f$&'0Às x ED1DB&9A! '0 E©!g9 4¸45f$&'I!#$18n'I182'!g%(45)6!#E f 1©'I"T!g'IE©!g9 f %(7gg13}¥'0ff2"¢!#%6B&$&1©'ÎÚyÎ!#9 45)(18'Á"T!g%(B&$&1©45187!#)ç}¥!g$&fQ245f1 $&))'0"¢'0hh459Cf 4 !g9Y1©)gy1©' ~5'09A!Û&)2)g B&9A!#"¢'0h5s 7zB&9A! f'0%Æ"¢45) '45§!g'0 g )1©9A!#$"T)gQ4nB&'0"T45f§45f$'G"T45ff'0 NaN 45) hB&'0"¢E )g¥18451&182' "¢!#%6B&$&1©4518!g)ç!gE&'0|ÀB9k'0nh!g) fÈ§'ÁÎÚrÎ45)2 B&9A!#$"T' NaN 9C4518'09p18245)645f1©)gDs v fn13!gEFh!g%('G!gED1©'Gh1©$&451©!g)z1©451D"T45)Å"T45$h'G4 NaN 459A' g§§'0)67)6r4 f' »©Òs0{'0)6}¥2'0) zero(f) B&9A! '0z!g$&1©h'18'G{!#%(457)6!gE f 1©' "T!g'IE©!g9 f }¥f7f29k'01©$&9C) NaN 45)2Þ182'I~5'09k!çÛ)'09u"T45)("¢!#)218)$'§s¼4513 zero(f) p)2!#1 è B&$&)n'0 é E©!g9°%(45È§)gç45)Q)"¢!#9C9k'0"¢1g§$&'0hÀs Ý 1©Q182p'0|À45%6B&f')Q%()2 1 '045 18!n'¼'}¥451D1©'G9A'¼n$&f1D!gED"¢!#% ))gQ4 NaN }¥1©645)6!g9A)459 ¾!g45187)gg» B&!g)1D)$&% '09un!g$&f '§s0qr$B&B&!#n'G18245131©'GÛ)45f&n18451©'0%('0)1D!gE f return(-b + sqrt(d))/(2*a) s x EDv ½£ÞÎ 18'0) f h!g$&fd9k'01©$&9C)64 NaN stqr)"¢' v ¾£QÎ sqrt(d) u4 NaN 45) -b + sqrt(d) }¥ff ' 4 NaN E18'Án$&% !gE&4 NaN 1 – 1 – 1 –

yï7ð#C Cñ u7 "ì"ì 45)@45) !g1©'09u)2$&% '¼9p 4 NaN s0qy%(f459Cf ED!g)'G!gB&'09Y45)2@!#EF4¸h!g) !gB&'09Y451©!#)Åz4 NaN 182'G¿§$&!g18'0)1Dn!#$f '4 NaN s x )6g§'0)'09Y45f }É'0)'0¸'¼9p4 NaN B&459C18"¢B&4518'0 )64y¾&!g451©)g§»©B&!g)1D!#B&'09C4518!g) 1©'G9A'0h$f1Dz45)!g1©'09 NaN s v )!g18'09p45B&B&9A!g45"T61©!}¥9C18)2g4y~5'¼9A!h!gf¸'¼9p1©451D!#'0n)Ät1D9k'0¿§$&79k'G1©'G$h'09p18! )B&$134!g%(45)(7ñ1©!ç$&h'Ghg§)45f5s0'I~5'09A!#»hÛ&){'¼9u"¢!#$fÞ)n1845ff&4ngg)45f&45)f'09 E©!g9p¾&!g451©)gg»hB&!#7)1D'¼|r"T'0B&1©!g)5st'0)6E f }É45z'0¸45f$&451©'¼d!#$18n'G1©z!g%(45)Å45) 9C45h'0Å45)'0|r"T'0B&1©!g) "¢!#)2189A!gf¸}É!#$f 'e9A'018$9Y)'0Å1©!¥182'e~5'09A!¥h!gf¸'09Cs¢{'eB&9A! f'0% }¥1©61© 45B&B&9A!#45"¢6 1©451D'¼¸'09 f45)g§$&45g§'G45 4E8E©'09A'0)1D%('0182!#@!gEF45)f)g hg§)45f Ì©EF1D45 4%('01©!#d451345f7f Í© 45)dh!1D45 )!!#B&'!#EFB&!#9C184 f1 s x ) xhÊÊÊ@ê5þ Ó NaN z459A'G!#Eh18'0)Å9k'0B&9A'0h'0)1©'0@45 ¾!#451©)g§»©B&!g)1D)$&% '09Y }¥1©61©' '0|ÀB!#)'0)1DW tp¦ P l 45)@)!g)~5'09A!hg§)Û"T45)Às x %(B&f'0%('0)1©451©!#) 459A'GE©9A'0'G18!B&$1 h1©'0%(»©{'¼B'¼)2'0)13)Eh!#9C%(451©!#)6)1©!ç1©'Ghg§)Û&"¢45) s0$z18'09A'Iz)!g134$&)¿§$' NaN § $1D9Y451©'09u4y}¥!gf'GE©45%6f !#E NaN 5s Ý 2'¼)Å4 NaN 45)d45)6!g9A)459 ¾!g4518)g§» B&!g)1&)$&% '09c459A'Á"T!g% 7)'0 1©'Á9A'0h$f1&h!g$&f 'Á1©'eh45%('e45 1©' NaN !gB&'09Y45) s $zEF1©'G9A'¼n$&f1D!gED4f!g)gQ"T!g%(B&$18451©!g)6 4 NaN 1©'G h1©'0%(»h'0B&'0)'0)1 )Eh!#9C%(451©!#)ç)É1©' hg§)Û"T45)Q}Éff ' 18' )2E©!g9Y%(451©!g)É1©451}É45°g§'0)'09Y451©'0Q}É'0) 1©'GÛ&9Ch1 NaN )Å182'G"T!g%(B&$&1©451©!#)Å}¥45 gg'0)'09C4518'0 s v "¢18$45ff T 18'09A'G 4"T45¸'0451D18! 1©'Gf45h1Dn18451©'0%('0)1s x E !g1©6!gB&'09Y45) 459A' NaN 1©'0)61©'G9A'0h$f1D}¥ff 'G!#)2'G!#E 1©!#n' NaN § $&1D1D%(g§1D)!g1 'G182' NaN 1824513}É45zg§'0)'09C4518'0@Û9Yn1Cs ¿5XÀÁ&l¢lÂ ¨ $h1D45 NaN B&9A!#¸'G4}É4 18!"T!g)1©)$&'G4y"T!g%(B&$&1©451©!#)Å}¥'0)Å'¼|rB&9A'0hn!#) fÈ§' ÎÚyÎ!#9 459A'G'0)"T!g$&)1©'09A'¼ )Û)187'¼ B&9A!g§'G4}É4 18!"T!g)1©)$&'}¥'0)645) !g§'09Y¾!g}Ë!g"T"¢$&9C5s0{z7z%($"TÅh45E©'09p18245)6h7%(B&f 9A'¼1©$&9C))gQ1©'Gf459Ag§'¼n1 C¢2¡uï Ã ¹ }#)æ%{~Äl)¡l}ö{~l2ºØ NaN Operation NaN Produced By P ∞ PçÌ8» ∞ Í | Î × ∞ Ú Î¸ÚÀÎ ∞ Ú ∞ Ù Ê i ÏÁÙ Ê i Î ∞ Ù Ê iÆ` Ì8}¥¸'¼) Ï¾QÎ Í x 1 –

« "ý"ý ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ 9A'0B&9k'0n'0)184 f'G)$&%
'09Cs v 45)6'0|À45%6B&f' "T!g)n'09u"¢!#%6B&$&1©)g }É'0) β l Î e Ò 45)6W tp¦ m5ú s x E ¥Ï Ò × l Î XÅ g 45)6` Ó × l Î XÅ g 18'0) þÏ }¥ff¸!g§'09C¾&!g} 45) 'G9A'¼Bf45"T'0 §m s m5m × l Î X s0qy%(f459Cf ` 45)2 ¢Ï P@` }¥ff&'045"¢6!g§'09C¾&!g} ) 1©$&9Y) 45) 'G9k'0B&f45"¢'0 gQm s m5m × l Î q s0qr!1©'GÛ)45f&9A'0h$&f1D}Éff ' }É"TÅ 9Y45n18"¢45ff }¥9A!#)2g301©'G"¢!#9C9A'¼"¢1D45)h}É'¼9 þ × l Î XÅ g s x ) xhÊÊÊ 459C182%('018" 182' 9A'0n$&f12!gE ãÏ ∞ 45u ` Ï P(` 45) s qy!y1©'eÛ&)45f¸9A'0h$f1& ∞ }¥7"Tun45E©'09c1©45)9A'01©$&9Y)2)g¥45)!g9A)459 ¾!g4518)2gg»©B!#)1 )$&% '09p18451D )!g}¥'09A'G)'0459p182'G"T!g9Y9A'0"T1D45)n}¥'09Cs '§n!#)Å!#EFÎ g Î9A'0h$f18 )Å4 NaN s v )!g)~5'09A!ç)2$&% '09p¸'0 g Î !g}¥'0¸'¼9 9A'¼1©$&9C) )Û&)21 l ÚyÎ ∞ » l ÚyÎ » ∞ s0'G9A'045h!g)6Eh!#9p1©'Gn18)"¢187!#)Å 1©507E 8äTÌ öÏ Í → Î45)ÉÌ nÏ Í → Î45 Ï45B&B&9A!#45"¢'0 h!g%('Gf%(71 1©'0) äTÌ nÏ Í ÚTÌ nÏ Í "¢!#$&f7 45¸'45) ¸45f$&'§s º !#9 '0|À45%6B&f' }¥'0) QänÌ nÏ Í n) øÏG45)¥Ì öÏ Í Ï 182'¼) QänÌ nÏ Í Ú¢Ì öÏ Í → l 45 QÏ → Îs å $&1{}É'0) QänÌ nÏ Í l »3"T!g Ï äTÌ öÏ Í ÚTÌ nÏ Í → Îs Ý '0)(18)2Èg)2gç!#E3ÎÚyÎ45ñ182' f%(718)g¥h1©$&451©!#)ç!gE&4¿§$&!g18'0)1&!#E&1©}¥!y§'09 h%(45f7fg)2$&% '09C ÎÚyÎe"¢!#$fQ9Y'¼B9k'0n'¼)21 45) 182)g3s2$&p)Q1©' xhÊÊÊ n1845)459A ÎÚyÎI9A'0h$&f718p)Q4 NaN s å $&1}É'0) p Î äTÌ öÏ Í → p RTr§vzÌ nÏ Í → Î 18'0) äTÌ öÏ Í Ú¢Ì öÏ Í → ±∞ E©!g9u45) 45)245f 1©"eE©$&)"¢187!#) ED45)2@gDs x E Ì öÏ Í < ÎE©!#9ph%(45ff ãÏ 1©'0) änÌ nÏ Í Ú¢Ì öÏ Í → » ∞ !#1©'09Y}Éh'G1©'If7%(13 P ∞ s0qy!ç182' xhÊÊÊ h1©45)459AÅ'0Û&)'0 p ÚyÎ ±∞ 45pf!#)2gÉ45 p ≠ Îs'hg§)Q!#E ∞ '0B&'0)p!g)1©'hg§) !gE p 45)@Îy)61©'G$&n$&45f&}É4 T h!18451D» l ÎÚrÎ » ∞ 45)@» l ÎÚr»hÎ +∞ s X³¸!g$Þ"¢45) n18)g§$&n '¼1©}¥'0'0)Ågg'01©18)2g ∞ '0"¢45$&h'G!gED!g¸'¼9C¾!#} 45)@g§'01©18)g ∞ '0"T45$h'G!gE ¸h!g) § ~5'09k! § "T'0"TÈ§)gç1©'n18451©$&ñ¾45g§pÌh}¥"¢Q}Éff 'n"T$&nh'0Þ)Q'01©45f )6n'0"T1©!#) è º f745gg é !#)ÅB&45gg' Ø l Î Í s0'G!g¸'¼9C¾!#} ¾45gQ}¥ff 'Gh'01D)61©'GÛ9Yn1D"T45h' 1©'G¸h!g) g ~5'09A!¾&45gQ)Å182'Gh'0"T!g) s '9A$f'GE©!g9u'01©'09Y%6))gQ1©'G9A'0h$&f1D!gED45)Å!#B&'09C4518!g)61©451D45 )Û)1 45 45) !gB&'09Y45)@ñn%(B&f'§¼9A'0B&f45"¢'I)Û&)21 }É1864Û&)218'I)2$&% '09 óÏ45)2Þ1845È§'I1©'If%(71{45 Ï → ∞ st$& ÒÚ ∞ Î § '0"T45$h' sGqy%(f459Cf T Ó» ∞ » ∞ 45) ∞ s Ý '0)Q18'f%(71!#'0n)Ät1'0|À7h1 1©'e9A'0h$f1p4 NaN n! ∞ Ú ∞ }¥ff '4 NaN Ì©r4 f' »©Òy!#)ÅB&45gg' l5m5m 45z45{18!g)45f&'0|r45%(B&f'0 Í s0{z45g§9A'0'0z}É18Å18' 9A'045h!g))gQ$&n'¼d18!"¢!#)"¢f$&'G1©451DÎÚyÎh!g$&f 'G4 NaN s Ý '0)64h$ '0|ÀB&9A'0hn!#)6'0¸45f$&451©'¼ 18!ç4 NaN 18'G¸45f$&'G!#ED1©'G'¼)2189A'I'0|rB&9A'0hh!g)6 45fh!4 NaN s x )61©'I"¢45h'I!gE ±∞ !#}É'0§'09 18'G§45f$'I!gED18'G'0|ÀB&9A'0hn!g)(%6gg1 'I45) !g9k)2459 ¾!#451©)g§»©B&!g)1D)$&% '09 '0"T45$h'G!gED9Y$&f'0 fÈ§' l Ú ∞ Îs 8m¥'09A'Gz4 R S g hg ` X b hg d z Yde b j e d e ` y ` c V Yd hg T T T V f hd e W ` dhd ha db f b g y b Ç ` f ½ b ` y g W e ` f a db ∞ i hd ha X b a a he Y` d b d e V g j ` de ` y ` c V Yd Àa ` ` Æ ¯ b g y hg j b y ` a Ç b g X V j ` ¡ Â x2 y2 + 9.99 1098 × 3.16 1049 × = x2 y2 + 3 x ⁄ x ∞ → lim 0 = ∞

yï7ð#C Cñ u7 « "ý¢ B&9C45"T1©"T45f&'0|r45%(B&f'G1©451D%(45È§'0z$h'G!gED1©'G9Y$&f'0 E©!g9u7)Û&)71 459Y718%6'018"ys0op!#)2h'09 "T!g%(B&$18)2gÉ1©'E©$&)"¢187!#) øÏÚyÌ öÏ P l5Í s¢{pñ4 45ÅE©!g9Y%6$&f4 # '0"¢45$&h')!g1!g)f }¥ff 1D!#¸'09Y¾!g}Ë}É'0) ëÏ7zf459Ag§'¼9p1©45) § $&1D)Û&)21 459C182%('018"e}¥7ff&g§§' 1©'G}¥9A!g)gQ45)h}É'¼9 '0"¢45$&h'13}Éff '0f@Î 9Y451©'09u1©45)Å4)$&% '09p)'0459 l Ú Ïs m¥!g}¥'0¸'¼9 ÏÚrÌ nÏ P l5Í "T45) 'G9A'0}¥9Y7181©'0)645 l ÚrÌ nÏtP¢Ï sh Í st %(B&9A!g§'0@'0|rB&9A'0hh!g) }¥f7f2)2!#1D!#¸'09Y¾!g}ËB&9A'0%(4518$9A'¼f 45) '0"¢45$&h'G!gED)Û&)21 459Y1©%('01©"Á}Éff&45§'G1©' "T!g9Y9A'0"T1D¸45f$&'G}É'0) ëÏ Î l ÚyÌ©Î tP@Î sh Í l ÚyÌ©Î tP ∞ Í l Ú ∞ Îs Ý 1©!g$&1D)Û&)21 459C18%6'018" 182'G'0|ÀB&9A'0hn!#) l ÚrÌ nÏtP¢Ï sh Í 9A'¼¿§$9A'0z41©'0h1DE©!g9 óÏ Î }¥"¢6)!g1D!#)2f 45 '0|À1©9Y4y)h1©9Y$&"T1©!g) § $&1D%(4 45fh!n9A$B&1D4B&B&'0f)'§st '0|À45%6B&f' ff$h1©9Y451©'0z4g§'0)'09Y45fE©45"T1 )45%6'0f 18451D)Û)1 459Y1©%('01©"e!#E©1©'0)645¸!#7z1©'G)'0'0 E©!g9phB&'0"T45f"T45n'G"T'0"¢Èg)g§Â§!g}¥'0¸'09 E©!g9Y%6$&f45 )'0'0@1©! 'G"T459A'0E©$&f7f 7)hB&'0"¢18'0@1©! %(45È§'Gh$9k'G1©' !)!g13245§'GnB&$&9C!#$ '0245§!g9u451D)2Û&)1 Ì©45 ÏÚrÌ nÏ P l5Í Í s È ÊÉ& ÛtËÍÌøÛ¨Þ~Ý Î '09k! 9A'¼B&9A'0h'0)218'0 § 1©'G'0|rB&!g)'0)1DW tp » l 45)@4~5'09A!hg§)Û"T45) stqr)"¢'G1©' hg§) 1D"T45)61©45È§'G!#)Å18}É!EûEh'¼9A'0)1D¸45f$&'0 1©'09A'G459A'G18}É!~5'¼9A!g PÎ45)d»©Îs x ED4 n18)"¢187!#)Q}¥'09A'e%(45'e}É'0)"¢!#%6B&459Y)2g P&ÎI45)26»©Î h%(Bf'e18'0n18pfÈ§' if (x = 0) }¥!g$&fd45§'G¸'09 $&)2B&9A'¼7"T1©4 f' '045¸!g9 '0B&'0))2g!g)61©'Gngg)Å!#E x s02$& 1©' xhÊÊÊ n1845){459kd'0Û&)'0 "T!g%(B&459Ch!g)6n!ç1©451 PÎ »©Î 9C4518'09p1©45)6»©Î ½£ PÎs v f182!#$ggÅ13}É!#$f 'B&!#nh f7'G45f}¥4 z1©!gg)2!#9A'G1©'Ghg§)6!gEF~5'¼9A! 1©' xhÊÊÊ h1©45)459A@!g'0z)2!#1D!h!3s Ý '0)Å4%($&f718B&f7"T451©!#)Å!#9p¸h!g)6)2§!gf§'0 4hg§)'0 ~5'09A! 1©'G$h$&45fhg§)69Y$&f'0 45B&B&f )6"¢!#%6B&$&1©)gQ18'hg§)6!gED1©'G45)n}¥'09Cs0{$& Ò ⋅ Ì öPÎ Í PÎ 45) ¢PÎÚy»©Ò »©Îs x EF~5'09k!d)!#1D45¸'G4ngg) 18'0)Å182'G9k'0f451©!g) l ÚyÌ l Ú Ï Í ÏG}É!#$&f7ÅEh45f1©!É!gfÞ}¥'0) Ï ±∞ s¼'I9k'045n!#)(ñ1©451 l Úy» ∞ 45) l Ú P ∞ !g18Å9A'¼n$&f1D)ÅÎ 45) l ÚyÎ9A'0h$&f1© ) ëP ∞ 1©'Gngg)Å)E©!g9Y%64518!g)6245§)g '0'0)Åf!#n1Cs Ð )'e}¥4 1©!¥9A'0h18!g9A'e182'e'0)1©1 Él ÚyÌ l Ú Ï Í ÏGp18!¥!g)f 45¸'e!#)2'eÈg)26!gE)2Û&)1 ¢ !g}¥'0¸'¼9p1©451D}¥!g$&f@9A'0h$f1D)61©'Gn45h1©9k!g$& "T!g)h'0¿§$'0)"T'G!gEDf!gh7)gQ18'Gng§)6!gE 45)6!g§'09C¾!#}É'0@¿§$&45)1©1 s v )!g18'09p'0|À45%6B&f'G!gED1©'G$&n'G!#EFhg§)'0@~5'09A!"T!g)"T'09C) $&)'09C¾&!g} 45)@E©$)"T1©!g) 1©451D45§'G4yh"¢!#)1©)$1 451DÎ h$"TÅ45zf!gg3s x ) x©Ê{ÊÊ 459Y718%6'018" 1Dz)24518$9Y45f&1©! '0Û&)2'ef!ggÉÎ » ∞ 45)Åf!#g ÏG18! '4 NaN }É'0) þÏ£6Îs¢qy$&B&B&!gh'e1©451 ¥ÏI9A'0B&9A'0h'0)1©p4 h%645ff2)2'¼g§451©§'G)$&% '09u18451D45z$&)'09C¾&!g}¥'0@1©!ç~5'09k!3s¼245)Ègz1©!çh7gg)'0@~5'09k! Ï }¥f7f 'G)'0g§451©§' n!çf7!#gQ"T45)Å9A'¼1©$&9C)64 NaN s ~m¥!#}É'0§'09 EF1©'09k'G}É'09k')!hg§)'0 ~5'09A! 1©'Gf!ggQE©$&)2"T1©!#)Å"T!g$&f@)2!#1Dn18)g§$&n645)Å$&)'09C¾&!g}¥'0d)'0gg451©¸'G)$&% '09 E©9A!g%ÆÎ 45)@}É!#$&f7@18'09A'0E©!g9A'G45§'G1©!ç9A'01©$&9C)6» ∞ s v )!g1©'09u'0|r45%(B&f'G!gED4E©$)"T1©!#) }¥1©Q4In"T!g)1©)$&1 451~5'09A!¥p182'ehg§)$% Eh$&)"¢18!g) }¥"¢9A'018$9Y)p1©'eng§)!gE{4 )$&% '09Cs ββe max 2 ⁄

« "ý¢« ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ Ã 9A! 4
f 182'%6!#n1{)218'09A'0h1©)gç$h'I!#EDhg§)'0Þ~5'09k!ç!#"¢"T$9Yñ)6"T!g%(B&f'0|459C1©%('01©"rs r!1845È§'G4n%(B&f'G'0|r45%(B&f' "T!g)h'09p1©'G'0¿§$&451©!#) s0{ "T'09C1845)2f 189Y$&'G}É'0) Ö ≥ Îs x E Ö » l5 1©'G! §7!#$& "T!g%(B&$18451©!g)6g§§'0 45) s0{$& pÀ{'GB&9A! f'0%Æ"T45) 'G1©9Y45"¢'0@1©!ç1©'GEh45"T1D1824513n¿§$&459A'G9A!#!g1D %($f18»h§45f$'¼ 45)(18'09A' )!}É4 18!n'0f'0"T121©' ¸45f$&'0 h!1©451212 "T!g)1©)$&!g$& ) 1©'G'0)1©9k'G"¢!#%6B&f'0|¥B&f45)'§s 8m¥!g}¥'0¸'09 h¿§$459k'9k!g!#1D "T!g)1©)$&!g$& ED4yXYRTr p P p f§S "T!g)nh1©)gQ!#EF45ff&)'0g§451©§'G9A'045f&)$% '09Cz '0|À"¢f$&'0dE©9A!#%Ç"T!g)h'09C45187!#)st f'045§'0 1©'GB&9A! f'0%Æ!gED}É451D18!!E©!g9u1©'G)2'¼g§451©§'G9A'045f&)$% '09C }É"TÅ459A'G!#E 1©' Eh!#9C% » öÏ¡P q Î }¥'09A' Ï ÎsTqrg§)'0(~5'09k!yB&9k!g¸'0 4eB'¼9CE©'0"¢12}¥4 18!9A'0h!gf§'Á1© B&9A! f'0%s0×¥$% '09Cz!gED1©' E©!g9Y% ÏtP q Ì öPÎ Í 45¸'G!#)2'Ghg§) 45)@)2$&% '¼9C !#EF18'E©!#9C% ÏtP q Ìh»©Î Í !g)61©' !g18'09ph{'G!#EF182' 9Y45)2"T6"¢$&1D45¸'G182'G!#1©'09phg§) s x )ÅE©45"T1 1©'G)451©$&9C45f E©!g9Y%6$&f45 E©!g9u"¢!#%6B&$&1©)g }¥f7f&gg7§'G1©'0h'G9A'0h$f18Às å 45"TÈQ1©! s x E Ö l » l P q Î 1©'0) l Ú Ö l ÚyÌ©» l P q Î Í ÏÌh» l » q Î Í ÑhÚ&ÏÌh» l P q Î Í Ì©» l » q Î Í Ñ Ì©» l » q Î Í ÚyÌ©Ìh» l5Í »FÎ Í » l P q Ì©»©Î Íh 45)@n! }¥f' s$& xhÊÊÊ 459C18%6'018"eB&9A'0h'09Y¸'0z1© '0)1©1 Eh!#9p45ff Ö s0qr!g%('G%6!#9A'Gh!gB&n187"T451©'¼d'0|À45%6B&f'0 459A'Ggg¸'0) § Üz4545)ªÏ l5m5ú5ê Ñs v f1©!g$&g§6n18)g§$&n)g '01©}¥'0'0) êP&Î45)2@»©Î245 45¸45)1845g§'0 1"T45)¥!g"T"T45n!#)245ff 'ñ"¢!#)Eh$&h7)g3s º !#9F'0|À45%(Bf' ngg)2'0ç~5'09A!'0h1©9A! 1©'Á9A'¼f451©!g) §Ï ` ⇔ l Ú Ï l Úr` }É"T Eh45fh'Á}¥2'0) §Ï P&Î45)2(` »©Îs m¥!g}¥'0¸'09 1©' xhÊÊÊ "T!g%(%6181©'0'G'0"T{'¼d18245131©'G45¸45)1©45gg'0 !#EF$&1©f~5)gQ1©'Gh7gg)Å!#EF~5'09k! !g$&18}É'0gg2'0@182'Gh45¸45)1©45gg'0Às Ï Ût Ý)Þ¨ß|nÐtÊÑ ÛtË9ÒÔÓ ßÕ¥Û'Þt} op!#)2h'09 )!#9C%(45f~5'0Å¾!g4518)2gg»©B!#)1{)$&% '09Yñ}¥1© β l Î e Ò 45)ÆW tp » m5ú s 'Á)$% '09C ¥Ï ø s ú5ê × l Î s¦Å&45)(` ø s ú5l × l Î s¦Å45B&B&'04591©! 'ÁB&'09CE©'0"T1©f !g9A)459 ¾!#451©)g§»©B&!g)1D)$&% '09Y }¥2"TÅ459k'G%6!#9A'G1©45)64Eh45"T1©!#9p!gE l Îf459Ag§'09u1©45)Å18' h%645ff'0h1D¾!#451©)g§»©B&!g)1D)$&% '09 l s Î5Î × l Î s s' 45§'4h1©9Y45)g§'GB&9A!gB&'09Y1 T 1 z ⁄ 1 z ( ) ⁄ = 1 1 – ( ) ⁄ 1 – i = = 1 1 – ( ) ⁄ 1 i ⁄ i – = = 1 z ⁄ 1 z ( ) ⁄ ≠ i x ( ) i x – ( ) 1 z ⁄ 1 z ( ) ⁄ = 1 z ⁄ 1 – i 0 – ( ) + i – = = 1 z ( ) ⁄ 1 i ⁄ i – = =

yï7ð#C Cñ u7 « "ý¢ò !g}¥'0¸'¼90 8Ï ` Îy'0§'0)61©!g$&g§ ëÏ ≠ ` )pÀ{'G9A'045h!g)6 18451 Ïe»&` s Î ø × l Î sÅ ø s Î × l Î s¦&°18!g!h%(45ffg1©! ' 9k'0B&9A'0h'0)1©'0Q45°4e)!g9Y%(45f7~5'0Q)$&% '09 45)Qh!%($&n1 'G¾$&n'0@1©!~5'09k!3s 8m¥!g}Ë7%(B&!g9Y1©45)1Dz1D1©!B&9k'0n'09Y¸'G182'GB&9A!#B&'09C1 Ï8@` ⇔ ÏD` 8@Î ù Ì l Î Í x 18Ät §'09 '045 1©!%(45g§)'G}É9Y1©)gQ1©'G"T!g'GE©9C45gg%6'0)1 if (x ≠ y) then z = 1/(x-y) 45)d%($&"¢6f451©'09u45¸)gQ4B&9A!ggg9C45%ÆEh45f&$&'G1©!4 hB&$9Y!g$&p¸h!g) § ~5'¼9A!3s¢r9C45"TÈ§)gÉ!g}¥) $&g§uf7Èg'e1©ppE©9Y$&h1©9Y451©)gÉ45)Å187%(' "T!g)n$&%()gDs Ð )64%6!#9A'GB&7f!#n!#B&2"T45f&f'0¸'0f "T!g%(B&$18'09ph"¢'0)"T'G1©'0|r1 !g!gÈg !#Eh18'0) B&!g)1D!#$131©451D'0§'0)Å182!#$&g§6713 "T$9Y9A'0)1©f %(B&9C45"T1©"T45f&1©!B&9A!#¸'Gf459Agg'GB&9A!ggg9C45%( "T!g9Y9A'0"T1 '0ngg)2)gQB&9A!#g§9Y45%( }¥718Å18'G7'04!#EFB&9A!#¸)gQ182'0%Æ!gE©1©'¼)Å9A'¼n$&f1©z7) '01©18'09p"T!g'§s º !#9p'0|r45%(B&f' )1©9A!#$"T)gQ)2§459C45)18 ¿§$&1©'G$&n'¼Eh$&f '0¸'0)6EF18' 459A'0)Ät13g§!g)gQ18! 'G$h'0@45 B&459C13!gED4yB&9k!g!gEs º f!#451©)g§»©B&!g)1D"T!g'G ½ $h1DfÈ§'G45) !g18'09p"T!g'§01D'0fB& 1©!ç245§'GB&9A!g§4 f'E©45"T1©z!g)6}É"¢61©!ç'0B&'0)2 s º !g9p'0|À45%6B&f' }¥2'¼)Å45)45f ~5)gQE©!g9Y%6$&f4Ì ø5Íh 1D}¥45 §'09 '0fB&E©$&f18!È§)!g}Ë1©451 ÏÚyØ ¾£` ¾£QØ 'Ï ⇒ Ï ` ÏG»3`¼sqr%6f459Yf ¢ È§)!#}É)gç182451Ì l Î Í ñ189Y$&'%(45È§'0p}É9Y1©)g 9A'0f4 f'I¾&!g451©)gg»hB&!#7)1{"T!g''045n'09Cs x E31{ñ!g)f 1©9Y$&'Eh!#9 %(!gh13)$&% '09Y 1{"¢45))!g1 'G$h'0@1©!B&9A!#¸'G45) 1©)2g3s ' xhÊÊÊ n1845)2459k6$&n'¼u'0)2!#9C%(45f~5'0 D)2$&% '¼9C }¥7"Tgg$459Y45)1©'0'ÁÌ l Î Íh 45u}É'¼f7f 45 !#1©'09p$&h'0E©$f&9A'0f4518!g)Às0' 459A'G182'G%(!gh1D"T!g)1©9k!g§'09Ch45fB&459Y1D!gED1©'Gh1©45)459A 45)ÞB&9k! 4 f 45"T"T!g$&)1©'0ÅEh!#9 1©'f!g)gç'0f4 )Qg§'01©18)g ê5þ Ó45B&B&9A!#¸'0 sij!#n1{gg B&'09YEh!#9C%(45)"¢'G459A}¥459A'G1©451D"Tf45%6z1©! ' xhÊÊÊ "¢!#%(B4518 f'G!#'0 )!g1Dh$&B&B&!g9C1 '0)!g9Y%645f~5'0@)$&% '09Y 9A'0"T1©f ¢§ $&1D9C4518'09p1©9Y45B& }¥'0)Å"T!g)h$%()gQ!g9 B&9A!#$"T)g¥'0)!g9Y%(45f7 45)(f'045§'0 7121©!h!gE©1©}¥459A'Á18!yh%($f4518'Á1©' x©ÊÊÊ n1845)2459k s 2 ''04 '0)2@'0)!g9Y%(45f7~5'0@)$&% '09Y g§!#'0 45"TÈQ1©!e!#f7 '09AgwÏ l5m5ø5ê ÑD45)d ¸'¼9 h%(Bf'§s Ý '0)61©'G'0|rB&!#)2'¼)2137zW tp 18'Gng§)Û&"¢45)@!g'0z)2!#1D45¸'G18! ' )!g9Y%645f~5'0 h!1©451D}¥'0) β l Î e Ò45)@W t¡ » m5ú5§l s Î5Î × l Î s )!f!#)g§'09 1©'Gh%645ff'0h1D¾!#451©)g§»©B&!g)1D)$&% '09 § '0"¢45$&h'Îs m5ú × l Î s 45fh!4¾!g45187)gg»hB&!#)21 )$&% '09Cs R S e ` f V f ` d V YY` y v É Ö t p q w hg i t p q w hg S S e ha ha d e ` d V a ` b c b g ` b c de ` W b a d df b e Y` a b W ` V a X ` d da b c de ` a dV g y V f y S × f b j f V W a d e V d c f ` Æ ` g d Yf g y ` f ½ b b c d ` g f g g b d hd ` V e Yf a Yb ` f b g e V f y V f ` d e V d a ` a a b c d V f ` d f V X a S

« "ýký ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ '09A'G 4h%645ff&n)45gQ}¥'0) β Øy45)@47'0)
1D '0)gQ$h'0 n)"T'G4 )$&% '09c}É1845)'0|rB&!#)'0)21&!#E{W l¡ }¥ff¸45f}¥4 u45¸'Á4Ih7gg)Û"T45)2(g§9A'04518'09c1©45)!g9 '0¿§$&45f&1©! l s Î '0"¢45$&h'G!gEF18'G7%(B&f"¢13f7'¼45{)g 1s0'Gn!#f7$&187!#)Åzn%(f459p1©!ç1©451 $&n'0Q18!9A'0B&9k'0n'0)1Î 45)2Q°n$&%(%(459C~5'0Q)Éy4 f' »hØe!#)çB&45gg' l5m5ê sn2' '¼|rB&!g)'0)1 W tp p$&h'0Å18!É9k'0B&9A'0h'0)1'0)2!#9C%(45f5sij!#9A'eE©!g9C%(45ff ¢ E{1©' 718p)Q182'eh7gg)Û"T45)2 Û'¼fd459A'G 5h … 5r¦suh 45)@1©'G¸45f$&'G!gED1©'G'0|rB&!#)'0)213 W 1©'0)6}É'0) W W tp » l5 1©'I)$% '09 '0)gç9A'0B&9A'0h'0)1©'0Þ l s 5hh … 5rtswh × Ø ° }¥'09A'045ñ}É'0)(W W tp » l5 1©'G)$&% '09 '0)gQ9A'0B&9k'0n'0)18'0dzÎs )h© … )rtsuh × Ø ° mØh st' P l )61©' '0|ÀB!#)'0)1D )'0'0'0 '¼"¢45$&n'G'0)!#9C%(45f 45¸'G45)6'0|rB&!#)2'¼)213!gEFW tp )!g1DW l¡ » l s Ùc'0"T45ff¸182'Á'0|À45%6B&f'Á!gE β l Î e Ò W tp » m5ú5 Ï ø s ú5ê × l Î s¦Å45)6` ø s ú5l × l Î s¦Å B&9A'0h'0)1©'0Å4511©' '0gg)2))gÉ!gE{1©pn'0"T1©!#)s Ý 1©Q'0)!g9Y%645f ÏG»2`G!#'0p)2!#1{¾&$h 1©!ç~5'09A! $&1D )n18'045@9A'0B&9A'0h'0)1©'0 g 1©'G'0)!g9Y%645f~5'0@)$&% '09Gs ø × l Î s s07 '045¸!g9u "T45ff7'¼dgg9C45$&45ffgr§v5WYX 9ubTps x 1D '045 1©!ç¸'09YE 1©451DÌ l Î Í 45f}¥4 !#f{ }¥2'¼)Å$&n)gQgg9C45$&45f&$)'09Y¾!g}ys î5 y5©¡uï7 %D£Ä¨{¤Ùð}n{ÛÚ{~ ¹ }n9Ü§ÝoÞ}o"lD"2ßól}làD{~ º gg$9k' »©ØÁff$&n189C451©'¼°'0)!g9Y%(45f7~5'0)$&% '09YÀsh'z1©!#Bç)$% '09Ff)'z)¥1©'zÛgg$&9A' h2!#}Éz)!g9Y%645f~5'0@¾!g4518)g§»©B&!g)1D)$% '09C5s0×¥!g1©"T'G1©'Gg§45B '01©}¥'0'0)6Îy45)@1©' h%645ff'0h1D)!g9Y%645f~5'0@)$&% '09 s x ED1©'G9A'0h$f13!gEF4¾!#451©)g§»©B&!g)1 "T45f"¢$&f451©!g)6E©45ff )1©!182zg§$&fE 1Dz¾$&n'0@1©!~5'09k!3s02' !g181©!#%Ç)$&% '09pf)' h2!#}Éz}É451D45B&B&'0) }¥'0)Å'0)!#9C%(45f 459A'G45'0@1©!ç1©'Gn'013!gEF¾&!g451©)gg»hB&!#7)1 )$&% '09YÀs02' è gg$fE é Û&ff7'¼d) 45)d}¥'0)Å18'9k'0n$&f1D!#EF4"T45f7"T$&f451©!g)6 f'0h 1©45) 1Dz9A'0B&9A'0h'0)1©'0 g 1©'G)'0459A'0h1D'0)!g9Y%(45fks Ý '0) '0)!g9Y%645f~5'0@)$&% '09Y 459A'G45'0@1©!18')$&% '09uf)2' 182'GhB&45"¢)g '018}É'0'0) 45 ½ 45"¢'0)1D¾&!g451©)gg»hB&!#7)13)2$&% '09Cz¸459Y'0 )649A'0gg$f459u}É4 045 ½ 45"T'0)1DhB&45"¢)g§z459A' '01©'09u1©'Gn45%('Gf'0)g§18Å!#9pE8E©'09 g 4E©45"¢18!g9p!#E β s Ý 1©!g$&1D'0)!g9Y%645f 1©' hB&45"¢)gQ4 9Y$&B&1©f "T45)2gg'0 E©9A!#% 18! }¥2"T67z4Eh45"T1©!#9p!gE 0 βe min βe min 1 + βe min 2 + βe min 3 + 0 βe min βe min 1 + βe min 2 + βe min 3 + 1.0 βe min × 1.0 βe min × β p – 1 + βe min βe min

yï7ð#C Cñ u7 « "ý"k 9C45182'09u1©45)61©'G!g9A'09Yf "T245)gg' g 4E©45"¢18!g9u!gE β s å '0"T45$&n'G!#EF1© %645) 45fg§!#9C182%( 18451D"¢45)645¸'Gf459Agg'G9A'0f451©§'G'09C9k!g9pE©!g9u)!g9C%(45f~5'0@)$% '09Cz"¢f!#n'G1©! 1©'G$&)'09C¾!#} 1829k'0n!#f7@459A'G}¥'0ff» '045§'0d)61© 9Y45)g§'G}¥2'¼)Ågg9C45$&45f $&){'¼9C¾!#} $&h'0 s Ý 18!g$&1Dg§9Y45$45f&$&)'09C¾!#} 18'h%(B&f7'G'0|ÀB&9A'0hn!#) êÏD`y"T45)6245§'G4¸'09 f459Ag§' 9A'0f4518¸'I'09C9A!#9uEh!#9u)2!#9C%(45f~5'0@)B&$18 45z}É45zh'0'0)64 !g§'GE©!g9 óÏ ø s ú5ê × l Î s¦ÅD45) ` ø s ú5l × l Î sÅhs Ô 459Agg'9k'0f451©§''¼9C9A!#9Cz"¢45)645B&B'¼)Å'0§'0)Å}¥1©!#$1D"T45)"¢'¼f7f4518!g) 45 1©'GE©!gff!g}¥)gQ'0|r45%(B&f'Gn!#}ÉÏ '0%(%6'¼f l5m5ú Ó5Ñs0op!g)h{'¼9p¸)gQ1©}¥!"T!g%(B&f'0| )$&% '09Y R üP q I45) p P q vs0'! ¸!g$& E©!#9C%($f4 ⋅ q h$EûE©'09CzEh9A!#%Ç18'GB9k! f'0%Ç18451DEF'0182'09u"¢!#%6B&!#)'0)213!gED1©'G'0)2!#%(7)4518!g9 p P q v7 f459Agg'09p1©45) 1©'GE©!g9C%($&f4}Éff&!g§'09C¾!#} '0§'0)61©!g$&g§61©'GÛ&)245f&9A'0h$&f1 %(4 Q 'G}É'0ff&}É187)69Y45)2gg'§s væ '0181©'09u%6'018!g@!gED"¢!#%6B&$&1©)gQ182'G¿§$&!g18'0)1© z1©! $&n'Gqr%618Ät E©!g9Y%6$&f4 Ì lTl5Í v B&B&f )gQqr%618Ät E©!g9Y%6$&f41©!Ì©Ø ⋅ l Î s P qtl Î s Í ÚrÌhÓ ⋅ l Î s P q Ì©Ø ⋅ l Î s Í©Í gg7§'0 18' "T!g9Y9A'0"T1D45)n}¥'09p!#EFÎs þ }É18Ågg9C45$&45f&$)'09Y¾!g}ys x 1 '0f{zÎs Ó}É18Å¾$&hÅ18!~5'09A! 45)'09Y9A!#9c!gE l Î5Î ulps s x 1&7 1 B&"T45f¸E©!g9c'0)!g9Y%(45f7~5'¼6)$% '09C 1©!yg§$&459C45)18'0'e'09Y9k!g9 !g$&) E©!g9p459kg§$&%6'¼)218 45ff&1©'G}É4 !g}¥)Å18! l s Î| s ¦BA¥û¡"¡¦¨§£¢¥©¡ á&ÿ ¡ £¤¡¤¥©¡©â¥D¤Fã¤¥©¡©¥ £¡D Ý '0)É45)¥'0|r"T'0B&1©!#)245f#"¢!#)718!g)¥f7Èg' §n!#) g ~5'09A!!#9F!g§'09C¾!#}æ!g"T"T$9Y°) x©ÊÊ{Ê 459C18%6'018" 1©'e'0E©45$f1&u1©!y'0f7§'09c4I9A'0h$&f71&45)6"T!g)1©)$&'§s¢ B&"¢45f¸!#E1©'e'0E©45$&f1 βp 1 – a ib + c id + ------------- - ac bd + c2 d2 + ----------------- - bc ad – c2 d2 + ----------------- - + = ββe max 2 ⁄ a ib + c id + ------------- - a b d c ⁄ ( ) + c d d c ⁄ ( ) + ---------------------------- - i b a d c ⁄ ( ) – c d d c ⁄ ( ) + ---------------------------- if d c < ( ) + b a c d ⁄ ( ) + d c c d ⁄ ( ) + ---------------------------- - i a – b c d ⁄ ( ) + d c c d ⁄ ( ) + -------------------------------- - if d c ≥ ( ) +       = βe min

« "ý"x ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ 9A'0h$&f1© 459k' NaN E©!#9pÎÚyÎ45)
45)2 ∞ Eh!#9 l ÚyÎ45)@!g§'09C¾!#}yst'GB&9A'0"T'0)2g h'0"¢18!g) gg45¸' '0|À45%6B&f'0°}É'09A' B&9A!#"¢'¼'07)gyE©9A!#% 45)É'0|r"T'0B&1©!#)ç}¥1©É1©'0h' '0Eh45$&f1 ¸45f$&'0°}¥45°1©'z9A'045h!g)4 f' 182)g1©!!{s Ý '0)É45) '0|À"¢'0B&187!#)É!#"¢"T$9Y 4Án18451©$&°¾45g p45fh!¥n'¼1s x %(B&f7'¼%6'0)18451©!g)p!#E{182' xhÊÊÊ n1845)459A6459A'e9A'0¿§$&9A'061©!¥B&9A!#¸'e$&n'09Y }¥1©64y}¥4 18!9A'045@45)d}¥9C18'G1©'Gn18451©$& ¾45ggÀs0'G¾45g§z459A' è n18"¢È §é )Å18451 !g)"T'Gn'01 1©' 9A'¼%645)6n'013$)187f&'0|ÀB&f7"T1©f "Tf'0459A'0 s0r'0h1©)gQ1©'G¾45gg 18'!#)f }¥4 1©!h1©)g§$&h l ÚrÎ }¥"¢6 4g§'0)$&)'G7)Û&)71 E©9A!g%Æ45)6!g¸'¼9C¾!#}¥s qy!#%('01©%('0 "T!g)1©)$&)gQ'0|r'0"T$&1©!g)6)Å18'GEh45"T'G!gED'0|r"T'0B&1©!g)6"T!g)1©!g)z7z)!g1 45B&B&9A!#B9Y451©'§s è5x )Û&)71 §é !g)6B&45g§' l5m5m gg45¸'G1©'G'0|À45%6B&f'G!gE ÏÚyÌ öÏ P l5Í s Ý '0) êÏ 18'I'0)!g%()24518!g9 ñ)2Û&)1©' 9A'¼n$&f1©)g)(4GÛ)45f245)n}¥'09 !gE3Î }¥"¢ 18!g1845f7f }É9A!#)gDs v f1©!#$ggÅE©!g9u1© E©!g9Y%6$&f41©'GB&9A! f'0%Æ"¢45) 'Gn!#f7§'0 § 9A'0}¥9Y718)gQ1D45 l ÚrÌ nÏtP¢Ï sh Íh 9k'0}É9Y1©)gQ%(4 )!g1D45f}¥4 zn!#f7§'G1©'GB&9A! f'0%s02' xhÊÊÊ h1©45)459A6h1©9A!#)g§f 9A'0"T!g%(%6'0)u1©451&7%(B&f'0%('0)1©451©!#)p45ff!g} 189C45B(45)f7'¼9C 1©! 'G)h1©45ff'0 s0{'0)6}É'0)645)Å'¼|r"T'0B&1©!g)6!g"T"¢$&9Y 1©'G189C45BÞ45)2f'09u7z"¢45ff'0 )n18'045Å!gE3h'0181©)gç1©'¾45g3s¼'¸45f$&'9A'01©$&9Y)2'0 g 1©'189C45B645)2f'09°}Éff '$&h'0 45 18'G9A'0h$f1D!#EF182'G!#B&'09C4518!g)s x 1D 18'9k'0nB&!#)n f71 !gED1©'G1©9Y45BÞ245)f'09p18! '01©'09u"¢f'0459u!g9ph'01D182'Gh1©4518$z¾45g§Â§!#1©'09C}¥n' 18'§45f$&'!#EF18'¾&45gQ 45ff!g}¥'0@1©! 'G$)'0Û&)2'¼us ' x©ÊÊÊ h1©45)459A@¸'0z'0|r"T'0B&1©!g)z7)18! þ "¢f45hn'050!g§'09C¾&!g} $&)'09Y¾!g} ¸h!g) § ~5'09A! )¸45f@!gB&'09Y451©!#)Å45)@)2'0|À45"¢1Cs0{'09A'Gz4n'0B&459Y451©'Gh1©451©$& ¾&45g E©!g9p'¼45"¢6"¢f45h !#EF'0|À"¢'0B&18!g)s0{'G%('045)7)gQ!#EF18'Û&9Ch1D1829k'0'G'0|r"T'0B&1©!#)2z h'0fEh» '0§{'¼)21Cs x )§45f7(!#B&'09C4518!g)ç"T!g§'09C 1©'Áh1©$&451©!#)ufh1©'0()çr4 f' »©Ò!g)çB&45g§' l5m5m5 45)@45) "¢!#%6B&459Yn!#)Å18245137)§!gf§'0 4×¥45×Þs'G'0Eh45$&f1D9A'¼n$&f1D!gED45)Å!#B&'09C4518!g) 1©451&"T45$&n'0u45)ç)¸45f6'0|À"¢'¼B18!g)ç7 1©!y9A'¼1©$&9C)ç4 NaN r $1&18'e"T!g)§'09Ch'e )2!#1&1©9A$'§s Ý '0)6!g)'G!gED1©'G!gB&'09Y45)2z1©!45)6!gB&'09Y451©!g)6 4 NaN 1©'G9A'0h$&f1D 4 NaN $&1D)! )¸45f@'0|r"T'0B&1©!#)Å 9Y45n'¼d$&)f'0nz1©'G!gB&'09Y451©!g)645fh!n4518nÛ&'0 !#)2'G!#EF182' "T!g)1©!g)z7)6r4 f' »©Ò!g)6B&45g§' l5m5m s C¢2¡uï§ äºØ ¹ æ%{~ %o2 ¡ "kký å Exception Result when traps disabled Argument to trap handler !y§'¼9A¾&!#} ±∞ !#9 ± ÏXp 9k!y$&)¸&Ì Ï5Ø s α Í $2)¸&'¼9Y¾2!#} Î !#93&'¼)¸!#9Y%Q4tf 9k!y$&)¸&Ì Ï5Ø α Í &tgt' g ~t'¼9k! ∞ !#B2'¼9A4t)¸& 1 – ββe max 2 ⁄ 2e min ±

yï7ð#C Cñ u7 « "ý¢¡ åæ %'æ¥ä¨)ºwl}n£DÝ{qçpæ¥{ ¹ }o)æ%{~ α "ì¢« ç{~}ø£r¦è'% ¹ }nºôr£%{~ t k¢ò "x ç{~}Hl{~¨ r " ÏXp¦ l s l¢l … l¢l × é 'G)2'¼|r45"T1D'0|À"¢'0B&187!#)6 9Y45n'0@}¥'0)61©'G9A'¼n$&f1D!#ED4¾!g4518)2gg»hB&!#)213!gB&'09Y451©!#)Å )!g1D'¼|r45"T1s x )61©' β l Î e Òy h1©'¼% Òs þ ⊗ Ós Ø l Ós ê '0|r45"T1 § $&1 Òs þ ⊗ Ós Ò l5þ s Îyz)!g1D'0|À45"¢1DÌ©h)2"T'GÒs þ ⋅ Ós Ò l5þ s Î þ5Íh 45)@9C45h'0 45)6)2'0|À45"¢1 '0|À"¢'0B&187!#)s è å )459 18! '0"T%(45f§op!#)2§'09Yn!g) é !g)ÉB&45g§' Ø5Ò ø n"T$hh'0 45)ç45fgg!g9Y718% 1©451D$&h'0 182'G)'0|À45"¢1D'0|À"¢'¼B18!g)s v n$&%(%6459 !#EF1©' '045¸!#9p!gED45ffÛ&¸' '0|À"¢'0B&187!#) gg¸'0)6)År4 f' »hÓs '09A'G 45)6%6B&f'0%('0)1©4518!g)67hh$'G"T!g))'0"T1©'0@}É18Å182'GE©45"T1D1©451D182'G)'0|À45"¢1 '0|À"¢'0B&187!#)Åz9C45h'0@n!!#Eh18'0)s x EF¾&!g451©)g§»©B&!g)1D459A}¥459A'G!g'¼ )!g1D45§'G¾45g§z!gE 1©z!g}¥) ¸ $&1D)n18'045d)18'09C9Y$&B&1©z1©'G!gB&'09Y451©)gQ h1©'0%Æ1©!hg§)45f&4¾!g45187)gg»hB&!#)21 '0|À"¢'0B&187!#) 18'G"¢!#n1D!#EF)'0|r45"T1D'0|À"¢'0B&18!g) "T!g$&f 'GB&9A!# 1©§'§s0{z"¢!#n1D"T45) ' 45¸!#'0 § 45§)2g1©'Gn18451©$& ¾45gg %(45)1©45)'0 § h!gE©18}É459A'§s0'GÛ9Yn131©%('45) '0|À"¢'0B&187!#)Åz9C45h'0 n'01D18'Gn!#Eh18}É459k'¾&45gQE©!g9p182'G45B&B&9A!#B&9C4518'"Tf45h 45)d18'0ff&1©' ¾!#451©)g§»©B&!g)1D459A}¥459A'G1©!%(45hÈQ!gEûEF18451D"¢f45h !#EF'0|À"¢'0B&18!g)Às0'0)Å45ff&E©$9Y1©'09 '0|À"¢'0B&187!#) }¥f7f&9Y$&)6}É182!#$&1D)218'09C9A$B&18)2g1©'G!gB&'09Y451©)gQ h1©'0%s Ý 2'0)64$&n'09 9A'0h'018 1©451Dh1©4518$z¾45g 182'G459A}¥459A'G%(45nÈ 9A'0»©'0)4 f'0 s ÙÜÞt|% ëêÔ|n&ËYÐ¤Û'Þt} Ð )'G! §!g$& $&n'GE©!g9u1©9Y45B@45)f'09Cz E©!g9 45"TÈ§}¥459A@"¢!#%6B&4518 f1 s Ð f@"¢!#'0 1©451{'0|ÀB&'0"¢1{18! '4 !g9Y1©'0Å}É'0)Q'0|r"T'0B&1©!g)ñ!#"¢"T$&9 "¢45)Q)n1845f7f4G1©9C45B645){f'09°1©451 4 !g9Y1© 18'GB9k!g"T'0n5s02z '0hB&'0"¢45ff $&n'¼Eh$&f&Eh!#9p"T!g'0 }¥1©64f!g!gBÞfÈ§' R S º b hg Ç V Yhy ` U d ` X dhb g ha f V ha ` y g Y` a a V Æ d f V X X hg j Ç º V º ha hg Ç b YÇ ` y hg d e ` b X ` f V d hb g S ` ` a ` d d hb g h S b c T T T d y ¾ R S ¾ ¾ T y S ) 4 f × 4 × ! B ' 9 4 ) ) ' | 4 " 1 9 ! $ ) Ì Ï Í 9 ! $ ) Ì Ï Í ¢ ¡ ï § ä º ¹ { ¡ k ý å Exception Result when traps disabled Argument to trap handler 2e max

« "ý"Ñ ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ do S until (x
>= 100) s¢qr)"¢'Á"¢!#%6B&459Y)2g¥4 NaN 18!¥4)$% '09c}É18 e£ ≤ ≥ !g9 Ì $&1D)!#1 ≠ Í 45f}¥4 z9A'01©$&9Y)2zE©45f7h' 1© "T!g'G}¥ffgg!)1©!45)6)Û)1©'Gf!g!#B@E x '0§'09 '0"¢!#%('0 4 NaN s '09A'G 4%(!g9A'G)1©'09k'0n18)gQ$h'GE©!g9p189C45BÞ45){f'09Y 18451D"¢!#%6'¼ $&B@}¥'0) "T!g%(B&$18)2gB&9A!g$&"¢18 h$&"¢645 1©451D"T!g$&fdB&!#1©'0)1©45ff !#¸'09Y¾!g}ys Ð )' h!gf$&1©!g)É°1©!$&n'zf!gg§459Y1©%( 45)Q"T!g%(B&$&1©' '0|ÀB )n18'045 sn' B&9A! f'0% }¥1©61© 45B&B&9A!#45"¢6 1©451D137zf'0nz45"¢"T$&9C4518' 45)2@1©451D13"¢!#n18 %(!g9A'G18245)61©' h%6B&f'G'0|rB&9k'0nh!g) '0¸'0)6EF18'09A'G )!!#¸'09Y¾!g}ys02'09k'G 45)!g182'09un!#f$18!g) $&n)gQ189C45BÞ45)2f'09Y "T45ff'0dbTV5WYX kf§r§v5WAX 9pbT p b¢fgr§S q r06182451345¸!g !#1©6!gED1©'0h' B&9A! f'0%(yÏqr1©'09 '0)~ l5m5ê Ó5Ñs ''04 45zE©!gff!g}¥Às0'09A'G 4g§f! 45f&"T!g$&)1©'09p)1©45f~5'0@1©!ç~5'09A!3s Ý '0)'0¸'¼9 1©'GB&459Y1©45f&B9k!g$&"¢1 !g§'09C¾&!g}¥ E©!g9ph!g%(' ùs 18'G1©9C45BÞ45)f'09 )"¢9k'0%6'¼)218 182'G"T!g$&)1©'09 § !#)'G45)2@9A'018$&9C) 182'G!#¸'09Y¾!g}¥'0@¿§$&45)2181 }¥1©61©' '0|ÀB!#)'0)1D}É9Y45B&B&'0d459k!g$&) s x ) x©Ê{ÊÊ@ê5þ Ón)ggf7'GB&9k'0"¢h!g) W tp¦ l Ø ê5 h!E e u l s Ó þ × Ø h ~ g 1D}¥f7f&!#¸'09Y¾!g}Ë45)2@"T45$h'G1©'G1©9Y45BÞ245)f'09p18! 'G"¢45ff'0 }É"T }¥f7f&}¥9C45BÞ1©'G'0|ÀB!#)'0)1 45"TÈQ)1©!ç9C45)g§' "T45)2gg)2ge Xì 18! l s Ó þ × Ø sí Ì©n'¼' '0f!#} Í s qy%(f459Cf T EFe u $&)2'09Y¾!g}¥ 18'G"¢!#$)1©'¼9p}É!#$&f7 'G'0"¢9k'0%6'¼)218'0 45)2@)'0g§4518¸' '0|ÀB!#)'0)1D}É!#$f@g§'¼1D}É9Y45B&B&'0d459k!g$&)d)1©!ç4yB&!#n18¸'G!g)'§s Ý '0)645f7f&182' %($f18B&f7"T451©!#)2z459A'G!g)' EF18'G"¢!#$)1©'¼9p ~5'09k!1©'0)61©'GÛ)45f&B&9A!g$&"T1D e Xa¢s x E 1©'G"T!g$&)1©'09pzB&!gh1©¸' 182'GB&9A!#$&"¢1D!#¸'09Y¾!g}¥'0 EF182'G"T!g$&)1©'09u )'0g§4518¸' 1 $&){'¼9C¾!#}É'0 s x ED)!#)2'I!#ED1©'IB&459C18745f2B&9A!g$&"T1©z459A'I!#$13!#ED9C45)gg' 18'G189C45BÅ245)f'09 )'0§'09p"T45ff'0d45)@1©'G"T!g%(B&$18451©!g)6)"¢$&9Cz)!'0|r189C4"T!gh1s Ê ¸'0)6EF18'09A'G459A' !g§'09¼Úy$&)2'09Y¾!#}É 1©'G"¢45f"T$f4518!g)6 %(!g9A'G45"T"¢$&9Y451©'G1©45)6EF1D45 '¼'0) "T!g%(B&$18'0@}É1©6f!ggg459C182%( ¸ '0"T45$&n'G'045"TÅe u }¥45 "T!g%(B&$18'0@Eh9k!g%Çe u suhF$h)gQ4 E©$ff&B&9A'0"Tn!#)Å%($&f1©B&f s å 459C)'0181ñÏ l5m5ú5ê ÑFh"¢$&nh'0z4yE©!g9Y%($f4}¥2'¼9A'G1©'GE©$ff 45"T"¢$&9C45" !#EF!#¸'09¼Úy$&){'¼9C¾!#} "T!g$&)1©)gQ1©$&9C)'0@$&B@45)6'09Y9A!g9u)Å'0459Yf'09p1©4 f'0 !#E 1©451DE©!#9C%($f4s xhÊÊÊQê5þ ÓÁnB&'0"TÛ'¼°1©451}É'0)É45)É!#¸'09Y¾!#}æ!g9F$&)2'09Y¾!g}æ189C45Bç245)f'09F°"T45f7f'0 1 B&45hn'¼d182'G}¥9C45B&B&'0»©459A!#$)@9A'0h$&f71345 45)6459Ag§$&%('0)1s02'G'0Û&)718!g)6!gE }¥9C45B&B&'0»©459A!g$&)@Eh!#9p!g§'09C¾&!g} z1©451D182'G9A'¼n$&f1D "T!g%(B&$&1©'0@45 ED1©!)Û)18' B&9A'0"Tn!#) 18'0)Å§7'0 g Ø α 45)d18'0)Å9A!#$&)2'0@18!1©'G9A'0f'0§45)1DB&9A'0"Tn!g)s º !#9 $&){'¼9C¾!#} 182'G9k'0n$&f1D %($&f1©B&f'0 § Ø α s0'G'0|rB&!g)'0)1 α l5m ØE©!g9un)g§f' B&9A'0"Tn!#)Å45) l5þ Ò ø Eh!#9p!g$ f'GB9k'0"¢h!g)s02z }¥ l s Ó þ |yØ h ~ g }¥45 1©9Y45)nE©!g9Y%('0d)18! l s Ó þ × Ø sí )61©'G'0|r45%(B&f'G4 !#¸'§s Π i 1 = n x i Σ logx i ( ) Πx i p k Π i 1 = k x i =

yï7ð#C Cñ u7 « "ý"ì îøÝnÓ&£ËYtYÉÍï ÝnËÜÛt} x )1©' x©ÊÊÊ h1©45)459A 9k!g$&){)gÉ!g"T"T$9Yp}¥2'0)'0§'09°45)Q!#B&'09C45187!#)Q45p4I9A'0n$&f11©451 )!#1D'0|r45"T1 h)2"T'GÌ©}É1©61©'G'0|À"¢'0B&18!g)6!gE )459 {'¼"¢%(45f"T!g)§'09Ch!g) Í '045"¢ !gB&'09Y451©!#)Åz"¢!#%6B&$&1©'0@'0|À45"¢18f 45)2@182'0)69A!#$)'0 s å '0E©45$f1 9A!g$&))2g%6'045) 9A!#$)@1©!#}É459kd)'0459k'0n1Cs0{'Gh1©45)459A@9A'0¿§$&9A'0 18451D1©9A'0'G!#1©'09p9k!g$&))2g%6!#'0 'B&9k!g§7'0 )45%6'¼f 9k!g$&)Å18!g}¥459AÅÎ 9A!#$)61©!#}É459k êP ∞ 45)Å9A!#$)61©!#}É459kÅ» ∞ s Ý '0)6$h'0@}É18Å182'G"T!g)§'09C1D18!)1©'0gg'09p!gB&'09Y451©!#) 9A!#$&)2@1©!#}É459kd» ∞ "¢45$&h'0 1©' "T!g)¸'09Y1D18! '¼"¢!#%6'G182'G¾!#!g9uEh$&)"¢18!g) }Éf'G9A!g$&)@1©!g}¥459A ¢P ∞ "T'0f)2g3s0{' 9A!#$))gQ%(!g'G45E8E©'0"T1©z!g§'09C¾!#} ¸ '0"T45$h'G}¥2'0)69A!#$)@1©!#}É459kdÎ!#9p9A!#$) 1©!#}É459kd» ∞ )Å'¼E8E©'0"¢1 45)Å!#¸'09Y¾!g}Ë!gEDB&!gh718¸'G%(45g§)1©$&'G"¢45$&h'0 182'G'0E©45$&f1 9A'0h$&f1D1©! 'G1©'Gf459Agg'0n139A'0B&9A'0h'0)1©4 f'G)$% '09 )!#1 P ∞ s0qy%(f459Cf T !#¸'09Y¾!g}¥ !#E )'0g§4518¸'G%(45g§)1©$&'G}Éff&B&9A!g$&"T'G1©'Gf459Ag§'0h1D)'0g§4518¸'G)$&% '09p}¥'0)69A!g$&) 1©!#}É459k P ∞ !#9p9A!#$)@1©!#}É459kdÎ )6'0E8E©'0"T1s Ð )'G45B&B&f"¢4518!g)6!gED9A!g$&))2g%6!#'0 !g"T"T$9Y )6)218'09C§45f&459C1©%('01©"ÁÌh45)!#1©'09p %('0)1©!g)'0@) è å )459 1©! '0"T%645f&op!g)§'09Ch!g) é !g)6B&45g§' Ø5Ò ø5Í s Ý '0)Å$&h7)g )1©'09Y¸45f&459C18%6'018" 1©'Gh$%Æ!gED1©}¥!)$&% '09Y Ï45)d` 45)6)1©'09Y¸45f }¥2'¼9A' Ï ⊕ `9A!#$)'0@1©!#}É459A@» ∞ 45) Ï ⊕ `9A!#$)'0@1©!#}É459A ¢P ∞ s0' '0|À45"¢1D9k'0n$&f1D!#EF1©'G451©!#)Å "T!g)1845)2'0@}¥1©)Å18')18'09C§45f s Ý 1©!#$1 9A!#$))gQ%(!g'0 7)18'09C§45f459Y1©%('01©"Á7z$&n$&45ff %6B&f'0%('0)1©'0 g "T!g%(B&$18)2g 45) }¥'09A' %(45"¢)' '0B&hf7!#)s 7z9k'0n$&f1©z)6!#¸'09A'¼n187%(4518'0zEh!#9p18'Gh~5'G!gED18'G)1©'09Y¸45f5s0qy)"T'G182'I9A'0n$&f13!gED45) !gB&'09Y451©!#)Å)6)218'09C§45f&459C1©%('01©"Á 45)6)218'09C§45f 7)6g§'¼)2'09Y45f&1©'G)2B&$&1D18!45) !gB&'09Y451©!#)Å}¥ff45fh! 'G45)Å)1©'09Y¸45fks x ED1©}¥!)1©'09Y¸45f 45) 459k' 45'0 1©'G9A'0h$f137 }¥'09A' }¥1©61©'G9A!#$))gQ%(!g'Gh'01D1©! 9A!#$)Å18!g}¥459AÅ» ∞ 45) }¥1©Q182'9k!g$&){)gÉ%6!#'n'011©!¥9A!g$&)Å1©!g}¥459A P ∞ s Ý '0)64y¾&!g451©)g§»©B&!g)1D"T45f"¢$&f451©!#)ÅzB&'09CE©!g9Y%6'¼d$&h7)gQ)18'09C§45f459Y1©%('01©" 18' Û)45f45)h}É'09°ñ45)Q)1©'09Y¸45f182451"¢!#)1©45)ñ18''0|r45"T1{9k'0n$&f1{!#E318'"¢45f"T$&f74518!g)s2 )!#1D¸'09 '0fB&Eh$&f&EF182'G)1©'¼9C§45f18$9Y) !g$&1D18! 'Gf7459kg§'GÌ©45 1D!#Eh18'0)Å!#'0 Í© h7)"T' 1©'G"T!g9Y9A'0"T1D45)n}¥'09p"T!g$&f 'G45) }¥'09A'G)61©451D)1©'09Y¸45fks x )1©'09Y¸45f&459Y1©%6'¼1©" %(45È§'0u%6!#9A'eh'0)2h'e}¥2'0)$&n'06)"¢!#) ½ $&)"¢187!#)Q}¥1©4I%6$&f1©B&f'eB&9A'0"Tn!#)Q¾!#451©)g§» B&!g)1DB&45"TÈ§45gg'§st'G"¢45f"T$&f451©!g)6 Û&9Ch1DB&'09CE©!#9C%('0@}É1©6h!g%('GB&9A'0"T7h!g)6es x E )1©'09Y¸45f&459C18%6'018"eh$ggg§'0h1©z1©451D1©'GÛ&)245f&45)h}É'09u%64 Q 'G)45"¢"T$&9C4518' 18' "T!g%(B&$18451©!g)¥°9A'¼!g)'z}É1©¥g§'09F45)2gg2'¼9FB&9A'0"Tn!#)2c$)18fg182'zÛ&)245f#)1©'09Y¸45f 49A'¼45n!#)24 f'Gn~5'§s R S W V f e ` j f ` V d ` f d e V g hc e b d e U V g y f V f ` g ` j V d hÇ ` S ¾ ¾ T y S z,z [ ] z z z,z [ ] z x y ⊕ ( ) 1 ε – ( ) = z x y ⊕ ( ) 1 ε + ( ) = ε z z x,x [ ] y,y [ ] z,z [ ] z x y ⊕ z z y ⊕

«¢ "ý ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ðÐl|%É&} ' xhÊÊÊ h1©45)459AQ45°4e)$&%
'¼9!#E2¾45g§°45)Q%(!g'05s v °h"¢$&hn'04 !#¸' 1©'09k' !#)'Gn18451©$& ¾45gEh!#9p'045"T6!gED1©'GÛ§'G'0|r"T'0B&1©!#)250$&)2'09Y¾!#} !#¸'09Y¾!g} §7h!g) g ~5'09A! 7)§45fd!#B&'09C45187!#)Å45)@)'0|r45"T1s02'¼9A'G459A'GE©!g$&9p9A!#$))gQ%(!g'0509A!g$&) 1©!#}É459k@)2'¼459A'0h1 9A!#$)@1©!#}É459k ¢P ∞ 9A!g$&)@1©!#}É459A@Î 45)@9A!#$)@1©!#}É459k@» ∞ s x 13 h1©9A!#)g§f 9A'0"T!g%(%('0)'0d18451D1©'09A' 'G45)6'0)24 f'G%(!g' 713Eh!#9p'045"TÅ!#EF182'GÛ&¸' '0|À"¢'0B&187!#)Às07zn'¼"¢187!#)Ågg¸'0zn!#%6'Gh%(Bf'G'0|À45%6B&f'0z!gED2!#} 182'¼n'G%(!g'0z45)2 ¾45gg "T45) 'GB&$131©!gg!g!g@$&n'gs v %(!g9k'Gn!#B&2h1©"T451©'0@'0|r45%(B&f'G h"¢$&nh'0@) è å )459 18! '0"¢%(45f&op!g)¸'09Yh7!#) é !#)ÅB&45gg' Ø5Ò ø s op!#)2h'09p}¥9C187)gQ4h$ 9k!g$&1©)'G1©!"T!g%(B&$&1©' ùÏ a }¥'09A'GrQz45)Å)1©'0gg'09Cs Ý 2'¼) r Î 4yh%(B&f7'G9k!g$&1©)'GfÈ§' x ED) Í£ÞÎ 18'0)Å4%(!g9A'G45"T"T$9Y451©'G}¥4 1©!ç"¢!#%6B&$&1©' ùÏ a )!g1D18!"T45ff PositivePower(1/x, -n) $&1D9Y451©'09 1/PositivePower(x, -n) § '0"¢45$&h' 1©' Û9Yn12'0|ÀB&9A'0nh!g)É%6$&f1©B&f'0 ry¿§$45)18718'0 '045"Tç!gE&}¥"¢ç45§'Á4e9A!g$&){)gy'09Y9A!g9 E©9A!g%Æ1©'G¸h!g)6Ìhks '§s §l Ú Ï Í s x )61©'Gh'0"¢!#)d'0|ÀB&9A'0hn!#)Å182'¼n'G459A'G'0|À45"¢13Ìhks '§s Ï Í© 45)@1©'GÛ)45f&7§n!#)Å"T!g%(%(1© ½ $&n13!g)'G45{18!g)45f&9A!g$&))2g'09C9A!#9Cs ÷I)E©!g9Y1©$&)451©'0f ¢ 182'¼n'G 4hfg§1Dh)245g7)61©zn189C451©'¼g s x E PositivePower(x, - n) $&)2'09Y¾!g}¥ 18'0)Å'018'09p1©'G$&){'¼9C¾!#} 1©9Y45BÞ245)f'09p}¥ff 'G"T45ff'0 !#9p'0fh' 1©'G$&)'09C¾!#} h1©4518$z¾45gQ}¥f7f 'Gn'01Cs0{ z)2"T!g9Y9A'0"T1 § '0"T45$h'GE ÜÏ sa $&){'¼9C¾!#}É 1©'0) ëÏ a }¥ff'0182'¼9p!g§'09C¾&!g} !#9 'G)Å9Y45)2gg'§s å $&1Dn)"T'G1©' xhÊÊÊ R S d d V g e ` hg f V g j ` e ` d V a ` hc § ñ R i t ñ ¡ V g y § sa ha ò a d V d hg f e hd a W V YY` f d e V g de ` g y ` f ½ b de f ` a e b Yy i d e ` g i V g y a b W V f g b d b Ç ` f ½ b i a hg d ` hg V YY T T T X f ` d ha hb g a i ¾ ñ é PositivePower(x,n) { while (n is even) { x = x*x n = n/2 } u = x while (true) { n = n/2 if (n==0) return u x = x*x if (n is odd) u = u*x } 2e min xn 2 e min – 2e max < ≈

yï7ð#C Cñ u7 «¢ C h1©45)459AÞgg¸'0ñ18'I$&h'09 45"¢"T'0hñ1©!É45ff2182'¾45gg 1©'n$ 9A!g$&1©)'I"T45)('045hf "¢!#9C9A'¼"¢1 E©!g9p18Às x 1Dh%(B&f 1©$&9C) !#E8ED1©'G!g§'09C¾&!g} 45)@$&)2'09Y¾!#} 1©9Y45BÞ'0)24 f' 1© 45) h45¸'0z1©'G!g§'09C¾!#} 45)@$)'09Y¾!g}Ën18451©$& 1©5s x 1D182'0)6"T!g%(B&$18'0 1/PositivePower(x, -n) s x E&)'0182'¼91©' !g§'09C¾&!g}ö)!#9$&)2'09Y¾!#}öh1©451©$& 12 h'01 1D9A'0h1©!#9A'0z1©'0%Ç18!ggg'01©'09p}¥1©61©'G1©9Y45BÞ'0)24 f' 1©5s x EF!#)2'G!#EF182'Gh1©4518$ 18 h'01 1D9k'0n18!g9A'¼ 1©'G¾45gg 45)@9A'0!g'0z1©'G"¢45f"T$&f451©!g)6$&n)g PositivePower(1/x, -n) }É"TÅ"T45$h'0z1©'G"¢!#9C9A'¼"¢1D'0|À"¢'¼B18!g) 18!!g"T"T$9Cs v )!g18'09p'0|À45%6B&f'G!gED1©'G$&n'G!#EF¾45gg !g"T"T$9Y }¥'0)Å"T!g%(B&$&1©)gQ459A"T"¢!# §4y18' E©!g9Y%6$&f4459A"T"¢!# Ï Ø459A"T1©45) s x ED459A"T1©45)Ì ∞ Í '0¸45f$&451©'¼ 1©! π ÚyØ 1©'0) 459A"T"T!ghÌh» l5Í }¥ff&"¢!#9C9A'0"T1©f '0¸45f$&451©'G1©!çØ ⋅ 459A"T1©45)Ì ∞ Í π § '¼"¢45$&n'G!#EF)Û)1 459C18%6'018"ys 8m¥!g}¥'0§'09 18'09A'G 4h%645ff&n)45g ¸ '0"T45$&n'G1©'G"T!g%(B&$18451©!#)Å!#EFÌ l » Ï Í ÚyÌ l P¢Ï Í }¥ff"T45$&n'G182'G§{' g ~5'09k!'0|r"T'0B&1©!#)Å¾45g1©! 'Gn'¼1 '0¸'0)61©!#$gg 459A"T"T!ghÌh» l5Í z)!g1D'0|À"¢'0B&18!g)45fkst'Gn!#f$18!g)61©!187zB&9A! f'0%Æ h1©9C45gg1©E©!g9Y}É459kus qy%(B&f h45¸'G18'G¸45f$&'G!gED1©'G¸' g ~5'09A!ç¾45g '0E©!g9A'G"T!g%(B&$&1©)gQ459A"T"T!g 45) 1©'0)69A'0h1©!#9A'G1©z!gf@¸45f$&'G45Eh18'09p182'G"T!g%(B&$&1©451©!#)cs y6ó ´µ x¡ô ´± ´ nõ x ¹°µ´ ''0hg§)6!gED45f%6!#n13'0¸'09 45hB&'0"¢1D!#EF4"T!g%(B&$18'09p h1©'0%Æ9A'0¿§$&9A'0zÈ§)!g}¥f'0g§' 4 !g$&1D¾!#451©)g§»©B&!g)1s0op!#%6B&$&1©'09u459A"T218'0"¢18$9k'0 $&h$45ff 45§'¾&!g451©)gg»hB&!#7)1 )n189Y$&"¢18!g) "¢!#%6B&f'09Yu%($&n12g§'0)'09Y451©'Á182!#n' ¾!g4518)2gg»hB&!#)212)n189Y$&"T1©!g) 45)(1©' !gB&'09Y451©)gy h1©'¼% %($&n1'0"¢' }¥45121©!!}¥2'¼)ç'0|À"¢'¼B18!g)É"T!g)1©!g)°459A' 9Y457h'0 E©!g9p18!gh'G¾!g45187)gg»hB&!#)2137)h1©9Y$&"T1©!#)25s0op!g%(B&$&1©'09p h1©'0%Æ'0h7gg)'09Cz9C459A'0f g§'01 g§$&45)"¢' Eh9k!g% )$%('09Y"¢45f§45)45f h 1©'0|À1© }É"Tç459A'Á1 B&"T45ff 45%('0(451&$&h'09C 45) }¥9C1©'¼9C°!#E2h!gE©1©}¥459A' )!g1451"T!g%(B&$&1©'09F'0hg§)'09YÀs v °45)¥'0|r45%(B&f'z!gE2!#}æB&f45$h f' '0hg§)6{'¼"¢h!g) "T45)Åf'045@1©!ç$)'0|ÀB&'0"¢18'0 '045§7!#9 "T!#)2h'09p182'GE©!gff!#}É)g å v q x o B&9A!ggg9C45%s Ý '0)6"¢!#%6B&f'0@45)d9A$)6$&n)g å !g9Yf45){Äw r$9 ! å 45n"Á!g)645) x å i Ã o 182' B&9A!#g§9Y45%ÆB&9C)1© Not Equal p5{z'0|À45%6B&f'I}Éff 'I45)45f ~5'0@)(1©'G)'0|À13n'0"T1©!#) è5x )h1©9Y$&"T1©!g)6qr'01©5s é q = 3.0/7.0 if q = 3.0/7.0 then print "Equal": else print "Not Equal" 1 x – 1 x + ----------- -

«¢ ¢« ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ x )"T'0)21845ff ¢ h!g%('IB'¼!gB&f'G182)È1©45131©'Gh!gf$&1©!#)61©!çn$&"T645)!g%(45f7'¼
z)'0§'09u1©! "T!g%(B&459A'G¾!g4518)g§»©B&!g)1D)$% '09CzEh!#9p'0¿§$&45f1 T¸ $&1D)n18'045@1©!"T!g)h7'09u1©'0% '0¿§$&45f§E182' 459A'Á}¥718)h!g%('Á'09Y9A!g9 !#$)(U sT2 u459Af 4"T$9k'0»©45f7f '0"T45$h'Á1 9C45h'0z45 %(45) ¿§$'0h1©!#) 45 1D45)h}É'¼9s Ý 451Dh!g$&fd18'§45f$&'!#EFU ' ùex E ÜÏ£@Î 45) Î459A'G}¥1©)ÅU h!g$&fd18' 9A'045ff Q 'G"¢!#)n'09A'¼d18! ''¼¿§$45f '0§'0) 1©!#$ggÅ18' 45¸'GE8E©'09A'¼)213ng§) ù º $9Y1©'09Y%6!#9A' 1©'G9A'0f4518!g)6'0Û)'0 § 1© 9Y$&f' R öQ ⇔ g4»D gT£ÅU z)!g1{45)6'0¿§$&¸45f'0)"T'I9A'¼f74518!g) '0"¢45$&h'IR ÷öÞe45)Þ Ûö p !g'0z)!g1D%(B&f 1©451DR ÷ö p s ø¨©üD¦¨ Ó û¦"§£¢H© ± ¡D¦" x 1&p¿§$&1©'Á"¢!#%6%(!g)E©!g9c45)45fg§!#9C1©% 1©!y9A'¼¿§$9A'Á4In!#9C1 $&9Ch1&!gE7gg'09°B&9k'0"¢h!g) )6!g9A'09u1©!B&9A!#$"T'G45"¢"T$&9C4518'G9A'0h$f18Às Ð )'G'0|À45%6B&f'G!g"T"T$9Y )61©'G¿§$459Y451©" E©!g9Y%6$&f4Ì Í ÚrØ5Rys v zn"T$&nh'0@!g)6B45gg' Ø5Ò5Ó }É'0)6 ≈ Ó5R pC 9A!#$))gQ'09Y9A!g9u"¢45)6"¢!#)1©45%()24518'G$BÞ18!45f7ED1©'Gg§18 )Å18'G9A!g!#1© "T!g%(B&$&1©'0 }¥1©61©'G¿§$459Y451©"eE©!#9C%($f4s å B&'09YEh!#9C%()gQ1©'Gh$ "T45f"¢$&f451©!g)6!gED »FÓ5R p ) !g$ f'GB9k'0"¢h!g) 45fEF1©'G!g$ f'GB&9A'0"T7h!g) 1© !#EF182'G9k!g!g13459A'Gf!gh1 }¥2"T %('045) 1©451D45ff&1©'Gh)2ggf'GB&9A'0"T7h!g) 1© 459k'GB9k'0n'¼9C§'0 s 'e"T!g%(B&$18451©!g)!#E »©Ó5R p )!#$ f'eB&9k'0"¢h!g)}¥'0)'045"T!gE1©'Á¿§$45)18718'0uR 45) p 459A'G)Åh)g§f'GB&9A'0"Tn!#)Å '¼45 ED1©'09A'Gz4y%($&f1©B&f"¢4518!g)67)h1©9A$"T1©!#)Å182451 1©45Èg'0z1©}¥!h)2ggf'GB&9A'0"Tn!#)6)2$&% '¼9Cz45)@B9k!g$&"¢'¼ 4!#$ f'GB&9A'0"Th7!#)69A'0h$f1Cs x ) !g9k'09p1©!çB&9A!g$&"¢'G18''¼|r45"T1©f 9A!#$)'0@B&9A!#{$&"T1D!gED1©}¥!e5»©g§1D)$&% '09Y 4 %($f18B&f7'09u)'0'0 1©!çg§'0)'09Y451©'G1©'G'0)1©9A'GØ5e 1©z!gEFB&9k!g$&"¢1 45f18!g$&g§61D%64 1©9k!g} 718ñ45}É4 45ñ1{B9k!g"T'0'05s¼2$& 459A}¥459A'I18!É"¢!#%6B&$&1©'4!g$ f'IB&9A'0"Th7!#) B&9A!#$"T1DE©9A!g%Æh7)ggf'B&9k'0"¢h!g)6!gB&'09Y45)2z}Éff&)!g9Y%645ff Q 'G!g)f 4f1©18f7'G%(!g9k' '0|ÀB'¼)2h¸'G18245)64n)ggf7'GB&9A'¼"¢h!g)6%6$&f1©B&f'09 45)d%($&"¢6"¢'045B&'09u1©45)Å4!#$ f' B&9A'0"Tn!#)Q%($&f718B&f7'¼9s '0nB&18'182 %6!#'09C)7)h1©9A$"T1©!#)Qh'01©p18'0)Å1©!¥B&9A!#¸'e!g)f )n189Y$&"¢18!g) 1824513B&9A!g$&"¢'G49A'¼n$&f1D!gED1©'Gh45%6'GB&9A'0"Th7!#)Å45z1©'G!gB&'09Y45)Às x ED45)Å)h1©9Y$&"T1©!g)61©451D"T!g% )'0 1©}¥!h)g§f'GB&9A'0"Tn!g)6!gB&'09Y45) 1©!çB&9A!g$&"¢'G4 !g$ f'GB9k'0"¢h!g)6B&9A!g$&"T1D}É45z!g)f $&n'¼Eh$&f&Eh!#9p1©'G¿§$&459C4518"eE©!g9Y%6$&f4 1 }¥!g$&f)2Äw1 'G}¥!g9Y1©645)2g1©!45)6)2h1©9A$"T1©!#)Åh'01s 8m¥!g}¥'0§'09 18 )n189Y$&"¢18!g)6245 %(45) !#1©'09$&n'05sTop!#)n'091©' B&9A! f'0% !#E&n!#f¸)gy4 h1©'0% !#E&f)2'¼459c'¼¿§$4518!g) R S e ha ha X f b e V e Yf e ` d V a ` y ` a hj g ` f a Yh« ` Æ b f d e b j b g V Y Ç hg a d f d d hb g a ` da i e ` f ` d e ` X f ` d ha hb g a b c V ½ b V d hg j ¾ X b hg d hg a d f d d hb g V f ` hg y ` X ` g y ` g d b c de ` V d d V Y b X ` f V dhb g S V « hg j V a X ` d hV Y d V a ` c b f W YdhX Yhd V dhb g y ` a d f b f a d e ha b f d e b j b g V Yhdf S b – b2 4ac – ±

yï7ð#C Cñ u7 «¢ ¢ò R hùhöÏXhÜP@R h ÏXúÜP ⋅ ⋅ ⋅ + R `¦a¤ÏXa )h R hnÏXhP@R ÏXúÜP ⋅ ⋅ ⋅ + R úa¤ÏXa ⋅ ⋅ ⋅ R alhöÏXhÜP@R a ÏXúÜP ⋅ ⋅ ⋅+ R ala¤ÏXa )a }¥2"T6"¢45) 'G}¥9C181©'0)6)Å%(451©9Y|¥E©!g9Y%Ç45 Ï }É'09k' qy$&B&B&!gh'G1©451D4h!gf$&1©!g) ëÏXûh¦ü z"¢!#%6B&$&1©'0 g h!g%('G%6'018!g B&'09A45B& e45$&nh45) '0f%()24518!g)s¼'09A'Iñ4Gn%(B&f'I}¥4 18!ç%(B&9A!#¸'1©'I45"T"¢$&9Y45" !gE318'I9A'0h$&f13"T45ff'0 q S©WYXYRTS q V5W q dIe5Xkb¢V5WAdIWYr§Ss º 9Yn1D"T!g%(B&$&1©' ξ Ïqûh¦ü »F Ì l Ø Í 45)@1©'0)Åh!gf§'G1©'G n18'0% Á` ξ Ì l Ò Í ×¥!g18'1©451E qÏXûhü p45)('¼|r45"T1{h!gf$&1©!g) 182'¼) ξ p1©'~5'09k!É§'0"T1©!g9 45pp`¼s x )Qg§'¼)2'09Y45f 1©'G"T!g%(B&$18451©!#)Å!#E ξ 45)@`}Éff&)"¢$&9p9k!g$&)2)gQ'09Y9A!g9 h! Á` ≈ ξ ≈ ÏXûh¦ü »& Ì öÏXûh¦ü » ÜÏ Íh }¥2'¼9A' ùÏ7z1©'GÌ©$)Èg)2!#}É) Í 1©9Y$&'Gh!gf$&1©!g)s02'0) ` ≈ ÏXûhü » Ï n!ç45)Å%(B&9A!g§'0@'0n18%64518'GEh!#9p1©'Gh!gf$&1©!g)6 ÏXû ü Ïqûh¦ü »` Ì l Ó Í A a 11 a 12 ... a 1n a 21 a 22 ... a 2n ... a n1 a n2 ... a nn             =

«¢ ký ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ '189A'0'Gh1©'0B& Ì l Ø
Í© Ì l Ò Í© 45)dÌ l Ó Í "T45) 'G9k'0B&'0451©'0 9A'0B&f45"T)2g þÏXûh¦ü }É18 êÏXû ü 45)2 ÏXû ü }¥1© ÏXû~ ü sn{°459Ag§$&%('0)12182451 )ÏXûmh¦ü °%6!#9A' 45"T"¢$&9Y451©' 1©45)É| %ûü °!g)f )E©!g9C%(45fks º !#9p%(!g9A'G)E©!g9Y%64518!g) n'0'QÏe!gf$ 45) Q345) Ô !#45) l5m5ú5m Ñs Ý '0)6B&'09CE©!g9Y%6)gQ18'09C4518¸'G%(B&9A!g§'0%('0)1 ξ z4¸'0"T1©!#9p}¥!gh'G'0f'0%6'¼)218 459k'G1©' E8E©'09A'¼)2"T'G!gED)'0459 g )'0|r45"T1D¾!#451©)g§»©B&!g)1D)$&% '09Y 45)dh!"T45)Åh$&E8E©'09pE©9A!#% "T451©45h1©9A!#B&2"ñ"¢45)"T'0ff451©!g)s¢{$&p18'09C45187§'e%6B&9k!g¸'¼%6'0)1u)2!#1¸'09 $&n'0E©$&f$&)f'0h ξ Ïqûh¦ü »F "T!g%(B&$18'0@)Å!g$ f'GB&9A'0"T7h!g)s Ð )"T'G45g§45) 1© 4"T45n'G!#E "T!g%(B&$18)2gÉ1©'B&9k!g$&"¢1!gE{1©}¥!Éh)g§f'B&9A'0"Th7!#)Q)$% '09CpÌæ45)2 êÏXûhü Í© }É'09A'e1©' E©$ff&!g$ f'GB9k'0"¢h!g)69A'0h$&f713 )'0'0'0 s r!h$%(%(459C~5' )n189Y$&"¢18!g) 18451D%6$&f1©B&f 18}É!ç¾!g45187)gg»hB&!#)213)2$&% '¼9C 45) 9A'018$&9C)64yB&9k!g$&"¢1D}¥1©61©}¥"¢'G18'B&9k'0"¢h!g)6!gED1©'G!gB&'09Y45)2z%645Èg'G4$h'0E©$f 451©!g)Q1©!É4¾!#451©)g§»©B&!g)13)h1©9A$"T1©!#)(n'01Cs¼qr!g%('I!#E31©'I%(B&f7"T451©!#)z!#E31©ñE©!g9 "T!g%(B&f7'¼9C 459k'G7h"T$hh'0d)61©'G)'0|r1Dh'0"T1©!g)s ý¡¤¥©¥ Ó ¤¡¡¤¥©¡© ú ¢¥«¡§£ ¤¡D ')18'09C45"T1©!g)6!gED"T!g%(B&7f'09Y 45)@¾!g4518)g§»©B&!g)1D h"¢$&nh'0@) º 459C)$&% Ï l5m5ú5ú Ñ 45)@%6$&"TÅ!#EF1©'Gh"¢$&nh!g)6)Å18 h'0"¢18!g)67z1©45Èg'0)ÅE©9A!#%Ç1824513B&45B'¼9s þ ßiÕùÉ&Ó£t¢dÂ x '045ff T 4f45)g§$&45g§'G'0Û&)1©!g)6n!#$f@'0Û)'G182'Gh'0%(45)218"¢z!gED1©'Gf45)2gg$&45g§' B&9A'0"Tn'¼f '0)!g$&g§61©!B&9A!#¸'Gh1©4518'0%6'¼)218 4 !g$&1DB&9A!#g§9Y45%65s Ý 7f'G182z $&n$&45ff 1©9A$'GE©!#9p1©'G)1©'0gg'09pB&459Y1D!gED4f45)g§$&45g§' f45)g§$&45g§'G'0Û&)718!g) !#E©1©'0)645¸'G4f459kg§' g§9k' 459A'04}¥'0)Å1D"T!g%('0z1©!¾!#451©)g§»©B&!g)1s Ã '09A45B& 187z $&'G1©!182'GE©45"T1D1©451 %(45) f45)2gg$&45g§'G'0ngg)'09C '0f'0§'G1©451D)!g182)gQ"T45) 'GB&9A!#¸'0)64 !g$&1D¾!#451©)g§» B&!g)1 h)"¢'G1D'0)1©45fz9A!g$&))2g'09C9A!#9Cs x EFh! 182'GB&9A'¼¸!g$&zn'0"T1©!#)2z45¸' '0%(!g)h1©9C4518'0@1©'GEh45ff45" )61© 9A'045h!g))gDs0 h'0"¢18!g)6{h"T$hn'¼ h!g%(' "T!g%(%6!#)Ågg9A' 459k'045 )6f45)2gg$&45g§'G'0Û)1©!#) )"¢f$&)2gn$&g§gg'0n18!g) 4 !g$&1D!#} 1©!ç'045f}¥1©61©'0%s Ùc'0%(459YÈ§4 f '0)!g$&g§ n!#%6'Gf45)g§$&45gg'0 !g)Ät13"¢f'0459Yf nB&'0"TE 1©451DE x z4 ¾!#451©)g§»©B&!g)1D§459C4 f'Ì©}¥1©6n4 4§45f$'G!#E 3.0/10.0 Í© 1©'0)6'0¸'09 !g"T"T$9Y9A'0)"T' !gEDÌ©n4 gÍ 10.0*x %($h1D45¸'G18'h45%('G¸45f$&'§s º !g9u'0|r45%(B&f' v 4 }É"¢6 45h'0 !g) å 9A!#}É)Ätz%6!#'0f h'0'0%( 18!%(Bf 1©451D¾!#451©)g§»©B&!g)1D459C18%6'018"e!g)f 45z1©! h451©hE å 9k!g}¥)2Äw 45|r!#%6 45)d18$z'0|rB&9A'0hh!g) "T45)Å45§'G!g)'G!gED%645) B&!#nh f' ¸45f$&'05s0{)È§)gQ4 !g$&1D¾!#451©)g§»©B&!g)1D)61© E©$&~5~ }É4 n1845)2z)Åh459CB

yï7ð#C Cñ u7 «¢ "k "T!g)1©9Y45n118!¥1©' xhÊÊÊ %(!g'0f }É'09A'e182'e9A'0h$&f1!gE{'045"TQ¾&!g451©)g§»©B&!g)1!#B'¼9C451©!#)Q B&9A'0"Tn'¼f '0Û)'0 s x )61©' xhÊÊÊ %6!#'0f }¥'G"¢45)6B&9A!#¸'G1©451 (3.0/10.0)*10.0 '0§45f$4518'0 18! 3 Ì©'0!g9A'¼% ê5Í s x ) å 9k!g}¥)Ät %(!g'0f }¥'"T45))!g1s v )!g18'09p45% gg$1 )Å%(!#n1Df45)g§$&45gg''0Û&)1©!g)z"¢!#)2"T'09Y)2z}É451D45B&B&'0) !#) !g§'09Y¾!g} $)'09Y¾!g}Ë45)d!#1©'09p'0|À"¢'0B&18!g)Às0' xhÊÊÊ h1©45)459A@B&9A'0"Th'0f hB&'0"¢Û&'0ñ1©' '045§!g9 !gE{'0|r"T'0B&1©!g) 45)Ån!Éf45)g§$&45g§'0ñ18451{$h'I18'Ih1©45)459kÞ45 4%(!g'0f&"¢45)645¸!#d45) 45% g§$&1 !#)Å182zB&!g)1s v )!g18'09pgg9A' 459k'04"¢!#)"¢'09Y) 182'I7)18'09CB&9A'018451©!#)Å!#EDB459k'0)1©'0n'¼Às $'I1©!9k!g$&){!#E8E '09Y9A!g9Y 1©' 45nh!g"T451©§'Áf45}¥ !gE&45fgg' 9C4e{!)!#1&)'0"T'0hn459Yf !#f(E©!g9¾!#451©)g§»©B&!g)1 )$&% '09YÀs º !#9p'0|À45%6B&f' 1©'G'0|ÀB9k'0nh!g) (x+y)+z 245z41©!#1©45ff E8E©'09A'0)1D45)h}É'¼9 1©45) x+(y+z) }¥'0) Ï l Î X~ g ` » l Î q~ g 45) Ö l Ìh1{ l )(182'Eh!#9C%('09 "¢45h' Î) 1©'Gf451©18'09 Í s02'G%(B&!g9Y1©45)"T'!#EFB&9k'0n'09Y¸)gQB&459A'0)18'0n'0z"¢45))!g1 ' !g§'09A'0%(B&45n~5'0 s0'G45f7gg!g9Y1©%( B&9A'0h'0)1©'0@)Å18'0!g9A'0%(zÒ Ó45) ø 45f7f&'0B&'0) !g)61s º !g9p'0|À45%6B&f' )6{'0!#9A'0% ø5 18'E©!#9C%($f4 tÏXi d Ïy»DÌ©d Ï» ÜÏ Í }É!#$f@9A'0$&"T' 1©! §ÏXi ÏEF1D}¥'09A'0)Ät13Eh!#9pB&459A'0)182'0h'0 1©'09A' g '0h1©9A! )gQ182'G'0)1879k'G45fg§!g9Y1©%s v f45)g§$&45g§'G'0Û&)218!g)61©451D!g'0z)!g1D9k'0¿§$9k'B&459k'0)1©'0n'¼ 1©! 'G!g)!#9A'0@ $&n'0f'0h E©!#9p¾!g4518)g§»©B&!g)1D"¢45f"T$&f451©!g)5s qy$ '0|rB&9k'0nh!g)6'0¸45f$&451©!#)Åz%6B&9A'0"Th'0f '0Û)'0@)Å%(45) f45)g§$&45g§'05sGqy$&B&B&!gh' 1©451 ds z{!#$ f'GB&9A'0"Tn!#) ¸ $&1 x 45) y 459A'Gh)g§f'GB&9A'0"Tn!g)s0'0)Å)61©' '0|ÀB9k'0nh!g) ds + x*y z1©'GB&9A!g$&"T1DB&'09CE©!g9Y%('0d)6n)ggf7'G!#9p!g$ f'GB9k'0"¢h!g) ù v )!g18'09p'0|À45%6B&f'§0) x + m/n }É'09k' m 45) n 459A'G)1©'0g§'¼9C z1©'G¸h!g)645) )1©'0gg'09 !#B'¼9C451©!#)(!#9 4G¾!#451©)g§»©B&!g)1{!#)2' ù '09A'459A'18}É!¥}É4 ñ18!É'045f}É18(187 B&9A! f'0% )'0182'¼9u!#E3}¥2"T(ñ"T!g%(B&f7'¼1©'0f n4518nE©45"T1©!g9 s{'Û9Yn1{ñ1©!É9A'¼¿§$9k'I1©451 45ff¸¸459Y74 f'0p)45)Q'0|ÀB&9A'0hn!#)Q45¸'e182'eh45%6'e1 B&'§s¢{pu1©'en%(B&f'0n1h!gf$&1©!#) $1{245ñn!#%('G9Y45} 45"TÈ§5s º 9Ch13!#ED45ff f45)2gg$45gg'0zfÈg' Ã 45h"¢45f21©451345§'In$ 9C45)g§' 1 B&'0 45ff!g}Ë%(7|À)gQn$ 9C45)g§'G§459C4 f'0 }¥1©6)1©'0g§'¼9p¸459Y4 f'0 n!ç1D h!g%('0}É451 ~5459Y9A'G1©!B&9A!# 1D%(|r)gQh)g§f'G45)d!#$ f'GB&9A'0"Tn!#)Å§459C4 f'0Às v )!g18'09cB&9A! f'0% "¢!#)2"T'09Y)2 "¢!#)n1845)218Às x )ç1©'Á'0|rB&9k'0nh!g) 0.1*x %6!#n1&f45)g§$&45gg'0 )1©'09YB&9A'01DÎs l 1©! 'G4h)2ggf'GB9k'0"¢h!g)6"¢!#)n1845)21Cs0×¥!g} h$&B&B!#n'G18'B&9k!gg§9Y45%(%6'09 '0"T{'¼p1©!¥"T45)g§'e1©'e'0"¢f459Y451©!g)!gE45ff182'e¾!#451©)g§»©B&!g)1§459C4 f'0pE©9A!g% n)g§f' 1©!ç!g$ f'GB&9A'0"Th7!#)s x EFÎs l h1©ff&1©9k'0451©'0@45 4h)2ggf'GB9k'0"¢h!g)6"¢!#)n1845)21 1©'0) 1©'09k'G}Éff 'G4"T!g%(B&f7'G18%6'G'09Y9A!#9s0'GB&9A!ggg9C45%(%('09p}¥7ff&45¸'G18!$)1D!#}É) 45)@"¢45)g§'G'0§'09 ¾&!g451©)gg»hB&!#7)1D"T!g)h1©45)1s 'h'0"T!g)@45B&B9k!g45"TÅz1©!45ff!g}Ë%(7|À'0@'0|rB&9A'0hh!g) )6}É"TÅ"T45n'G9A$f'0zEh!#9 h$ '0|ÀB9k'0nh!g)6'0§45f7$&45187!#)Å%($&n1 'GB&9A!#¸'0 s02'¼9A'G459A'G4)$% '09p!gEDg§$&)g '0|À45%6B&f'05s0{'G!g9Yg§)45f&{'¼Û)1©!g)6!gEDo 9A'0¿§$&9A'0@1©451D'0§'09 ¾!g4518)2gg»©B!#)1 '0|ÀB9k'0nh!g) 'z"T!g%(B&$18'0Q)É!#$ f'zB9k'0"T7h!g)ÞÏ¤Üñ'09C)g§45)¥45)2Ùc1©"T' l5m5ê5ú Ñsn f'045 18!45)!g%(45f'0 fÈ§'G182'G'0|À45%(Bf'G451D18' '0gg)2))gQ!#EF1©zn'0"T1©!#)cs0'

«¢ "x ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ '0|ÀB9k'0nh!g) 3.0/7.0 "T!g%(B&$18'0@7)6!g$
f'GB9k'0"¢h!g) § $1DE q 4n)g§f'0» B&9A'0"Tn!#)ç¸459Y4 f' 1©'Á¿§$&!g18'0)1& 9A!#$)'0(18!n)g§f' B9k'0"¢h!g)çE©!g9h1©!#9C45gg'§sTqy)"T' ÒÚ ê z4y9k'0B&'0451©)g )2459 E©9C45"T1©!#) 18 "T!g%(B&$18'0d§45f$&')6!g$ f'B&9k'0"¢h!g)6 E8E©'09A'¼)21&E©9A!g% 18un18!g9A'0(¸45f$&'e)h)2ggf'eB&9A'0"Tn!#)s¢2$&u1©'Á"¢!#%6B&459Yn!#) ÒÚ ê E©45fÀs02zn$&g§gg'0h1© 18451D"¢!#%6B&$&1©)gQ'0§'09 '0|rB&9A'0hh!g)6)Å182'Gg§'0h1DB&9A'0"Th7!#) 45¸45f4 f'G )!g134ygg!g!#d9Y$&f'§s v )!g18'09g§$&7)gy'0|À45%6B&f' 7 ))'09B&9A!#$"T1©5s x E&1©'Á))'09B&9A!#{$&"T12245 18!g$&n45) !gED1©'09Y%( 1©'G9A!#$&)2)gQ'09Y9A!g9u)Å182'Gh$&%Ç"T45) '0"T!g%('Gn$ n1845)1©45fks Ð )'G}É4 18! 9A'0$&"T'G1© 9A!#$))gQ'09Y9A!g9u 1©!ç45"¢"T$%($&f451©'G1©'Gh$%(z7)6!g$ f'GB9k'0"¢h!g)6Ì©1© }¥f7f 'Gh"¢$&hn'0@)6%(!g9A'I'01©45f&) è Ð B&1©%(~5'09Y é !#)6B45gg' Ø l5m5Í s x E d 4!#$ f' B&9A'0"Tn!#)(§459C4 f' 45) x[] 45) y[] 459A'h7)ggf'IB&9A'0"Th7!#)(459Y9C4 1©'0)Q1©'I))'09 B&9A!#$"T1f!g!gB(}Éff¸f!g!gÈÉfÈ§' d = d + x[i]*y[i] s x E{182'e%($f18B&f7"T451©!#)Qp!g)'e) h)2ggf'eB&9A'¼"¢h!g) 18245)ç%6$&"T!gE&1©'e45§45)1©45g§'Á!gE&!g$ f'ÁB&9A'0"Tn!#)45"¢"T$&%6$&f451©!#) f!#n1 ¸ '0"T45$&n'G1©'GB&9A!#$"T1D 189Y$&)"¢4518'0d18!h)2ggf'GB9k'0"¢h!g) ½ $&h1 '0Eh!#9A' '0)g 45'0@1©!4!g$ f'GB9k'0"¢h!g)6¸459Y4 f'§s v 9Y$&f'G1©451D"T!g§'09C !#1©6!gED1©'GB&9A'0§!g$& 1©}¥!'0|À45%6B&f'0z 1©!ç"¢!#%6B&$&1©'G45) '0|ÀB9k'0nh!g)6)Å18'G2gg2'¼n1DB&9A'¼"¢h!g)6!gED45) ¸459Y74 f'G1©451D!#"¢"T$&9Cz7)61©451 '0|ÀB9k'0nh!g)s02'¼) q = 3.0/7.0 }¥ff 'G"T!g%(B&$&1©'0@'0)1©9A'0f )6h7)ggf'B&9k'0"¢h!g) ¥ 45)@}Éff&245§'G1©' !g!gf'045)6§45f7$&'G1©9A$' }¥'09A'045 d = d + x[i]*y[i] }Éff ' "T!g%(B&$18'0@7)6!g$ f'GB9k'0"¢h!g) g§45))gQ1©'GE©$ff&45¸45)1845g§'G!gED!g$ f'GB&9A'¼"¢h!g) 45"T"¢$&%($f4518!g)s 8mÉ!#}É'0§'09 1© 9Y$&f'G 18!g!h%(Bfh1©"Á1©!"T!g§'09p45ff&"¢45h'0 "Tf'045)f s x E dx 45) dy 459A'G!#$ f'GB&9A'0"Tn!#)Å§459C4 f'0 182'G'0|ÀB&9A'0hn!#) y = x + single(dx-dy) "¢!#)1©45) 4!g$ f'B&9k'0"¢h!g)6¸459Y4 f' § $&1DB&'09CE©!g9Y%(7)g 1©'Gh$%Æ)Å!#$ f'GB&9A'0"Tn!#)Å}¥!g$&f 'GB&!g)18f7'¼n § '0"T45$&n' !g18Å!#B&'09C45) 459A' h)2ggf'GB9k'0"¢h!g) 45 z1©'G9A'0h$f1Cs v %(!g9A'Gh!gB&h1©"¢4518'0@n$ '0|rB&9A'0hh!g)6'0¸45f$&451©!g)69Y$&f'G 45zEh!#ff7!#}É5s º 9Yn1D45hhg§) '045"Tç!#B&'09C4518!g)É418'0)1©451©§' B&9A'0"Tn!#) }¥7"TÉ 1©' %(45|r%($%Ë!gE21©' B&9A'0"Th7!#) !gE 1©z!gB&'09Y45)Às0 45hngg)2%('0)1D45z1©! 'G"T459Y9C'0@!g$&1DE©9k!g%Æ1©'Gf'045§'0 18!182'G9k!g!#1 !gE318'I'0|rB&9k'0nh!g)(189A'0'§s¼'0)(B&'09CE©!#9C% 4h'0"¢!#)ÞB&45nñE©9A!#% 1©'I9A!#!g1{1©!É1©'If'045§'05s x )61© B&45h 45hh7gg)Å18!'045"TÅ!#B&'09C4518!g)É1©'G%(45|r%($%Æ!gED1©'G1©'0)1©4518¸'GB&9A'0"Th7!#) 45)@1©'GB&9A'0"T7h!g)6'0|rB&'0"T1©'¼ g 182'GB&459A'¼)21Cs x )Å18'G"¢45h'G!gE q = 3.0/7.0 '0§'09 f'045E{ph)2ggf'B&9A'¼"¢h!g) h!¥45f7f¸1©'!#B&'09C4518!g)p459A'e!g)')n)g§f'eB&9A'0"Tn!#)cs x )Q182' "T45n'!gE d = d + x[i]*y[i] 1©'1©'0)1©4518¸'IB&9k'0"¢h!g)(!gE318'I%($f187B&f !gB&'09Y451©!g)( h)2ggf'GB9k'0"¢h!g) § $1D)61©'Gh'0"¢!#)dB&45h 1Dgg'01©zB9k!g%(!g18'0d18!!g$ f'GB9k'0"T7h!g) R S e ha V a a W ` a d e ` d b W W b g d b g Ç ` g dhb g d e V d 3.0 ha V a hg j Y` ¾ X f ` d ha hb g d b g a d V g di e hY` 3.0D0 ha V y b e Y` X f ` d ha hb g d b g a dV g d S

yï7ð#C Cñ u7 «¢ ¢¡ '0"¢45$&h'G1© B&459k'0)213!gB&'09Y451©!g)6'0|rB&'0"T1©z4{!#$ f'GB&9A'0"Tn!#)Å!#B&'09C45) s v )d) y = x + single(dx-dy) 1©'G451©!g)6 !#)2'G)6n)g§f'GB&9A'¼"¢h!g)s º 459C)$&% Ï l5m5ú5ú ÑFB&9A'¼n'0)1©z'0¸'0)"T'G1©451D1©z45f7gg!g9Y1©%Ç)6)!g1DE8Û&"T$f1D18!%(B&f7'¼%6'0)1Cs 'Án45§45)1©45g§' !gE&182 9A$f' u18451&18'Á'0§45f7$&45187!#)ç!gE&4en$ '0|rB&9k'0nh!g)ç'0B&'0) !g)61©'G'0|ÀB&9A'0hn!g)6)6}É"¢61D '¼% '0'0 s0{ "T45)6245§'Gn!#%('45))! )g "T!g)n'¼¿§$'0)"T'0Às º !g9p'0|À45%(Bf' h$&BB&!#n' !g$Þ459A'G' $&ggg§)gQ4B&9A!ggg9C45%Æ45)2@}¥45)1 1©!çÈ§)!g}Ë1©'G¸45f$&'G!gED4yh$ '¼|rB&9A'0hn!#)s q³¸!g$Þ"¢45))!g13n%(B&f 1 B'G18' h$ '0|ÀB9k'0nh!g)Q1©!¥1©'' $ggg§'09°45)2Å45hÈç1{18! ''0¸45f$&451©'0 # '0"T45$h'1©'§45f$'!#E 1©' h$ '¼|rB&9A'0hn!#)É)ç18' B&9A!ggg9C45%Ë'0B&'0)°!g)É1©' '0|ÀB&9A'0hn!#)É12°'0% '0'0Q)s v Û&)245f&"T!g%(%6'¼)213!g)6n$ '0|rB&9A'0hh7!#)À0h)2"T'G"¢!#)¸'09Y1©)gQ'0"T%645f&"T!g)n1845)1© 18! )2459 45)6!gB&'09Y451©!#) 182'G'0§45f$&451©!g)69Y$&f'G45fn!45EûE©'0"¢18 1©'G)1©'09YB&9A'018451©!g)6!gE '0"T%645f#"¢!#)2h1©45)18Àsh{°°'0hB'¼"¢45ff %6B&!#9C1©45)1E©!g9F"T!g)n1845)1©°fÈ§' 0.1 }¥2"TÉ459k' )!g1D'¼|r45"T1©f 9A'0B&9k'0n'0)184 f'G) )459 s v )!g18'09pB&!g18'0)1©45f&g§9A' 459A'04!g"T"¢$&9Y }¥2'¼)Å4f45)g§$&45g§'G)"¢f$&'0 '0|ÀB&!g)'0)1©45187!#) 45 !#)'G!gEF18 $&7f18»h)6!gB&'09Y451©!#)25s0÷I)fÈ§'G1©' 45n"Á459C1©%('01©"Á!gB&'09Y451©!g) 18' ¸45f$&'!gE{'0|rB&!#)'0)218451©!g)Qñ)!#1{45f}É4 ñ! §!g$&ÏÜz4545)Q45)2Åop!#!g)'0) l5m5ú Ø5Ñs x E ** 18''¼|rB&!g)'0)1©4518!g)6!gB&'09C4518!g9 1©'0) (-3)**3 "¢'09Y1©45)f 45 182'G§45f$'G»©Ø ê s m¥!g}¥'0¸'¼9 (-3.0)**3.0 pB&9A! f'0%(451©"T45fks x E{182' ** !gB&'09Y451©!#9 "T'0"TÈ§pE©!g9°)218'0g§'09 B&!g}¥'09Y 1D}¥!g$&fd"T!g%(B&$&1©' (-3.0)**3.0 45 »©Òs Î X~ »©Ø ê s Ð )61©'G!g182'¼9p45)2 E 1©'GE©!g9Y%6$&f4 tÏXÿ W tÿ¡ ¢¤£¦¥F $&n'0@18!'0Û)' ** Eh!#9p9A'¼45f459kg§$&%6'¼)218 182'0)6'0B&'0))2g !g)61©'Gf!ggEh$&)"¢18!g) 1©'G9A'0h$&f713"¢!#$f 'G4 NaN Ì©$&n)gQ1©'G)451©$&9Y45f'0Û&)718!g)6!gE f!gggÌ nÏ Í NaN }¥2'0) ëÏ£@Î Í s x EF18' º Ð Ù3Ù v × CLOG Eh$&)"T1©!g)6 $&n'¼d!g}¥'0§'09 1©'0)61©'G45)n}¥'09p}¥ff 'G»©Ø ê5§ '0"T45$&n'G182' v ×Éq x º Ð Ù3{Ù v ×Ën1845)2459A@'0Û&)'0 CLOG(-3.0) 1©! ' q π P@f!#gQÒ6Ï v ×Éq xFl5m5ê5ú Ñs0{'GB&9A!#g§9C45%(%()gQf45)2gg$&45g§' v 4 45¸!#ñ182pB&9A! f'0% g !#)f '0Û))gç'0|rB&!#)2'¼)218451©!g)QE©!g9 )1©'¼g§'09 B&!#}É'09Y }¥f7' v ×¥q x º Ð ÙDÙ v × B&9A!# 18 9C45h)gQ4)2'¼g§451©§'G)2$&% '09p1©!ç4y9k'045fB&!#}É'09Cs x )6E©45"¢1 18' º Ð Ù3Ù v ×Ëh1©45)459A@n4 182451 v ) 459Y1©%('01©"e!#B&'09C4518!g)6}É!gh'G9A'0h$&f713 )!g1D%(451©'0%(451©"T45ff '0Û)'0@ B&9A!#2 1©'0 sss ÷I)E©!g9Y1©$&)451©'0f ¢ }¥1©6182'G)189A!g$&"¢18!g)6!gE ±∞ g 1©' x©Ê{ÊÊ h1©45)459A 1©'G%('045))g !gEDrgbTSDdIR¢S©P5WYdIRTS qtp RTutut`v5W 1 r§WYv )!çf!g)g§'¼9p18!g1845f7f "Tf'0459p"T$&1s Ð )'G'0Û&)218!g)6%(g§1 'G18!$&h'G1©'I%6'018!g@h!g}¥)6)6n'0"T1©!#) è5x )Û)1 gé !#)6B&45g§' l5m5m s º !#9p'0|À45%(Bf' 1©! '01©'¼9C%()2' 182' §45f7$&' !gE&R¨§ "T!g)h'09)!g)»©"¢!#)2h1©45)1245)45f 18"pEh$&)"T1©!g) äu45)É}É18 1©'GB&9A!#B&'09C1 1©451 8äTÌ öÏ Í → Re45)ÉÌ nÏ Í → I45 Ï → Îs x E ~änÌ nÏ Í¡© û¥ ü 45f}É4 45B&B&9A!g45"T'0 1©'Gh45%6'Gf%(1 1©'0)61© h!g$&f 'G1©'G§45f$'G!#EFR¨§Cs0 '0Û&)218!g)6}É!#$f@n'¼1 Ø ∞ ∞ }É"TÅh'0'0%( ¿§$&1©'G9A'045h!g)4 f'§s x )61©'G"¢45h'G!gE l s Î ∞ }¥'0) äTÌ öÏ Í l 45) Ì öÏ Í l Ú ÏG1©'If%(1{45BB&9k!g45"T2'¼ l5g $1{}¥'0) QäTÌ öÏ Í l » qÏG45)2yÌ nÏ Í l Ú Ï182'f%61

«¢ "Ñ ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ ñW n!#Ts¼qy! l s
Î ∞ n!#$f 'I4 NaN s x )(1©'I"T45h'I!gE3Î g änÌ nÏ Í¡© û¥ ü W © û¥ ü ¢¤£ û¥ ü sqy)"¢' ¥ä 45)¥ 459A'G45)45f 18"e45)@1©45È§'G!#)Å182'G§45f$'GÎ451DÎ äTÌ öÏ Í R höÏ h P@R Ï P … 45) Ì öÏ Í 5hnÏ h P Ï P … s0$&zf% ¥ → g¤Ì öÏ Í f!#g óäTÌ öÏ Í f% ¥ → gnÏef!gggÌ nÏ5Ì©R `ãPR ÏüP … Í©Í f% ¥ → gnÏf!#g§Ì©R `öÏ Í Îs0qr! óäTÌ öÏ Í¡© û¥ ü → W g l E©!#9p45ff äÁ45)2É }É"TÅ%('045) 18451 Î g l s q2e÷In)gQ187z'0Û)1©!#)Å}¥!g$&f@$)45% 7gg$&!g$&nf '0Û&)'G1©'G'0|rB&!g)'0)18745f E©$)"T1©!#)QE©!g9c45ff¸459Ag§$&%('0)1© 45)Å)B&459Y1©"¢$&f459c}É!#$f6'0Û&)2' (-3.0)**3.0 18! ' »©Ø ê s ÙÚÜÛ¾¿5o È ¢|n&ËY|)ÞtË qy'¼"¢18!g) §è {' xhÊÊÊ qy1845)2459k é h"¢$&hn'0@%(45) !gED1©'GEh'¼451©$&9A'0 !#EF18' x©ÊÊÊ h1©45)459A s 8m¥!g}¥'0¸'09 1©' xhÊÊÊ n1845)459A@n4 )!g182)gQ4 !g$&1D!g}Ë1©'0h'GEh'04518$&9A'0 459A'G18! '45"T"T'0nh'0@Eh9k!g%Æ4yB&9k!gg§9Y45%(%6)gQf45)g§$&45g§'gst$& 1©'09A'Gz$h$&45ff 4 %(n%(451©"T '018}É'0'¼)Å¾!#451©)g§»©B&!g)1D459A}¥459A'G1©451Dh$&B&B!#9C18 182'Gh1©45)459A@45) B&9A!#g§9Y45%6%()gQf45)g§$&45g§'0zf7Èg'Go Ã 45h"¢45f&!g9 º Ð ÙDÙ v ×@s0qr!g%('G!gED1©' xhÊÊÊ "T45B&4 f1©'0z"¢45) 'G45"¢"T'0nh'0@1©9A!#$ggÅ4f 9C459 !gEDn$ 9A!#$18)2'G"T45ffÀs º !#9p'0|À45%6B&f' 1©' xhÊÊÊ n1845)2459k69A'0¿§$&9A'0 1©451&n¿§$&459A'Á9A!#!g1 'Á'0|r45"T1©f 9A!#$)'0 45)2(1©'Án¿§$&459A' 9A!#!g1DE©$&)"¢187!#)Åz!gE©1©'0)6%(Bf'0%('0)1©'¼d9A'0"T1©f 7)6459A}¥459A'§s02zEh$&)"¢18!g)45f1 '045hf 45"T"T'0nh'0Þ§44f 9C459 n¿§$&459A'9A!#!g1{9A!#$18)2'§s ¨m¥!g}¥'0¸'09 !g1©'09 45hB&'0"¢18ñ!gE318' h1©45)459A@459A'G)!g1Dh!'045hf %6B&f'0%('0)1©'0@45zn$ 9A!g$&1©)'05s º !g9u'0|r45%(B&f' %6!#n1 "T!g%(B&$18'09pf45)g§$&45g§'0znB&'0"TE 451D%(!gh1D1©}¥!¾!#451©)g§»©B&!g)1D1 B&'0 }¥f'18' xhÊÊÊ h1©45)459A@45 E©!g$&9pE8E©'09A'¼)213B9k'0"¢h!g) Ì©45f1©!#$ggÅ18'9k'0"¢!#%(%6'0)'0 "T!g)Ûgg$9Y451©!#)2z459A'Gh)2ggf'GBf$& h)g§f'0»©'0|r18'0)'0d!#9ph)2ggf' {!#$ f' 45)@{!#$ f'0» '0|À1©'0)'0 Í s x )Û)1 B9k!g§{'¼ñ45)!g18'09 '0|À45%6B&f'§sop!#)n1845)218ñ18!É9A'¼B9k'0n'¼)21 ±∞ "T!g$&f 'Án$&B&B&f'0 g 4h$ 9A!#$&1©)'§s å $&1&18451&%6gg1&%645Èg'Á1©'0% $)$&n4 f'Á)B&f45"T'0u182451 9A'0¿§$&9A'G"T!g)h1©45)1D'0|ÀB9k'0nh!g) n$&"¢645 18'G7)18745f~5'09u!gED4y"T!g)h1©45)1D§459C4 f'§s v %(!g9A'Gh$ 18f'Gn1©$&45187!#)Åz%645)B&$&f451©)gQ1©'Gh1©451©'G45hh!g"T74518'0@}É18Å4 "T!g%(B&$18451©!g) }¥2'¼9A'I1©'In18451©'I"T!g)h7h1©ñ!gE31©'I9A!#$))g%(!g'0 1©9C45BÅ'0)4 f' 18 1©9Y45BÞ245)f'09Y 45)@'0|r"T'0B&1©!g)6¾45ggÀs Ð )'G45B&B&9A!g45"T6 1©!çB&9A!g§'h$ 9k!g$&1©)'0 E©!#9 9A'045)g¥45)(}¥9C1©)g¥18'Án18451©'§s x )ç451©!g) 4h7)ggf'Á"¢45ff§1©451&"T45)ç451©!g%("T45ff n'0124 )'0} §45f$'45)Å9A'018$9Y)Q1©'!gfÅ¸45f$&'ñ!gE©1©'¼)($&n'¼Eh$&fks v ñ182''0|À45%(Bf'0p) è º f45g§ é !g)ÉB45gg' Ø l Îh!g} 4§'09 "T!g%(%(!g)çB&451©18'09C)É!gE&%(!gE )g x©ÊÊ{Ê n18451©'Á 18!y"T45)g§' R S e ` d b g d Y a hb g d e V d ¡ g ¢ R y ` X ` g y a b g de ` f ` a d f hd d hb g de V d e ` g b g d b g a dV g d S c de ha f ` a df hd d hb g ha f ` W b Ç ` y i d e ` g Y` dd hg j e ` d e ` hy ` g d hd V YYf ¡ c g d d hb g j hÇ ` a ¡ V a V X b a a he Y` Ç V Y ` c b f i V g y a b ¡ g b Yy e V Ç ` d b e ` y ` w g ` y db e ` V NaN S S g d e ` d V a ` b c ¡ i X YV a he hYhd f V f j W ` g d a d V g e ` W V y ` i e d de ` d b g Ç hg d hg j V f j W ` g d ha c b g y hg Æ b g d f ` d` V de ` W V d hd a Ç e f f V e V W i g de V g y × V d V a e g h« i V g y V f j ` a d e V d ¡ ¢ R c b f de ` e hg b W hV Y de ` b f ` W d b b f « S ¾ ¾ T y S f x ( ) x 0 → lim g x ( )

yï7ð#C Cñ u7 «¢ "ì 1D!#)2f }¥1©)61©'Gn"T!gB&'G!#EF4 f7!#"¢È!g9ph$ 9k!g$&1©)'§s0{$& 18' $&9A'0)6 !g)61©' B&9A!#g§9Y45%6%('09p18!Û)@'045"TÅ'0|À1DE©9A!g%Æ1©' f7!#"¢È 45)2@%(45È§'Gh$9k'G1©'Gn18451©'G 9A'0h1©!#9A'0 s Ô 45)g§$&45g§'Gh$B&B&!#9C1DE©!g9un'0181©)gQ1©'Gh1©4518'B&9k'0"¢h'0f )61©'Gn"T!gB&'G!gED4 f!g"TÈQ}É!#$&f7 'G¸'09 $&n'0E©$&f&2'09k'§s0ij!g$&f45»©Òyz!g)'Gf45)g§$&45g§'G1©451D%(B&f'0%('0)218 1©z7'04E©!g9u1©9Y45B@45)f'09CÏ¤×¥'0fh!g) l5m5m5l Ñs '09A'G459A'G4)$% '09p!#EF%()2!#9pB&!g)18 1©451D)'0'0@1©! 'G"T!g)h{'¼9A'0@}É'0) %(B&f7'¼%6'0)187)gQ18' xhÊÊÊ h1©45)459A@)64yf45)g§$&45gg'§stqr)"¢' ùÏ» ÜÏ P&ÎEh!#9p45ff ãÏ Ì öPÎ Í »&Ì öP&Î Í P&Îs "mÉ!#}É'0§'09 »hÌ öPÎ Í »©Î 1©$&p» nÏIh!g$&fÅ)!#1 ''0Û&)'0645pÎe»2|&s¢{' )1©9A!#$"T1©!#)Å!#E NaN z"¢45) '"T!g)E©$&n)g § '0"T45$h'G4 NaN z)2'¼¸'09p'¼¿§$45f&1©!ç45) !g18'09p)$&% '09pÌ©)"¢f$&)gQ45)!g1©'09 NaN Í© h! þÏ Ï )!f!g)g§'¼9p45f}É4 189Y$&'§s x ) E©45"¢1 1©'G'0|rB&9A'0hh!g) êÏ ≠ Ï 182'Gh%6B&f'0h1D}¥4 1©!ç1©'0h1DE©!g9p4 NaN EF182' x©ÊÊ{Ê 9A'0"T!g%(%('0)'0dE©$&)"¢18!g) Isnan z)2!#1DB&9A!#¸'0 s º $&9C18'09C%(!g9A' NaN z459A' $&)!g9A'09k'0@}É18Å9k'0hB'¼"¢131©!ç45f7f&!#1©'09u)$% '09C h! þÏ ≤ `"¢45))!g1 'G'0Û)'0@45zrgb¢S Ï `¼s0qy)"T'G1©'G)2189A!#{$&"T1©!g)6!gE NaN "T45$&n'0z¾!g4518)2gg»©B!#)1D)$% '09Cz1©! '0"T!g%(' B&459C1845ff !g9A'09k'0 4 compare E©$&)2"T1©!#)Å18245139A'01©$&9Y)2z!g)'G!gE '£ !#9pfgr§bTXYv5WYXkWYv "T45)Å%(45È§'G1D'045h'09pE©!g9u1©'GB&9A!ggg9C45%(%('09p18!'045f}¥1©6"T!g%(B&459Ch!g)Às v f182!#$ggÅ18' xhÊÊÊ h1©45)459A@'0Û&)2'0z1©' 45n"Á¾!g45187)gg»hB&!#)213!gB&'09C4518!g) 18! 9A'018$&9C)64 NaN ED45) !gB&'09Y45)2@ 4 NaN 1© %(g§1D)!#1D45f}É4 'G1©' '0n1 '0Û&)218!g)6Eh!#9p"T!g%(B&!g$&)2@!#B'¼9C451©!#)Às º !#9p'0|À45%6B&f'G}¥2'0)6"T!g%(B&$18)2g1©' 45B&B&9A!#B9Y451©'Gh"¢45f'GE©45"¢18!g9u1©!$&n'G)6B&f!g1©18)gQ4g§9C45B& 1©'G%(45|r%($%Æ!gED4n'013!gE ¸45f$&'0z%($&n1 'I"¢!#%(B$&18'0 s x )(1©z"T45n'I13%(45È§'0zh'0)n'IE©!g9u18'G%(45|y!gB&'09Y451©!#)61©! h%6B&f gg)2!#9A' NaN 5s º )45ff ¢ 9A!#$&)2)gQ"T45) 'G4B&9A! f'0%s0{' xhÊÊÊ n1845)459A@{'¼Û)'0 9k!g$&){)gQ§'09 B&9A'0"Tn'¼f T 45)d1D'0B&'0) !#)Å18'G"¢$&9C9k'0)1D¸45f$&'G!gED1©'G9A!g$&))gQ%6!#'0Às0 h!g%('01©%('0 "T!g)¾"T1©z}É18Å182'G'0Û&)718!g)6!gED%6B&f"T7139A!g$&))2g)Å1 B&' "T!g)¸'09Yh7!#) !g9u1©'G'0|rB&f"T1 round Eh$&)"¢18!g)6)Åf45)g§$&45g§'05s02z%6'045)z1©451 B&9A!#g§9Y45%6z}É"TÅ}¥n61©!$&h' xhÊÊÊ 9k!g$&)7)gQ"T45)Ät1D$&h'18'G)24518$9Y45f&f45)2gg$45gg' B&9C%(1©§'0 45)@"¢!#)¸'09Yn'0f 18'Gf745)gg$45gg'GB&9C%(1©¸'0z}Éff 'G7)'0EûÛ"T'0)1D1©! %(B&f7'¼%6'0)1D!#)Å18''¼¸'09p)"T9A'045h)2g)2$&% '09p!gE xhÊÊÊ %645"T)2'¼Às § 6¢¤ßùÑ Û¨Þ%} op!#%6B&f'09°1©'0|À1©p1©'0)Å18!Égg)2!#9A'e1©'h$ ¼½ '0"¢1!gE{¾!#451©)g§»©B&!g)1s º !g9°'0|À45%6B&f' v 2!¥'01 45fks#Ï l5m5ú5ø Ñ{%('0)1©!g)ñ9k'0B&f45"¢)g x/2.0 }¥718 x*0.5 f'045)gç1©'9A'¼45'09 1©!¥45nh$%(' 1©451 x/10.0 h!g$&f 'e9A'¼B&f745"T'0 g 0.1*x s "m¥!#}É'0§'09 182'0h'Á1©}¥!¥'0|ÀB&9A'0hn!#)2u! R S ± g Y` a a d e ` f b g y hg j W b y ` ha f b g y d b V f y ¾ ∞ i hg e hd e d V a ` § ¾ § ¢ ¾ ¡ S

«¢« "ý ¬ 5uC¢h&Cu¢57C7® °¯ 5¢ )!g1D45§'G1©'Gn45%('Gh'0%645)187"T !#)Å4 )459
%(45"T2)' § '0"¢45$&h'GÎs l "T45))!g1 ' 9A'0B&9k'0n'0)18'0d'¼|r45"T1©f ) )459 s0{z1©'0|r1 !g!gÈ45f7h!h$&g§g§'0h1©z9A'0B&f45"T)2g x*y-x*z g x*(y-z) '0¸'0)61©!#$ggÅ}¥'G45¸'Gh'0'0)Å18451D1©'0h'G1©}¥!'0|rB&9A'¼nh!g) "T45)6245§' ¿§$&1©'GE8E©'09A'0)1D§45f$&'0 }¥2'¼)Å` ≈ ~s v f1©!g$&g§61D!g'¼ ¿§$&45f7E 18'h1©4518'0%('0)1D1©451 45) 45fg§' 9Y45"e'0)1©1 "¢45) 'G$&h'0d}¥'0)6!gB&1©%(~5)2g"¢!#' g )!#1©)gQ1©451 !gB&18%6~5'09Y h!g$&f@)!g1D§!gf451©'I1©'Gf45)g§$&45g§'G'0Û&)218!g) 1Df'045§'0 18'G%6B&9k'0nh!g) 1©451D¾&!g451©)gg»hB&!#7)1Dh'0%(45)1©"T 459A'G)!g13¸'09 %6B&!#9C1845)21Cs Ý 2'¼1©'09p!#9p)!g1D182' f45)g§$&45g§'Gh1©45)459A@nB&'0"TÛ'¼ 1©451DB&459k'0)218'0nz%6$&h1 '!#)2!#9A'0 (x+y)+z "¢45) 45¸'G41©!#1©45ff EûEh'09k'0)1D45)n}¥'09p18245) x+(y+z) 45 h"¢$&nh'0@4 !g¸'gst'09A'G 4 B&9A! f'0%Æ"¢f!#n'0f 9A'0f451©'0@1©!çB&9A'0h'09C§)2gB459k'0)1©'0h'0 1©451Dzff7$&h1©9Y451©'0 g 1©' E©!gff!g}¥)gQ"¢!#'§ 7z '0hg§)'0@1©!gg¸'G45)6'0n18%64518'GEh!#9p%(45"¢)'G'0B&nf!g)s x ED45)Å!#B&1©%(~5)2g "T!g%(B&f7'¼9p)!g1©"T'0 18451DWYe s P l l ⇔ WAe s Î 18'GB9k!ggg9C45%Æ}Éff 'G"T45)2gg'0 "T!g%(B&f'01©'0f s x )h1©'045Þ!#E3"¢!#%6B&$&1©)gç1©'Ih%(45ff7'¼n1{)$% '09 ¥Ïh$"T(182451 l ⊕ Ïñh1©ff g§9k'0451©'09F1©45) ÏeÌ nÏ ≈ W ≈ Í© 12}¥ff§"T!g%(B&$&1©' 182' f459kg§'0h12)$&% '09 ÜÏE©!#9}¥2"T ÏÚyØ 9k!g$&){'¼d18!ÎÌ nÏ ≈ Í s v ¸!#)2g1© Èg7)@!gE è !gB&187%(~5451©!#) é zn! %(B&!g9C1845)1D1©451D1Dz}É!#9C18ÅB&9A'0h'0)1©)gQ!#)2'G%(!g9k'G¸'09 $&n'0E©$&f45fgg!g9Y1©%Ç1824513 1©!#1©45ff 9A$)'0 § 1s ij45) B&9k! f'0%6 n$&"TÅ45z)$%('09Y7"T45f&)218'0g§9Y451©!#)Å45)@1©'G)$%('09Y7"T45f&n!#f$18!g)6!gE E8E©'09A'¼)21845f'¼¿§$4518!g)p)¸!#f¸'e"T!g%(B&$&1©)gÉh$%(p}¥1©%(45) 1©'09Y%(Às å '0"T45$h'e'045"T 451©!g)("T45)6B&!g18'0)1©45ff )1©9A!#$"T'I45)6'09Y9A!#9u45zf459Agg'I45sþ ulp 4h$&%Æ)¸!#f¸)g 1©!#$h45) !gED1©'09Y%( "T45)Å45§'¿§$&1©'G4 1D!gED9A!#$))gQ'09C9k!g9Cs v n%(B&f'G}É4 1©! "T!g9Y9A'0"T1DEh!#9p182z 1©!çn18!g9A'G182'GB&459Y1©45f&n$&%(%645)@)Å4!g$ f'GB&9A'0"T7h!g)6¸459Y4 f7' 45)(1©!B&'09CE©!#9C% '045"Tç45{18!g)ç$&n)gy!g$ f' B9k'0"¢h!g)s x E18'Á"T45f7"T$&f451©!g)ç '0)g !g)'G)6n)g§f'GB&9A'0"Tn!#) B&'09YE©!g9C%()gQ1©'Gh$%Æ)Å!#$ f'GB&9A'0"Tn!#)Åz'045 !#) %(!gh1D"T!g%(B&$18'09p h1©'0%(Às 8m¥!g}¥'0§'09 ED1©'G"T45f7"T$&f451©!g)6 45f9A'045 '0)gQ!#)2'G) !g$ f'GB9k'0"¢h!g) !g$ f7)gQ18'B&9k'0"¢h!g)6 )!g1Dh!h%6B&f'§s Ð )2'G%('01©!#d18451D h!g%('01©%('0 45§!g"T451©'0@ 18!n!#9C131©'G)2$&% '09Cz45)2@45@1©'0%ÇE©9A!#%Çh%(45ff7'¼n1D18! f459Agg'0n1Cs 8mÉ!#}É'0§'09 1©'09A'Gz4y%($&"¢6%(!g9A'G'0EûÛ"T'0)1D%('01©!g@}¥2"T6{9Y45%(451©"T45f7f %(B&9A!g§'0 182'G45"T"T$9Y45" !gEFh$&%6 )45%6'0f eps = 1; do eps = 0.5*eps; while (eps + 1 > 1); β p – βe min

!#"$¦%'&)(10 2)3456$¦%8759A@B$¦C56@ED¦750 45FEGIHPC50 QSRUTV0 45W5FE0 6@EC¨X`Y6ab0 @EC5 Rc%¦@E62)$¦6@E9
dde fhgpirqtsuirvxwytgtUUvvtAqtqtsrvUt dfe#g5g5hikjmlon5pl qri)sVhtug5eflojwvefikqrxuylon5jbzh{'{rh|cqrxyp{ yth}wqrlon5t ~ n5jwxlon5j)sVhtug5eflojwviweftdqri)jwefp{Pl¤h |cn5jw}Aj u¨tott5o#w1 fotb11ouüf151o Pk¢¡ δ£ ¡¥¤¦ x§¨©oj uª #«15w6t¬t 6¬¬tu kbmot®U5t51«5¬# o#w1t U¯w55¬°«1k#fu«ut u± 51««5P1uwo1 ²#u²#³#t³ ´ j #töu5#I1uwo1 ²#5obtU5c5k#u5 xfj «¬Pµ«1u¤b «1 ¶ u¬5U5Aut · ± A#w¸¤¹`1««5o#tº#15f´5»5¼ ½ ¾b¬¿5uÀot5²f6ufuÁb5ot § 1#tt5wot«uoÂ#²#u³fu¬tU ¥#t5#u²fw #1bP1ot5ÀÃÄÆÅE~ §odfÇÀ§ÀÈ ÄÉÅE¦ d#ÊÈ¥©o§odfÇÀ§ÀÈ ÄË5fkutu tot5fb ¬P #«1uou³1üu uÌ tuüÍ85«1u5²#¨««5 ¿5u²#³5³#tot5 #1¬¿5uw¨t#1Î²#uÍ8¬ku«u³5¬#1 tu511³ft5#u²fw5°uto¬u ot5ÀtbÎo#)ot5¬tu5¬51ku51t1²#uwouÍ81ku¨¨#ttbtÁb5ot § 1#tÀf5 5²#u ½ t#¬tu 5³ot5Àb1o«6¿5 5t5tf¬Pü«5P¬U#ÀÁ#5¬P § 1bPIb tbuUbPö5t¬ ¸¤otuÍ81Au¨# 1.0E-40*x fotuk5«16Àu5 o²ft5w³btuwbb 5ob¬t5Àbtuwo¬tÏ«5t1«²fu)o²#t5 ³ #t¬5t uÐ u51öt#tö5tÀ5PtbÀ²fAu1kuütouuÍ85o³²#6t5w³fot tuÍ85tuÍru1¬6bÑ¨t#1²fUw5u ¸¤¾5o#t5otU1PuwÁ Á5¨t#1¬h²# üÀp¬thÍ81ku¨bUu515outf1kub ¹8tot#tö5tÀ tuÍ85oroÒuÍ85P#tfu ¨#Ñoh²ft5 ³u1utÏÒ#Ñot1 AuPt51#Pot¸ ±±± kbtt«#u uÌ t15#1o«¿5um¬t5Àbtuwo 1.0E-40 ¬²#6t5w³5 #«16¬ #1²ft5P#Potu«5tobp¬P1kbfw5 uÓ ufu¬ #t¬5toÔ#´5Õ Ö t5kuÍr5¬6³ku1kuüto5²#¬P¨«56À5f55²f 1kub5²#¨5ou³#tfu ¬u5À#«1o«uf¨tuotu³5Au5 5³fUuÍr5¬ S = X[1]; C = 0; for j = 2 to N { Y = X[j] - C; T = S + Y; C = (T - S) - Y; S = T; } Σ j 1 = N x j Σx j 1 δ j + ( ) O Nε2 ( ) Σ x j , + δ j 2ε ≤ ( ) Σx j Σx j 1 δ j + ( )

ddd ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ 5PtbÀ 5¨5t³uÍ8u1obPf5t5k1P5¡ou¬uÎ²#³¬thA#1t6t«bÏ ª
bPö5t¬ot5À5A6t¬utÏÎ¬²#h#tfu ¬u5Àb1o«¨Pb²##t o¯#tö5tÀu5 5¬6bt¨15 const pi = 3.14159265 ª b«#Ñ¨1²fuÍr1ku¨bÑut5obÑÂ5tbotupuÍ85«1#P5Ñb¬¿55¬#Ñot5 5t5P#Á#5¬P § btÀu«5tof5U61öw5ou²f³ot¤b PbÏ ½ ¬Pb# A*B «5³¾51u5o¯²#°µ#«b¨²#uÍr1ku¨#t816ÂPb1²#u51¨°¬P kbtt«#¡¤uAutÀ5Àot uf55¬#¬u uÌ tkuÙt51uÍ85«1u5Ú Û Ä Û 5tt#I²#uk165u²#³¬Pu²#b#u5«bt¬5P true #²fu51uÏ¤5 tu Û ) NaNÜ § Û Ä«Ë § Û o5)o# Û Ä Ê Ë Ü 5t Û ¤ÝuÂPb1otS#11#öSb Û ≥ Ýw`²#u51 NaN 5AP6¬tumfAu5ou)ot5tbmuöt5#kt5w³Á#5¬P § btÀt«²#u ¶ u1¬h¬tuuÍr51u¨fotuk5k1üo1#1¬¿55¬6btot5À5²fbPb Áb5ot § 1#tÀ# uÞ wÀbp5fotuk5A5fu²# 5S6ut¬ou¬t5À5k5o# Áb5ot § 1#tÀt1²#uw ¹`b«hÍ85«16¸ ±±± 5wot«uo°5A Û Ê ÝÄÝ Ê Û ´ × Û Ä Û Ê Û rß × Û Ä Û r5tË Ö × Û Ä Ûtà ´ Ó ufu¬ru¬P¨1uto¬6 5o5#o 5tPu)15 ªm¶uª 5t ª 5³¾11uk#«11ouw ¸¤töw1o# ¨tu1P5P § t1k#u1k¨²#öo1ob5A #¬Pu)1#oP¬56³üo #1¬¿55¬#t á ½ mÙ1P5tuÍr5«u`bPüÂ¬tuÍ81Au¨# dx = x*y tuk x 5P y 5kµ¨P# 1kubf5 5²#u¨5t dx #1²#h1Au¨# uâ 5ttuot5Àt5f5 t¬w1¬#ot5À«¬16uU ¨P#Au¨#t«²#u Uo1kb1#1²#6 1kubt1«²fu¬ dx = x*y 5fÀ511uoot5Àt¬w1¬#tf 5¬Pu)ot5 #«16Î¬uwubpt¬w1¬#t¬t5ÀbtuwÀot#1uw5toÏb1²f5t otu1u ob ¯b²#ob²#1kub«1o1³ ¹`b«b1u ¬u U kuöw6obt P1A#t²fIbPfu ¬t¦ Û Ê Ý ©ÀÊã o Û Ê ¦¤Ý Êã© 5Â Auu5tor#1P¬uku¨1bt³¾o1kbf 5««u t¯1t1#wo5²# owÔ# uä ukt51¬tu³t5tot5ÀÁb5ot § 1#tÀt1²#uw«#u1ku5 t1²#uw5tÎ¨t#1#²#u³ot¨5 ¬P5Iku51P1«²#u UÏ uÌ t1kb²fu oku5fP1«²#uÒü«5tom)¬P51¬Pu³¾5kSuÍ8¬ku«u³ÑÍ81utf°¬¾«1uP ± ³¬ x ²#Àt1«²fuw5k«1o1ufot1k#1 1t5´ x ²#o å æ ç è é ê ë ì í î ï è ð ñ ò ó î ó ñ é ô õ ö ï è é ê ÷ ø ù ô é î ú é î û ü õ ó í õ ü ñ ö ý ð ñ ö é þ ó õ ÿ é ù ó é ô ù ò ô ï ñ ï ù ï ñ õ ö ¡ ñ ö ï è î ï ï è é ¢ ù ô é ï è é ñ ö é £ þ é ö ô ñ ¤ é ¥ ù í þ ï õ ô ù ò ó õ ù ï ñ ö é ÿ î ð ð ó î ï è é ó ï è î ö ï è é ô ð õ ú é ó CALLS î ö CALLG ñ ö ô ï ó ù ÿ ï ñ õ ö ô æ C = A*B; RndMode = Up D = A*B;

dd§¦ ± ³¬ x ²#Àt1«²fuw o u³¨15ÎuÍr1#tutoU5k5u¬t t1²#umbp²#oU¬P¨1 Ê ¬P¨15²#u uuotuÍ81btuP¬ Ì th¨1 #1t5µ1¬©¨ j§Ïj V©IÊ ²#o¨b#Òk#1#P³Ñ´ ⋅jÊ ²f¬ ½ 5fb ot ot5Àtf#otb¨5t#À#1uw5o#tU¦¤¨15¨#fP¯tu5m¨³föu © ¨##²##1uw5ot#t«²#u ¬«5t³¨6#tÙ5PI²f¬f5t²ft#1uu¨¨6³ ¨ Ì PS1u«uto5¬##²# 5w³Ño1t¬6btm¨15)¨5P«bm)ufubA ¬Ù11of²#u5öt51#Àotuöw5PüPut¬5o1t¬6bt5ku rI 5¬6bt5 t1²#uw u± Í85Àtou#um5w6¬tu¬°boou1k#fuÎ²#³¨¨³föu«U5P t5t³¤bm¨#«1k#²#u« uÓ ufuofuÍr5IÁ#5¬P § 1bPI5 ot«uoSw5ku³ 1uo1 Ì to5tÒot5totuA5kU1uo1#5fb ¬P«Ò¦oÔf¬t®c5P5¾1«5¬#Ño#w1 © ot51uÍr1#1otSo51¬t51¦ Û Ê Ý ©Ê«ã ≠ Û Ê ¦oÝ Ê«ã© 85t #wÔ tutufu¥otS²##1t p ⊕ ! Ä¦ p°Ê#"© ¦ 1 + δ) t#¦o5 u15¨5)²fbtob § × 5t à © ¹`t¬tu²##1tt#o# 5«#¨À5#««uk5t5A 5kuf #1²fo#b¤bmt1uw5 1kbfw5«uwo6#t#k¨15f#wot«f5t b²# ku¨1#t¨²fo# #«16 wouw¬u¬kb³otu¨56##wot«²#³1ku¬uPt¬P5IÁ#5¬P § 1bP 5w5²#ut5Au5t1«²fu)u«5to $&%('0)§132547698A@CB983D(E74F8(G Ì t¬#1Ò¨1ü1oht P5f1w5w³bPuwP¨³föu«Ò15o#t #À51 5³5t1ku6¨# uÌ 5t5Puw5¨ 5¨b«ht¬ukuöt¨³föu« ¨1u Ì tu¸ ±±± ö5t5kökbtf³Aub«utU¬t5I1uwc²fu5²#uoÑ1uo³ ow5t5tuÂo#Âu5«#ÏotuÙ1µ65¨üc#IÍ8u1o#t¨#5t · Ì w5 Ó 5tu ¨ºhb 15f´5Ë5Õ5f#5¨#«515obPU#ÀümuÙ1tuÎ¬ 5t5t6 ¸o¬th5¨# t5#1uw5o#5tÎfb²#³¿5ukuÍ8u1o#t¨fott5Pu)t#1²f 1kbu ¬¬Pb1u 5t¨botu ü ootuÍr5¬³k#1tuku¨ ¶ u1utÏtb¬th1A#fw5«t5tf15f²#ut1uf¬P¬ 5t5Pu)#P ²#u5²#uoÑ5u¨c#otuÂf5 5²#uU«¬tu1k## 5 5 u Þ b)5PuÍru1¬#tbot ow5t5Pu1¨1²#S5²fS¬¯utoo³ P5P#1uw5o# 5)²#ut¾1u o#w«u«5P ot1ku¨6bbp¬uöt5o# Ì th¸ ±±± ö5t5k5¨1«uot5Àb1u 5¬#t5Ah#tu1o15³ü 515tÎ¬P5 P5P¬u k1I#1 ¨fÀ1#¨¨²foÏP¬¤³ot#1uw5o#5tÎ¬ #1uw5t uâ «5ttu tt511uPtbm«¬165wot«uo°1to¨

ddFH ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ P5uÍ8u1obbw¨À«5³tbÀ²futb#¬¨6«1³t5¬th¬ 5 t5t6muÍ85«tot1k## 5bt¬u
Ó 5k 5k¨11b Àobmutoo³ft uÍr5¬6³ tb 5ob¬ 511u«5³²ftuu¨5w³ ½ t#¬tum1kb²fu1¬ 5oÎ²#³¬Po# t1kbfw5 o 5#ut ¹`111#ötübtÎ«1o1³ 5ü5uÍ8u1obt5totow5P5tu 5to o1S¬P51S# aâ t5k 5kSot5P5¯55«5t«1o1³¾15 5u 5b1o«6¿5 #1«1kb²#5²f³¯«#fSot56¬#b1u 5¬#5tu5«b1otu#t «¬16³fot5Àot551k#uu15w5u ootÙ1 ¨À«¬61³ uÌ t1 P¬Pü#tÎ«1o1³ow51 a = b + c t55k5Ï³²fuuÍru1oÏ 1#¬uto5³P5t#6t¬tku¨Àb a ¸¤ bt#À²#ku5¨#t5²#¤bmb1u o5b6¬PUÔftÎbpb1o«¿55o#tf²#u51¨huw³Á#5¬t § 1#tÀ#1uw5o#5 1#¬uto5³ow51f5Pot1f ¬5³51t¬w1¬#tut#1¬¿55¬#t #1Î²#u«6t5¬u Ì t1kb²#6 5²#5buÎ²#³1kbt²f¬6t¬ 5 t5t6 okb 5u¨t5t³5w5²fuU#Àot1k## 5kuo³ ¸¤töu5fot t5t6m5²#f¬Pb1u 5tbmku¨1À555kf1«ut Ð 1Àotuk5A¬611kb²fu« ¸o¬thow5f«ut otU hPökobtÒ#P u#²#UuÍ8u1ou¾15 5u ¸oPotU1o1³ow515o 5k#«ut z #1Î5Au5³t5²#uubu o¬u²#³¬th5¬6bt1u56³ ¨P5o#115³o5¬um¬P5«¬16³ uª #«11oum¨³föu«Uot5À¨11#w ot¸ ±±± ö5t5k1¨À1k#fb 5³o5fot51# z 6¬tum t5k 5k#)²f³t5fPotb1u)5#¨¯ö15obotÙwI165 I ®U5t5t51kb1#üÎ1¨P g5}Ajwike0"iwloqrl¤e#loqrhx 6töu5#Àow5P5tuwo5# otu¨1k#²#u« ¸o¬PUu¬t#fot1u)1uÙuU5uÍru1¬#5P¬Pf5 t 5t¬o²f1Î5Uotku¨ tuotuÍ8u1obbw ½ 5uÍ85«1u ¨11bot5À#¤bm#«1¬t¦¤¨ ÛQP¨à8Û ¬tümuu¬P5 Û ÄËU 5Aot5À #16«1kb1uw¤b «5t¬5bÎ¬uÀob Û ÄË5f5Pt¬u5 t5t6¬t6U5¨ PuË à Ëow5#1 u¨P¸ ±±± ¬ 5t5tÏuw¨¬th1ü #1 woµt5t6Âot5kuo1wPmf561µbIß5tt¬5t6²#u¤bkµ#«11ot x = y*z; z = x*w; a = b + c; d = a/x; x = y*z; z = a + b;

dd§R ¨ Ûtà8Û µt1ku¨1²fö¬¬#tfotü #11uo³ot5 tu5t5 #1uw5ob#1 ¨fotf51ßtb²#1 ®U5t55UotUAu¨1²fö¬¬#t ²#u51öt51o²#1u«¨À²#1uÙÎ²#uo#k¬thÍ8u1o##1w I tu1¨6t¾¬ 5«t5tu ¨8ot°f5So¯²#Sku¬wtu5²#°#«11ou tu¬P ow5#1 Ì th5f5to5fbp1k1²#¬6¬1o#¬P5I6IP5U¬ 5ft¬¤b 5kt5k 5A «16ut¬5o# á ½ mb#5cotµ¬³f1µ#ÏuÍ8u1o#t5c²#uuuouwtubÏ5 ²#üotuÍ¾o5²# P#t¬5PUotuAuÎAu1Àbp¬Pb1u 5¬6b ½ ¬Pb#1Au1²#¬o1o#t5¨#«5o¬ 5ofh5¬ow²f1ofot Ï1ku5 5u1¬5P#Àot¸ ±±± ¬5tÏ5AÎ«5Ô#uÀ1t6Ô#u³o²f u³1u«utou ²#³t5k 5k5t1o5¬Au SUTWV&XYVa`cbedgfch ½ t1²#umbp5Ut5²#uu«5¬t151um#tu tt1k#1uwouU# Áb5ot § 1#tÀ5w6¬tu¬ I t 1k#uuot ot5ÀÁb5ot § 1#tÀtb ²#5Ô«5#6f²#Àw5otu)6U¬ 5ft¬¤b 5k1²pi¬ t#¨5«5²# Ù1uÎ«5otu«5o5³ uÌ t6U¬6bUuP¬otkh15wo uÌ tÙ1 ¨À15 1kuütoÂ5P¬k#1o#ou A#5P5³#¨85t1kbuÂotSu¬56Âo#¹ùo#Ñ ·Qq btt ± A#¬Eº#15fß5Õ5» uÌ Pü#t5wÀuÍr16bkuU²ft5w³ou«5 #tuw¨6btÙ1P¨#«f51okb ¹8uo#Ñ · Ì P¸ ±Ï±± ¹8o5t5kEº# 15fß5¼Qr uÌ totk15 À¨1üUot®U5t51«5¬#¤b «15 t 5 1u5U5uÍr51¹ùo#Ñ · ¹`³#öu« ½ ¨1uoËº¯b15#´5ßß st6uv83D(478(GA$&wxwy69w ¸¤ot1¨##Àk#1ttu A#¬f 5U¬5ouot5À¨tff15kfÀ ut#1fof15 5toh¬t5À5o#5tÎ¨1²f¬ 5ob 5 5³fU²f51w5o ¦ Ì tubku ´ © I °P 1kbu¬Ñfu o³otm¤5 Ì tu#k ´t5c ¾15 o¨b#t o#m¨1²f¬ 5ob5t#to#m5ob uÌ t15 Ào#)1²#ow5¬6b å æ ç è é ñ ü ÿ ù ð ï ¢ ú ñ ï è þ ó é ô ù ò ô ï ñ ï ù ï ñ õ ö ñ ô ï è î ï ñ ï ó é ù ñ ó é ô é ñ ï è é ó ñ ó é ÿ ï è î ó ú î ó é ñ í þ ð é í é ö ï î ï ñ õ ö ¡ õ ó ÿ õ ö ï ñ ö ù î ò ð é õ î ï ñ ö ý þ õ ñ ö ï ï ó î þ ô ñ ü ñ í þ ð é í é ö ï é ñ ö ô õ ü ï ú î ó é æ ý ý æ

dd§ ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ fhgpirqtsuirv z Û
px#v Ý p}Aj)g5hiwqrloq5j)hploqrxuy og5hqrxflÀxfeft3"jw}kimq'xp1zh}wtuplp|cqrlong5p}wptujkl¤jw}wi β pxfvgQ pxfvq zSiwe"lo}wpsVloqrhxqri)v5hx#j)|Uq'l¤ngtv5q ytqrloi ¨ qEjfhx#jÏytefp}wvv5q ytqrl P flon5jwxlon5j)}Ajw{rpl¤q5j }Ahe#xfv5qrxyjw}w}Ah})qrxl¤n5j)}Ajwiwe#{rlpq'im{'jwiwi)l¤n5px j ≤ d j ehsuqtqgf ¸¤t¬ukP5t# Û 5tÝ6Àtu¨5 ³¨¬P5 Ûih ³ ¸¤ÀU5t5wu¨¬¨5 Û 5tÝ¾ot5 Û mku1Auut¬u²f³ Ûkj Ûkl … ÛkmQnl × β j ¸oÏÝ6mkAuP¬u5cÝ j Ý l … Ý mon l ¬tu¬th¡¤kutUuÍ85 ¸¤ÀÝ¯Uku1kuütou5Wp Ý l … Ý m fotuot f15k6#1P¨1kuÂot5PotS#«11ou«¡¤kut f²#S¬tSuÍr5P¬oukut kbtuo¯Á1#5ot § 1#tÀt1«²fu¬fötkbttuwk#)6U5À«#¨ j ¸¤ fPuw5fuÀÝÄË Ë … Ë5Ýkqpr l … Ýkqkr m f5t ²#hÝ¬w1P5oÎ¬ g°Ê ßf¬ Ì tu ³ § ¤¦ β § ß © ¦ β n myncl Ê β n myncs Ê … Ê β n myn q © ¦oß Ö © Þ A#otuÙtobbp#5AÎf¬fotb11ouf5b Û § Ý Û § kbtuo²fÁb5ot § 1#tÀt«²#u¬fot5À¨¦ Û § ©ÀÊ δ tukot kbttuwk# δ ¨5o¨Ùu ¡ δ ¡ ≤ ¦ βà ´ © β n m ¦oßQt © Ì thÍ85À¬oukut6 Û § Ýwfötuwk#)6U¦ Û § Ý ©p§ ¦ Û § Ê δ© Ä § Ý Ê δ Ì tuk5kotAu5ü ¸opÍ § ³ ≥ ßotuotku5ofu kbmU²fbtu²#³ ≤ β n m ÅE¦ β − 1)(β−1 Ê … Ê β n q ©ÀÊ βà ´ Ç ¤ β n m ¦¤ß Ê βà ´ © (17) ¹`bP³ Û § ¤ß5fotu δ ÄË ¹8Pot¨5u¨Àot5 Û § Ý¯5²f h ¦ β § ß © ¦ β n l Ê … Ê β n q © PuA ρ Ä β § ß5 ot5¬PAu5ouwkbm²##1tuÎ²#³ β 2 -- - 1 +     β p – 1 2 β -- - +     = y y y y y y y y – δ + 1 ------------------- - y 1.0 0. k 0…0     p ρ…ρ     –      

dd§u 18) Ì thÙ1t55¨6 tu Û § ³¤ß¯²#1 Û § ≥ ß uÌ t#t³ 5³¬PUb t511uU Û § Äß5f t5 δ ÄË Ð 1À δ ÄË5fotu¦oß5¼ © 511uf¨ ot5À5#5¬thAu5ofh kbm²##1tu²f³ β n m ¤ β n m ¦¤ß Ê βà ´ © I tu β Ä´5#¬tµ²fbtcuÍr5¬³´ j #5totc²##1tc5Pufu¤b Û Äß Ê ´ svnFm 5tÎÝÄ´ lwncm § ´ lwncsxm ¬P«À5 g → ∞ I tu5tt«²#u U#Àot5« ¨ftf¯#5AÎ#Àt#Àtuu¨5w³o5tu##bÎ51w5³f5oto# t ku¨1Àt fhgpirqtsuirvyaz z Û ≥ Ë px#v Ý ≥ Ë #lon5jwxlon5jÂ}Ajw{'ploq5jÂjw}k}kh}Âq'xsVhtug5e#l¤qrxy Û Ý qriÂplÀtµhiwl ´ ε bj5jwxq zxfh ytefp}wvv5q ytqrloi)p}kj)e#iwjwvQ ehsuqtqgf Ì th5##wot o#)5Ï¬# ¬|{Sf15k6#oU6U«5m¬ot5Ào# ¨²#ow5o# ¸¤ Û ≥ Ýk8to1ÝµwftPtof¬tS 5ÍubtoÂ# Û 5t«Ýµ5AS5ftu ¶ 5k5P³Ñ#6¬m¨Poo15öt gÊ {U1b¬6b ª b11o°¬P°¨ #Ï¬tu gÊ {S6#Àt«²#u UuÍ85o³ uÌ tuk#1to g f¬ I fu o³¬P¬P#ku tutf15kf¬5k1 Ü ¬P#utu 515 Â«5 Ì PuASÂP#¨Â#1fPuw5o³¯5¨1t¯ot5 Û ≥ Ý ≥ Ëµ5t«ot5 Û ¨56Î¬²#hbp¬thob vQEv5v …v × β j Þ öf5¨1«otukt5w ³b Ì tu¬Pu#6¬U¨t¤¬ub¬Iotuut#ÀÝht5uhf5uuÜ¬P5 β n m r l f5tot ¨5Àu5¨Àß5fötku5¬6fu A#mUuUot5 β n m r l à ßÄ´ j ¸op¬PuA6U 5 w³#1of¬tu¬tu kbmokb ¨t¤¬P1¨À²#5ÏÎ¬otkb1Ptuwkb # uÌ t1 U5Àu5 βf¬thAu5ofh kbmu¨¬t5 ≤ ´ ε y y – δ + β 1 – ( ) β 1 – … β k – + + ( ) ------------------------------------------------------------ - β 1 – ( ) β p – β 1 – … β k – + + ( ) β 1 – ( ) β 1 – … β k – + + ( ) --------------------------------------------------------------------- - < β p – = y y 1 2 -- -β p – 2 + β p – 1 + 1 2 -- -β p – 2 + +     β ⁄ 1 β 2 ⁄ + ( ) β p – =

dd§} ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ ¸¤I6U#²#bUot5À#«²#Pt¬P ¬P#ku«#u Ì tubku´
Ì tu#Au ´ ffu)ot°ku5¬°uwk#¥o#¥1u o#w«P¾#t°b1u 5¬6b ª #«15 t¾ot°k#1tt uwk#¥b Û s § Ý s 5P«¦ Û Ê Ý © ¦ Û § Ý © kuf1ku)Ô#t 6t¾otSku5of°uwk#¥#«1o1 #1uw5obP uÌ tAu5ofuwk#)b Û Ý δl ÄÉÅ ¦ Û Ý ©À§ ¦ Û § Ý ©¤Ç à ¦ Û § Ý © t ¨5o¨Ùu¡ δl ¡ ≤ ´ j uâ )o ¬6I5Pbotu 5³ Û ÝÄ¦ Û § Ý © ¦oß Ê δl © ¡ δl ¡ ≤ ´ j ¦oßQr © ¹`«5 ³ Û ⊕ ÝÄ¦ Û Ê Ý © ¦oß Ê δ~ © ¡ δ~ ¡ ≤ ´ j ¦o´5Ë © ½ ¨1«tot5À«16¬165obU1u o#wÎ²#³#«1¬totuÍ85À1kb 5totuA#1t6t#¬tku5ofu kbm5Ib Ö ulp f¨ e ⊗ Ä e ¦oß Ê δ © ¡ δ ¡ ≤ j ¦o´5ß © o#)5t³Áb5ot § 1#tÀt1«²fuw e 5t uä 1oototu¨otkuuuf5¬#tUo##uotu ¦ouo¬6t e Ä Û Ý¯5t Ä Û ⊕ Ý © #6fu ¦ Û Ý © ⊗ (Û ⊕ Ý © Ä¦ Û § Ý © ¦¤ß Ê δl © ¦ Û Ê Ý © ¦oß Ê δs © ¦oß Ê δ © ¦o´5´ © ¹`otku5¬6fu A#mtwku P#«1¬t¦ Û § Ý © ¦ Û Ê Ý © ¦ Û Ý © ⊗ ¦ Û ⊕ Ý ) − (Û s − Ý s ) = (1 + δl ) (1 + δs ) (1 + δ ) − 1 (23) ¦ Û ~ § Ý ~ © Ì t6Uku5¬6fu A#mUuf15¬ δl Ê δs Ê δ Ê δl δs Ê δl δ Ê δs δ Ê δl δs δ t ²##1tuÎ²#³ Ö ε Ê ¼ ε ~ ¸¤#¬tu #k¨fot5Ír1ku5ofu kbm5²#b Ö kbttuwk#w¦o¨6t j «51P1«²#u¬ j s 5«#¨Àtuff²# ©

dd§ ½ ¨5m5t5³f¨bp¦ Û ⊗ Û © ¦oÝ ⊗ Ý © 5tPbÀku¨1À¨5151o#m¬t ku5¬uwk#¬f²#u51¨ tu tu5 ²#³f5uU# Û 5tÝ5k11ffuto Û s § ³ s fotku5ofu A# 1¨5³²#f1o5kf ½ Pbotu 5³¬h¬t oow³5tÎ115¬h¬t5P5³#Uot5 # Ô#u#¦ Û Ý © ⊗ ¦ Û ⊕ Ý © f³fuP ¦oÍ ⊗ Í © ¦¤³ ⊗ ³ © Ä Å Û s ¦oß Ê δl ©p§ Ý s ¦oß Ê δs ©oÇ ¦oß Ê δ © Ä¦o¦ Û s § Ý s © ¦oß Ê δl ©pÊ ¦ δl § δs © ³ ~ © ¦oß Ê δ © I tu Û 5tÝh5kuPu5w²#³f¬tuwk#)¬u ¦ δl § δs © ³ s 5²#5U5k#5Uotuku¨ Û s § ³ s uÌ tu#«1¬5obPU¤b «5³i¡öo³b)5 ¬t5À¦ Û § Ý © ¦ Û Ê Ý © «#k5w5oot5 Û s § Ý s I °tuÍro1 ¬¾55P5³#)#ot°o# «1u¤bÒ¬P°5AuubÏuow5tf ¸¤#AuÒ¬ uö«5oot«5Í8«1 uwk#)ot5À5b1 tu#«1¬P o¦oÕ © f¬P o# to5 ²#tuuu fhgpirqtsuirvy y zSiwe0"lo}wps l¤q'hxq'img5jk} zh}ktujwv|Uq'l¤np¥ytefp}wv¯v5q ytqrlpxfv Ý d ≤ Û ≤ d Ý lon5jwx Û Ý qri sVhtug5e#l¤jwvj Û psVlo{ Ý ehsuqtqgf ¾#¬ot5À Û 5tÝP5fot5«uÍrbtutofotuuwo5t³ Û Ý6UuÍ85 â ¬tu u8okb otSbt6¬##1otS¬tu#ku«8otSuÍr1#tutoÂ5¬ou²f³¯5 «#Ïß ¹`5u5PtouAt5tf Û 5tÝItu¨5 ³¨¬P5ÏË ≤ Ý ≤ Û #5t Û ku1Auut¬uÎ5 Ûkj Ûkl … Ûkmyncl 5tÎÝ5UË Ý l … Ý m uÌ tuot5fb ot¤b #«1¬P Û Ý #«11o Û § ÝuÍ85o³5Pkbto¯Á1#5ot § 1#t t1²#u ¸¤¬t°¬oukPS)#¬PSo#w Ë v l … v m 8otS¬oukut 5ku5³ ²# g #obP#f5Ptkb1Pttuu¨5w³ ¹`t Û ≤ ´5Ýk Û § Ý ≤ Ýkf5P ¨PÝ#Àoto#w Ë v l … v m f¨ Û § Ý I tu β h ´5fott³f1botuU# Ì tu#Außß5tt#À²#Au15u²#³Ý à β ≤ Í ≤ βÝ Ü otökbP#u#to#ÑÝ à ´ ≤ Û ≤ ´5ÝµÂö#Puu¨5w³ Ì P5t5³f¨Â#Potu kb ¦ Û § Ý © ¦ Û Ê Ý © f«u5ou³¤b6 t¬th1A##À# Ì Pubkuß5Ë5«1üÎ¬t¤5 ot5À¬tku5ofu kbmot²f5¨°b1u 5¬#tbp5o#5t1²#o 5ob ¨5P¦oP5«u³uf15obPc¦¤ßQr © 5t¦o´5Ë ©o© Ì t6cUotu«#Ïb«bÔft#Iu kb 5t5³f¨ uÓ uof5t5³f¿5to#w«¦oÕ © kfk¨#«uott«#kuft5«u³ Ì tu#kußß5f5Uot¤b PA#b 61¨t

d§¦§ ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ fhgpirqtsuirvya zSiwe0"lo}wps l¤q'hxe#iwjwi)pÒyte#p}wvv5q
ytq'lgfpxfvq zSp"upxfvscp}Ajmlon5j)iwqrv5jwi)hrzSp°lo}wqrpxuyt{'j ¨ p ≥ " ≥ s P flon5jwxlon5j)}kjk{rploq5j)jw}w}Ah})qrxs htug5e#l¤q'xuy ¨ p ¨ "s PP ¨ s ¨ p" PgP ¨ s ¨ p" PP ¨ p ¨ "1s PgP qri)plptuhiwl ßQt ε fg5}Ah5qrv5jwvjt pQpQ ehsuqtqgf u'uÍr5toto5¬#w#t²#³#t uÞ A# Ì tubkuß5Ë5 " ⊕ s Ä¦ "cÊs © ¦oß Ê δl © Pk δ á Uotku5ofu A#¬f5t ¡ δl ¡ ≤ ´ εuÌ tuot 51u#ÀotuÙwIo5¬#)U¦ p ⊕ ¦ " ⊕ sV©o© Ä¦ pSÊ ¦ " ⊕ sV©o© ¦¤ß Ê δs © Ä¦ pmÊ ¦ "UÊ © ¦oß Ê δl ©o© ¦¤ß Ê δs © f5tÎ¬P1 ¦ pSÊt"uÊs © ¦oß § ´ ε© s ≤ Å p°Ê ¦ "µÊsV© ¦oß § ´ ε© ] ⋅ (1−2ε) ≤ ⊕ ¦ " ⊕ s © ≤ Å p°Ê ¦ "µÊsV© ¦oß Ê ´ ε©¤Ç ¦oß Ê ´ ε© ≤ ¦ p°Ê#"uÊs © ¦oß Ê ´ ε© s Ì t6U«u5PUot5À¬Pk5 η á ¨¬P5 ¦o ⊕ ¦¤² ⊕ ©o© Ä¦¤ Ê ² Ê © ¦oß Ê ηl © s h¡ ηl ¡ ≤ ´ ε ¦o´Q © Ì thtuÍrÀouw tf#ot1#¬uto5³5o5ökb1PS1²#ow5¬6bb s 5t p b f²fu51 p " «5³t5kbttuwk# Ð u51¨5²5tÎS5kot ¨Ï#Àow65t#u ≤ ² Ês f5tb²#tPot oot#kuwt s ≤ " ≤ p ffu p ≤ "µÊs ≤ ´ " ≤ ´ p ¹8 pS§" ¨5o¨ÙucotubtÏ¬#tUb Ì tu#ku ßß Ì t «u5tUot5 pS§" Ä p " UuÍr5¬ftut s ¦ pS§ ² © Ut5 «uÜ1²#ow5¬6b P5²fuö«5oÎokb Ì tu#kro²# ¦ s ¦ p "©o© Ä¦ sm§ ¦ p°§"©o© ¦¤ß Ê ηs © h¡ η~ ¡ ≤ ´ ε ¦o´ Ö © Ì th¬tkouw ¬th¨1 bp uÍr5I1#¨6¬f15toou¨f ¦ s ⊕ ¦ p "©¤© Ä¦o Ê ¦ p°§"©¤© ¦¤ß Ê η © h¡ η ¡ ≤ ´ ε ¦o´Qt ©

d§¦e Þ t5³f¬P5¨Ào 6 ¦ p ⊕ ¦ " s ©¤© Ä¦ p°Ê ¦ "u§ ©¤© ¦¤ß Ê η © s ¡ η ¡ ≤ ´ ε ¦o´5Õ © 1t²##¬ Ì tu#Aur5t Ì tubku ß5Ë ¸op«¬165obU5¨«uo²f uÍr5¬6³¾kbtu`¨¾ot5 Û ⊗ ÝuÄ Û Ý5¦oß Ê ζ© ¬ ¡ ζ ¡ ≤ ε`¬P#«²#Pt¾¦¤´Q © ¦o´ Ö © f¦o´Qt © 5t¦o´5Õ © #6fu ¦ p ⊕ ¦o² ⊕ ©o© ¦o ¦o ² ©o© ¦o ⊕ ¦¤ ² ©¤© ¦o ⊕ ¦ " s ©¤© ≤¦ p°Ê ¦ "µÊ ©o© ¦ sc§ ¦ p°§a"©o© ¦ scÊ ¦ p°§"©¤© ¦o Ê ¦¤² §ps ©¤© Pk Ä¦oß Ê ηl © s ¦oß Ê ηs © ¦oß Ê η © ¦oß Ê η © s ¦oß Ê ζl © ¦¤ß Ê ζs © ¦¤ß Ê ζ © ½ 11u)²fb1Po# Ε ¦oß Ê ´ ε©x ¦oß Ê ε© PÍ815t#1À¬ ß Ê ß Ö ε Ê â ¦ ε s © ¹8#« wouw¨6«1³ftbkot â ¦ j s © o «f²f1ÀÀUu5³¬ 5bt1¤bÒ I ¬tÑ¦oß Ê ´ ε©x ¦oß Ê ε© Äß Ê ß Ö ε Ê ε ¦ ε© ¦ ε© )u1#³ftb5 j o1#öfh#¬Ùuto¨f6I56tku5¨t¤1to## ε ¹`t ¦ Ë5Ë Ö © Ä Ö Ë Ö ¦ ε© ¤ß¯o#)5 ε ¤ Ë5Ë Ö f5ttuP ≤ ¦oß Ê ´ ε©x ¦oß Ê ε© ¤ß Ê ßQt εuÌ #uÀ um²##1tÎb ft#¬ot5Àß § ß Ö ε § ε q ¦ ε© ¤ ± f5P¨ tu ε ¤ Ë5Ë Ö fß § ßQt ε ¤¦oß § ´ ε©x ¦oß § ε© ª #«²fttotu¨ ²##1t³#66Uß § ßQt ε ¤ ± ¤ß Ê ßQt εuÌ t1¬Pk5ouwk#)5À«#¨ ßQt ε Ì tu#kuß5úwo5t³¨t ot5À¬tukt5o5ökb1t6S5tu5o# o#w1h¦¤Õ © ¹`5¬Pb1fIcPbItu¨5 ³¬t o#w1h¦¤Õ © Ut«uw5³ ö5²#uf6I¨5oo³#6t¬t5²##1tÎo#)otutoko#w«5 P t5 Ì tu#ku»# · ª 5tu5obtº¯b15#ß5ÕQr¯#u ehsuqtqgfqgffhgpirqtsuirv u Ä¦ pSÊ ¦ "uÊs ©¤© ¦ sc§ ¦ p°§"©o© ¦ smÊ ¦ pS§a"©o© ¦ p°Ê ¦ "µ§ps ©¤© 5t Ä¦ p ⊕ ¦ " ⊕ sV©o© ⊗ ¦o ¦ p "©¤© ⊗ ¦ s ⊕ ¦ p "©o© ⊗ ¦ p ⊕ ¦ " sV©o©

d§¦d ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ Ì tuP Ì tu#k
ß5´¨P Uot5 Ä ¦oß Ê δ© o δ ≤ ßQt ε ¸¤ÀUu5³otuÔ ot5 ¦o´5¼ © 1kbu δ ≤ ËQ à ¦ Ö ´ © s ≈ ß Ö f5tÎ¨t¢¡ δ ¡ ≤ ßQt ε ≤ ßQt5¦ Ë5Ë Ö © Ä Ë5¼5 δ Ïbu 5¬o³¬P#tob uÌ t o ¡ δ á ¡ ≤ Ö ´¡ δ ¡ ≤ ¼ Ö ε ¸¤Àf15kkb#¬5k#«1¬uo ot Ö ulp fotuot u A# tub11ot U¦oß Ê δl © ¦oß Ê δs © o ¡ δ2 ¡ ≤ ε ¸o β Ä´5f¬P otuA6Utowotumuwkbm#«¬ou tut²#³ uâ otu üf#t «#ko5obmß Ê δ ¬ ¡ δ ¡ ≤ ε tuu¨5w³o#m¬tÏfbPf5t1t¬P «uot#ot1k#bpb Ì tu#k ß5´5fotÙt51uwk#)²fbt#À¦¤ß Ê δl © ¦oß Ê δs © ¦oß Ê δ © UÏb«6t5¬uÎ²#³ß Ê δ o ¡ δ ¡ ≤ ßß ε Ì «5Ôf¬Ptu1 öSuÍ815t5o#«u5¬u³¤b Pot¬5ou«utÀ# Ì tu#ku1kuufottuÍ8Iotu#Au u¨w²#uei¡¨Àt bu³ µ¦ Û © 511kbÍ8«5ou#tö5t fhgpirqtsuirvya z µ¦ Û © Ä6t¦oß Ê Û © à`Û flon5jwxmzh} Ë ≤ Û ≤ ≤ µ¦ Û © ≤ ß px#v¯lon5j)v5jw}wq5ploq5j)iwploqri¡ Òjwi ¢ µ£ ¦ Û © ¡ ≤ ehsuqtqgf ¾#¬ot5 µ¦ Û © Äß § Ûtà ´ Ê Û s à » § … U55¬u t5otüw6 6¬uku5¨t o «fö# Û ≤ ß5 µ¦ Û © ≥ ß § Ûtà ´ ≥ ß à ´ ¸¤ÀUuuu5ümou¬P5I²fu51 otü uU¤b µ U5ouwP5¬P# µ¦ Û © ≤ ß uÌ P Ì 5³##mü uU# µ r¦ Û © 6U5 5¬u t5¬P#b5t Û ≤ t5muku5¨t¬u «¨b¨ § ≤ µ r¦ Û © ≤ § Ê ´ Ûtà »5`# § ≤ µ '¦ Û © ≤ Ë5fot¡ µ r¦ Û © ¡ ≤ 1 0.52 δ 1 δ – 1 δ + 1 0.52 δ + ≤ ≤ ≤ – Q q 1 δ + ( ) q 1 δ 1 + ( ) = = Q 3 4 -- - 1 2 -- - 1 2 -- - 3 4 -- - 1 2 -- - 1 2 -- - 1 2 -- - 1 2 -- -

d§¦§¦ ehsuqtqgfqgffhgpirqtsuirv¤ ¹`tot Ì 5³f#)uwuo#) m55o t5otÑuwu¨bË¤ Û § t¦¤ß Ê Û © ¤ Û ~ à ´5`¾ot°ku5¬°u kbÒP1 Au P511kbÍ8«5ott¦¤ß Ê Û © ²#³ Û 6U²##1Pu²#³ Ûtà ´ ¸opß ⊕ Û Äß5fotu ¡ Û ¡Ï¤ εfötku5¬uwk#)²##1tÏÎ²#³ εà ´ I tuß ⊕ Û ≠ ß5#uÙt hß ⊕ Û Äß Ê uÌ tu¨tuË ≤ Û ¤ß5f¦oß ⊕ Û © ßÄ ¸op¨#5tÎbf5w6¬tU5k#«1¬uÎ¬ ot ulp f¬tu¬P #«1¬u51#p¬tuÍ81ku¨¨6bt¦oß Ê Û © à ¦o¦¤ß Ê Û ©À§ ß © P¦oß ⊕ Û © t¦¤ß Ê © ¦oß Ê δl © ¦oß Ê δs © Ä ¦oß Ê δl © ¦oß Ê δs © Ä µ¦ © ¦¤ß Ê δl © ¦oß Ê δs © ¦oß ⊕ Û © ß ¦o´Qr © Pk¢¡ δl ¡ ≤ ε 5t ¡ δs ¡ ≤ εuÌ uö«5o µ¦ © 1öt5f5¬Pubku« P5³#ot5 µ¦ ©p§ µ¦ Û © Ä¦ § Û © µ′¦ ξ© ¦o»5Ë © o#m¨#« ξ ²#u u Û 5t uÞ A#otuÙtobb fÀ¤b ot5«¡ § Û ¡ ≤ εf5tb²#tPot o Ì tubku ß5»#u¡ µ¦ ©p§ µ¦ Û © ¡ ≤ εà ´5f# ¡ µ¦ © à µ¦ Û ©P§ ß¡ ≤ εà ¦¤´¡ µ¦ Û © ¡ © ≤ ε t6Ñ«u5tÒot5 µ¦ © Ä µ¦ Û © ¦oß Ê δ © 6¬ ¡ δ ¡ ≤ ε Þ t56³f1o1³ft²#³ Û tokbuU¯Ù1t5 δ föt#«1¬u 51# Û ⋅t¦¤ß ⊕ Û © à ¦o¦oß ⊕ Û © ß © 6 ¸¤c5³Ño¾tuÔ¬P5 ε ¤Ë ß5òtu¦¤ß Ê δl © ¦oß Ê δs © ¦oß Ê δ © ¦oß Ê δ © Äß Ê δ o ¡ δ ¡ ≤ Ö ε ln 1 x + ( ) x x2 2 ---- - – x3 3 ---- - … – + = x ˆ x ˆ x ˆ 1 2 -- - x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ln 1 x + ( ) 1 x + ( ) 1 – ---------------------------- 1 δ 1 + ( ) 1 δ 2 + ( ) 1 δ 3 + ( ) 1 δ 4 + ( ) , δ i ε ≤

d§¦FH ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ ½ toukuötuÍr5«bpuwk#)5t56³#U¨to#w15¦oßQr © ¦o´5Ë
© 5P¦o´5ß © #1wotf5w5o°ob « · ª 5Pu5obPºb 15fß5ÕQr5fuÍr156tut k ot¬thf5¬# 1ut5o¬P1bouto5 5Pu5ob51u²#³ot ± #1uw5o# uÐ 1ÀotukU5Pbotum1bouto5 5Pu5ob¬P5I5#1 tub11ot v Ä " s § ps uÌ t#t5tPbÀ²# u«P5¬u«²#³¯µ¨1ku5w 5tfu«utP#1¬tSo#w1 q #1ft³¯1u5Ô#6t# P " s ≈ ps A#1tPu kbm5#t¬5t5o1ot5p¬tÏ#ootkb#¬ #«1¬u ootf15 5¬°o#w1 uÓ k5Po#w«51A##À¦o5Pbotu 511kb5ouö«5ot¬th kbmotf5w5o°ob «511u5w®c5t5 ÅEßQr5Õ5´ Ç¤© z¥" s ≈ psc`}khe#xfv5qrxyjw}w}Ah}ÒsVpxÑsVhxfloptµqrxfplojÒefgulohUn5p{zclon5jÒv5q ytq'l¤i¥qrxÑl¤n5j¥}AhhloiÒs htug5e#lojwv |cqrlonl¤n5j)efpv5}wpl¤qrs#zh}wtµef{rp ä k#b¨Ú I o¦ " ⊗ ² © ¦g p ⊗ sV© Ä¦ " s ¦¤ß Ê δl ©À§ ps ¦oß Ê δs ©¤© ¦oß Ê δ © tuk ¡ δ¦ ¡ ≤ ε á ut v Ä " s § ps b¬Pm5²fµA wo¬u5c¦ v ¦oß Ê δl ©Ï§ ps ¦ δ~ § δl ©o© ¦¤ß Ê δ © Ì ¾fu5uö«5o°o#Òot°¿5°#Ï¬Pmuwkbo`ft#Aµü#t#AuÒouwm δ¦ t5öt5²#b¬u A# v ¦ δ á Ê δ§ ©1§ 55 δ tuk ¡ δ ¡IÄ ¡ δl § δs ¡ ≤ ´ ε ¹`t ¬thÙ1 ¨À¬u v ¦ δl Ê δ © 5²#ft#k uÌ uö«5o¬thü#t o «föto5À¬t5 p Û s Êt" Û Ês Ä p ¦ Û §p} l © ¦ Û §p} s © f¨ p} l } s Ä s ¹`t " s ≈ ps #¬Pu } l ≈ ¨ s #ÑotuübPuwkb)¬u Ì P1cot #«1¬u 51# uÌ ttuf156¬³ ¨P Uot5 Pk f¬t 5²#b¬u A#m p 5²##1 ¹`t δ ≈ β n m f5t ot1 otu5²f¨#1ouu A#)# u¬kb³fUot²#boobt56ÀbÀot²#oUbÀotkb#o } l ≈ s ¸obotu bk¨ftot515ob#ÀotA##¬U6tf#fu#«11ot o å æ © ö ï è ñ ô ñ ö ü õ ó í î ð þ ó õ õ ü ¡ î ô ô ù í é ï è î ï β ª « ô õ ï è î ï í ù ð ï ñ þ ð ñ ÿ î ï ñ õ ö ò ¢ ¬ ñ ô é £ î ÿ ï î ö õ é ô ö ï ó é ù ñ ó é î δ¦ ® b – b2 4ac – ± ( ) 2a ⁄ b – b2 4ac – ± ( ) 2a ⁄ d 4ac « 4acδ 4 4a2r 1 δ 4 2 ≈ d d 4a2r 1 2δ 4 + p q p2 q2 – p2 q2 + p q p q 0 > ≥ , + ≤ ≤ ≤ – d 4a2r 1 2δ 4 + d E + = E 4a2r 1 2 δ 4 ≤ d 2 ⁄ r 1 δ 4 δ 4 β p 2 ⁄ – ≈ r 1 δ 4

d§¦§R 5totuÍr1ku¨bbut#Àt5fu5ttfo1²#oU¬t 1#öb#wku¨1#tt¯¬ot up#Aut5P# } ¦ rotuÑot up#k²#oÒ# } ¦ 5tt#À²#«u5t6t#¤1 Þ t5³ o1 oot1k#bpb Ì tu#kut ¸¤ÀU²f5ü#ot¤b P¤5o P6U1k#fu · Ì tu#kußQ5t Ì tu#Au ¼5º#15f´Q5» fhgpirqtsuirvya¤ ¯ jkl Ë { gQpxfviwjwlÀt±° β q ß fpxfvpikiweftuj)lon5plkmhpl¤q'xuy og5hqrx#lphg5jw}wpl¤q'hx#i)p}Aj j Û psVlo{ Ý }Ahe#xfv5jwvQ ~ n5jwx ¨ t ⊗ ÛQP ¨ t ⊗ Û ÛQP qri¥j Û ps l¤{ Ý jwe#p{floh Û }Ahefx#v5jwvul¤hcg { ikq ytxfq ÒsVpx#lpv5q ytqrl¤i7v² h}kj)g5}Ajws qriwjw{ Ý Û qri)}khe#xfv5jwv" Ý lop { qrxyl¤n5j)iwq ytxfq Âs px#v¯hrz Û qrtupytqrx#qrxuypU}wpv5q Û g5hqrxfl`ëfiwl{rj zlÏhrzcl¤n5j { {rjwpiwlÏiwq ytx#q Âs px#lÏv5q ytqrloiÒpxfvu}Ahefx#v5qrxuyl¤hUpx qrx#l¤j¡ytjk}o ehsuqtqgfqgffhgpirqtsuirv³ Ð ³ Ì tu#k ßQ5 Ûk´ Û kb1Puo g¯§ ÔÄ g à ´  15u ¸¤p¬tukt5 w³ #1¬¬tuu ¬5t6³ Ûk´ 5²#ku1kuütou o g à ´  ¨6#tÙ5PI6#o ¹`111#ötukÒµ5w ³ § b ¸o Û Ä Ûvj Ûkl … Ûkmkn7l × βµ r¬tuA#1tÏt¯5Òßµ¬ Ûkmkn q nFl `5Potµbt³ 5³Ñotuk°5²f°5 w³ § #1m ÛkmQn q nl Ä β § ß5b²#1otu otµ6 #kuÂfÏ# Ûk´ cß Ê Ûkmyn q ngl ÄË5#5t¨5#5 Ûv´ cku1kuüto5²#u g à ´  #o Ì u5 ¬ Ûx¶ f¨5 Û o²f5P¬ufu)5¬o³ft β myncl ≤ Û ≤ β m § ß u tuk Uot gh§ {St##Au)f¬U# Û f5t Uot·{S #Aumf¬ uÌ Pk5kotkh5ü¬btu ¸¤ fotu kbtt Û o g§ {°15u¬th¨5«5Pb11P5P 5t ¹`t t5m5Ï«#¨{Uf¬bÏmuutb¬Pu t5c5«#¨ÏÔÑÄ g à ´  Ä g à ´  6#o â otu ü β Ä«´µ5t )ku1kuüto5²# ¬ { § ß ≤ g à ´  ¨ftÙ5tÏ²#o Ì tµübP5¨µ tu b5t d ( ) 2a ( ) ⁄ x x h x l + = x h x l x l β 2 ⁄ ( ) βk 1 – < x h x h = x l x l = x l x l x 1 2k 1 – < x β 2 ⁄ ( ) βk 1 – >

d§¦§ ×)452)$¦%¦@E9 F¨(10 2)3456 6@B0 CÒØp45@EG$ otu#«1¬P Ûv´ Pf#fukb1Pt11f Ûk´
Ä Ê β q f5t Ûk¸ Ä Û § Ûk´ Ä Û § − β¹ Ä § β q uâ t5f5t P5U5À«#¨{°#6¬f¨ ku1Auut¬5²f o  à ´  fo uÞ t5³f Ä¦ βà ´ © β q ncl fotu Ûk´ Ä b Ê β q u1utt# Potum¬PuAkbt ¹8 Ûkº 6¬tum¦ βà ´ © β q ncl #)¦ βà ´ © β q ncl § β q Ä § β¹ à ´5²##¬b t5kku1Auut¬u oßf Ì tu#kutffu 5³ouÍr1kuöt1kbÀbp #wÔft1kub t1²#uwcuÍr5¬6³Ñ5c¨ Ì tukµc#«15Pb«ob «o#ÒuÍ81ku¨t¨1 uÍr5¬6³ ¸¤«¡ Û ¡ ≥ ¡AÝ¡¥otu Û Ê ÝÄ¦ Û ⊕ Ý ©ÀÊ ¦ Û ¦ Û ⊕ Ý ©¤© ⊕ ÝÅ ¶ uÔfÔ#umßQr5Õ5ß Ü ®Ut1oßQr5¼5ß5 Ì tubku ª 6ü¬#» ´ ´ Ç uÓ ufuo tu1¨PuÍ85o³ kbtu#1uw5obP¨foto#w«bP³ow1o# β Ä´5f5tPbÀo# β Äß5Ë¯5 otuÍr51 Û Ä rQrQrQr5¼5fÝ°Ä rQrQrQr5Õ¯¨t ¼&4F83B9w5½A256¾C)x'047¿CB9EaÀt6983Á()xwÂx47698 ¹`ttf1ku¨#P5 g Ä´Q5f5P´ s ¤ß5ËkÃ¤f³fb«ftÀuÍr1uIot5 #tuwot²#P5w³t1²#um¬¼u«51fo b²#1¬Ù1utÀ¬kbu otb ft51²ft5w³t«²#u uÓ ufu¬fotUPbÀot5 fhgpirqtsuirvyaÄ Ån5jwxpW"q'xfp} ÝÆ 99iwqrxyt{rjÂg5}Ajws qriwqrhxxfe#t·"jw}ÂqriÂs hx5jw}wl¤jkvhl¤hlon5jÂsV{rhiwjwiwlIjwq ytn5lIv5q ytqrl v5jwsVq'tup{Px#eft·"jw}xfqrlpqri)x#hlÀp{'|Up Ý i)g5hiwiwq"{rj)l¤hSefx#qrefjw{ Ý }AjwsVh5jw})lon5j"qrx#p} Ý x#eft·"jw} z}Aht l¤n5jÒv5jwsVq'tup{hxfjÇuh|Uj75jw}obqzUxfqrx#jÂv5jws qrtup{v5q ytqrloiÂp}AjÂe#iwjwvQbl¤n5jwxs hx5jw}wloqrxuyl¤n5j v5jwsVq'tup{`x#eft·"jw}Àloh)lon5jps {rhiwjwiwl"qrx#p} Ý xfe#t·"jw}À|Uq'{r{`}kjws h§5jk}plon5jÀh}kq ytqrxfp{Fmhpl¤qrxy og5hqrx#l xfe#t·"jw}o ehsuqtqgf Ð t5 ³¨tf1kubt1²#uw³ftott5p#1utouw5ÒÅEß5Ë f´ loj © Ä ÅEß5Ë5Ë5Ë5fß5Ë5´Q © t5ß5Ë²#o¬otuoÀ#Àot²#P5w³1#t¬5tßQ²f¬oot #tÀ#Àot²#6t5w³1#t uÌ tUotuk5Ah¦o´ lxj § ß5Ë © ´ l Ä»Qr5»5E´5ßQt¬oukut ²#P5w³t1²#uwot5Àtouw5 ¸¤Àu«51P1«²f 5Aku1kuütou o¼ f¬fotuotuk5Ah¦o´ lxj § ß5Ë © ß5Ë Ä´Q5Ë5EË5Ë5ËÏ«5t1«²fuwot¨5 touw5 uÌ Pkt 5³ot5À´Q5Ë5EË5Ë5Ëu51t«²#u Ub16kAuut »Qr5»5E´5ßQt¬oukutÀ²#t5 ³t1«²fuw ¹8¼u«51foU5kt#Àutb#¬ 1t6f1u³Au1kuütÀu5¨P#Au¨#²#6t5w³t«²#u x h x h x l x l x l x h x h

d§¦§u Ì ¨t ¬t5rÏ#o5A1¬Ù16P¬f6Iut#1fot ot5À¬P¨15t ²#u uu²ft5w³t«²#u U5 5³##ku5omot5ot15t²f uu u51t«²#u uÌ P 1ut1k¬P5I¤bmu5u51t1²#uÈµ¬t touw5ÒÅÉÈ § ulp 0È Ê ulp Ç bto5t5À«#¨ÀbP²#t5 ³t1«²fu uÌ t u5²#t5 ³t1«²fu)k#1t¬tf1u«51P1«²f to1 kbtUo1Pf1²#P5w³t1²#u Ì ht ot5totU15P²#u uu¾²ft5w³¯t1«²fuwÒÒ5 5³fÒfAu5¬upot5Ñ¬P ¨15tÑ²#u uuu«5t«²#u ¨`btu¥5touw5pÅEß5ËkÊA`ß5ËkÊ r l Ç â ¬t6 touw5fot¨15t²#u uubPü¬u51t1²#uwß5ËkË Ê r lxÌ7nFÍ uâ ÅEß5ËvÊAf´kÎ Ç tuk t ¬P¨«56Àtofum¨¬P5Iß5ËvÏ)¤´kÎ8fot¨15tb ²#P5w³t1²#uw´kÎ ncs f5Pot¨15t#uo5Afumo1 ¬tum#¬t touw5 uÌ P1U6Iut#1fotuÔot5Àß5ËkË Ê r lxÌ7nFÍ ¤´kÎ ncs uÐ 1Ào5of¨6t ß5ËkÊÒ¤´kÎ8f¬Pß5ËkË Ê r lxÌ7n7Í Äß5ËkÊAß5Ë n Ãp¤´kÎ8ß5Ë n Ãp¤´kÎr´ n s Ì th¨5«5kf1«utÀ5116Î¬#1²#61Au¨#t ¬t5Àß5Õu«51Ï#o 5kkfAuÎ¬ku#fum#1²#Au¨#t«²#u Ð t5 ³ § u51#t ¨#56¨1kbuU5PbotumÍ85«16bp¬P1¨#pÁ15f q u51¤kb · ä ku¨6btº¯b15#ßQr5ß5fot5Àoku#fum²ft5w³t«²#umokb o u51uÍ815t¨#tfotu«51o²#t5 ³#tfu ¨#1¨À²#b11ou uÍr5¬6³ uÌ t5À#tfu ¨#1uw¤b «u²f³1o1³ft¬thf15to¬uWÈ5t ß5ËÐ ÑÐf¦ P5A²fbouÍ85À g ¤ß5» © ¨tf § uÍ8¬utuÎ1Au¨#5tÎ¬P kbttotUo¨P#1ku6¨#¦o#mfÏt g ¤Ë Ü ²fbo5ü5A«5 © â À#1 ötb11o5¬##È ⋅ ß5ËÐ ÑÐ#5tt#À²#uÍr5 Ü ÀUotb«²ftu #1uw5obkb1P¦È ⋅ ß5ËÐ¡ÑÐ © ot5À«1À²#uÍ85o tukotkbttokb ¨P# § Í8outuotf1kub Ì ü P³À#tÀ¤51o²fuÍr5¬fo5Ôf ot¨15b β Äß5Ë5 g Äó#)tfuf5t g Ä»o#)tf § uÍroutu ¸¤Àot 1kbIco²#uß5´ Ö ß5#otu«ot #1²#uk#1tuoß5´ Ö 5c15 IbÀ¬tutf § uÍroutu«16¬1³b1u 5¬# q bttot#6u1kub #1ffß5´ Ð 1 ot5À5t mt#Ib kuof²#u5¨kbttot1kbÏ1Àotf1kub ¨Pb16#6fß5» uÌ tuwk#)1h¬#1²#k#1tP Ð ³1t¬th¸ ±±± Á5#fb²#k#1tt5²#5#u5ob6 ¹`5fot wkutÀf5bp¬PtuÍ85ÀÁ5#f5P¬PukuüÀ ¹8uÀ¬PA#1tÏt«#¬ kbt § o § ¿5uA uÌ P1uw¤b ot«¬615o#»È ⋅ ß5ËÐ¡ÑÐ ¹8o#Aott f561 #ÏotµPuÍr5ÏÁ15 ixflag #5tkuöbkµotuA#1tÏt«#µ5ttuÍ85ÏÁ5 ¸¤ ixflag UË5#otuÒÈ ⋅ ß5ËÐ ÑÐ#cuÍ85o#¨k#1t¦È ⋅ ß5ËÐ ÑÐ © 1²#u# AuÏ oot5À²# ¸o ixflag ß5fotub«h#o uk¬w1P5oÏf¨tkbt § o § ¿5uk5 5³#okt5ou uÌ t¨ftÙ5tÎbp¬P1kb b#ÔÔf 1 2 -- - 1 2 -- -

d§¦§} ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ ß " l … "
sos ! s …!y l ½ #1²fkb1Ptuwkbm«5³#1m " s … " l Äß5Ë …Ë ½ ¨1 5³¯o¯5bt1o#²fbo5ü))¬¯1u o#w µ##5 OR b ixflag ¬ " l uÌ tuA#1t¦gÈ ⋅ ß5ËÐ ÑÐ © 61²#b11oÎ#wkuo³55ü $&wxwy69wÂ×58AØ3u3¿Ù¿CB2§47698 · â 1¬¿5uwºb15#´5ßQr5«utobtuÎ¬th1A#²#u bp51w5ou³#«1¬t ³Ñ#t¨1 Ì P°«1u511k#5o¾1A#f6tÑ51 5³mo¾#1²f°ot 1kub Ì #uÀkb#uö«5obpt «b²#tot1kub «1k#fucotµ51 5³#Ïh¨1bu i á Ä Ûkl i s Ä i l ⊕ Ûks … i ¦ Ä i ¦ § ß ⊕ Û ¦ Ì tu i ¦ Ä¦oß Ê δ¦ © ¦ i ¦ ngl Ê Û ¦ © tuA δ  ≤ εf5t6#t#w6tübt#Au)ouwU δ¦ ffu ¦o»5ß © Ì tÙwIuf5¬³#À¦o»5ß © t Uot5À¬P#«11ouf5b Uot¨5u5 I5«Í85I¨1«5¬6b 5c1u o#w«ub«1u ¬w²#uf5ucb Û £ Ì tuÙ1 ¨Ïouw Ûkl U o1w²fu²#³ x εf¬th5¨À¬u Û Ê ²#³bP³ εuÌ tu#tuf15o³¦¤»5ß © ¨P Uot5Àuwk#)ouw U²fbtu²#³ u¶ #1²ft¬P1Au¨#P5Uot u¡¤uÀbp¨f5wt ε ¸¤Àot1 U²f6tbth5¸ ±±± b²#1kub o#w5¬`ß à ε ≈ ß5Ë l `¨¾¬P5 o#¥5P³¾ku5¨#t5²#°51°# x Ì t1`b²#t ¬tS1ku¨6bo5Ô#uÂot5Ír1 1uwo1 ²#5ob# x ε 5t«t5tfuÂPo Ì tUot´ ε u kb)²fbto#)ot®c5P5¨1«5ob¤b «1¦ Ì tu#Au¼ © t# 5##bÎ5U¨t#1²#h1Au¨#tfuuotb#ÀU1²#u¬oum¬t5¨P# 1kub Þ bm5P¬16¬uÍr165t5¬## t³ot®c5P5¨««5o#o#w1 b Ô# #tu)ot¤b PÏ5# 5#Àot1k#u1k s n x j 1 δ k k j = n ∑ +       j 1 = n ∑ x j x j δ k k j = n ∑       j 1 = n ∑ + j 1 = n ∑ = = Σx j nεΣ x j nε 1 « nε2 ε «

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§¦§ ± 5¬6«1««5P5ufotukUb
kuobo5o#)Ã t ²# 511uÎb¬tPuÍrÀ#b ¹8Ù1 ¨À¨²#ow5Iotb k¬#Ãb11ouot 1kuf#1b#¤A#Û £ fff6t¬tb kuoÎ¨1«5t È uÌ tu5Î¬t ¨««5to¾¬tµktttÑ1 d Ì tµ #AÏÒ²f¬m# È ¦ot5«u³ È ¸ © 5k°#¨ ot°1 ¾uÍ8#«1¬µ¬P°t6##AuÂ²#omb È ²#³Ñb11otÑ~ §Ïd I P È ¨²#ow5ou¤A#otf¬P #Aum²#ob È ²#ku#fuku uÌ tu5Ah¬t ²#oUot5 uk#¨ÀotÙw¨À1¬tÏ5# 5 uÌ Pu³²fb¬P#wkuo# o5¬#ÒobÒot°tuÍ8b# ½ o# «51k#bÏb Ì tu#ku ¼5`o5Ô#uokb ®Ut1o ÅEßQr5¼5ß Ç 15f Ö Õ5´5f511u5 U¹8uo#Ñ · Ì PubkußQ5P Ì tu#ku¼ º ÜWÝUÞßÞ beàâá ¸¤I6Ut#À1tb«bo#)b11ou)³#¬u u¨ftuwoPu#uIot15 ¬#À ¨³f¨ouku5ou¬Á#5¬P § bt uÌ tU6U1k#²#5²#³1h¬oto5Iot5ÀÁb5ot § 1#tI#ufu ³o¬6u¦¤À5t³ © 5¬outobot#«1¬u)utu1 w11 Ì t6U¬wt551u¬t5115kuto³ 6u¨1ku5²#uuÀot5IÁ#5¬t § 1#tÀ t#Ïf15toÙ5²#µ1²pi¬uob5tÑ¬Pkµco¬6µ1b6tÏ¤1¨tbuÂ¬tuu¬5 #Àt5k 5k5tÎ¨#o 5k¬P5IÏ5 o S S T T Yh Yl Yh Yl −Yl Yh - - = + C

dFHx ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ Ì t6U151umt5u«#töw5ouot5ÀÀU1#¨²#ok5bwf#A#1³5²#b Áb5ot § 1#t
uÞ bmuÍr51ufÁb5ot § 1#tÀ5fb ¬P«tf#t5tu5¬# 5²#1k#fuot5fh¨«5Au5ouwkb U6Àot1tÏ ³ftt5k 5kt5 f15kf¬5totukU5u¡ÙutÀ5fb ¬Po#m²#t5 ³ § u51#t ¨# ot5À5²f1kbuo²ft o²#uf1k#fuot5ÀuÍ8¬utuÎ1Au¨# ¨11b o uÌ to5¨Ô#Àbt¬w1¬6tAu5²fÁb5ot § 1#tÀ¨#o 5k«5 «u5u tuot1Puw³ft#«1¬um¨³föu¨11b of#ÀÁ#5¬P § 1#t ¸o5ob¬ot uÍr51uai¬1Iut¬#tu¦¤#5AÎ#6¬5t uÍroutu1k¨# © 8¹ùobÑ · ¹`³#¬u« ½ 1uoËºub15fU´5ßßubot)151u¥t5 uÍr51uU 5tftok#t¬w1¬#uIÏfob1umb1o«¿55o# öw5ott o²#u¬oum¨111# IÁ#5¬P § 1bP Ì tU6tku5¨t5u1o5tU#totU¸ ±±± Áb5ot § 1#ttö5t5k«u5tÒot5bÏ ot5Ï1o¿5uou5¬kc#I¬Pµ¬5t5k5kµ²#ub«6tufuÂbkµb ¬5²f ¹ùobÑ · Ì th¸ ±±± ¹8o5t5kEº#15fß5¼Qr5ff5ft«ukb1uÍ85«1u1¬ 5¬PP otU¤5o1kuÒ#tot¸ ±Ï±± ¬5P5A5¾²fU1u6 wotw5o5bÁ#5¬P § bt #u ãßäåeæiçeè fcVâéeê Þ V æ `ch Ì t6U5 ¬ 5t1kÎ²#³#1 ¨f²#³ I ®U5t55À¹8»ë k#¨³fö ok#ë 5³otk#1#|ìo16³#ÀßQr5¼5¼5 t 5Uuw³5²#³#Af5t¿5u²f³ ¶ 5f6 Ó #1f#À¹`1 ë ³t#1Uout5²##otuw¬u5 5²##1Àottouw5¬## Áb5ot § 1#tÀ5tb11ou)³#¬u ¬t#1Àt5tofuÀ1¬¬ 5o¬ut¼tÚ Ë5Ë uo1ku Ì P5tÔ#c5Aµ1µo¾®U5t55t«5t³bÏ³Ñbu5f1u 5íuk#Í ä ½ q ª ¦¤u¨1u56³îì¤bP»ï°²fuw © ¤bmk5Ïtw5¤¬bp¬PU15m5t 1kbt«5P³üo1#«P¬ q uf ok# ä 51 Ó 6Ù1tfu)5P5 5t#t³f«bUkuouku5¨tu1uÎ«1kbot1kuüto5¬# ðñVòaVâàV æUä Vâh ½ t# ½ okuÒó 8¹òt q `5tµ6«5&ì ¶ ßQr5¼Qt Ã htug5qr{rjw}wiôwõ}wqrx#sVqrg5{rjwi7 ~ jwsVn5x#qrefjwi pxfv ~ hh{ri ½ ¨# §I uu³ q u5tf0ë ½ ½ ¾¹8¸1ßQr5Õ5¼ Ôö tµjw}wqrsVpx È ploqrhx#p{bd#l¤pxfv5p}wvvõ}AhAyt}wptutuq'xuy ¯ pxuyte#pAytja÷0ø ~ ö Èµ ½ ¾¹`¸ ¹`¬5t5ktí¥» r § ßQr5Õ5¼5 ½ uw5»¾5obP51¹8o5t5k¸oPö¬¬u0¾ ßù #wÔf0 ù Ð 5 tu¬o ¶ 5ßQr5¼5Õ cö õh}wl¤p"{rj÷f{rhpl¤qrxy õÏhqrx#lax05q'}khx#tujwxfl ft11²#6¨tu «5t¨w1

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 dFHè Ð k
t I ¹ ßQr5¼5ß gö dfqrtug5{'j"e#l jwp{rqrikl¤qrs²¢hv5jw{Phzú÷#{rhploqrxy gõhqrxfl Ã htug5eflopl¤q'hx ½ ª ë Ì w5t #»ë 5¬ ¹8#o 5kÕ5¦ © 1 Q Ö § 5¼5Ë ª bÏ³ I ìpu 5 ßQr5¼Q gö õ}khg5hiwjwv pv5q Û ppxfv»ÅÑh}wvQo{rjwxytl¤nQoqrx#v5jwg5jwxfv5jwx#lpd#l¤pxfv5p}wv zh}÷f{rhpl¤qrxy og5hqrx#l ö }wqrlon5tujwloqrs f¸ ±±Ï± ë 6k&5¦g © f1 ¼Qt § ß5Ë5Ë ª bÏ³ I ì ßQr5¼5¼ ÷f{'hploqrxuy õhq'xflpd#lopx#v5p}wv5iû ~ n5jwh} Ý pxfvõ}wpsVloqrsVj f ·Qq u5²#6¬³ ª #«1¬PIÚuotk#bptouw51«uot#Uüt¬Ù°#«11otfº5fu ²#³ q 5«# ±Â ë #bkuf1 rQr § ß5Ë5Õ5 ½ 5u« ä ku¨ Ð #öbP0ë ½ ª b#tutâìouA#«ßQr5¼Q Ã hx#l¤}wqü"e#l¤q'hx#i)l¤hSpvõ}Ahg5hiwjwvdflopxfv5p}wvpzh}ýqrx#p} Ý ÷#{rhploqrxuy õhqrxfl ö }kqrl¤n5tµjwl¤q's ä ¶ Ì tu¨fut b ª 5o#wt65 Ð uwÔfuu³ ¶ uÔfÔ#uo ÌÏ ì ßQr5Õ5ß Ôö ÷f{rhploqrxy gõhqrxfl ~ jwsVn5x#qrefj5zh}e Û lojwx#v5qrxuylon5j ö 5pqr{'p§"{rjeõ}AjwsVq'iwqrhx ¾1u ë 5o ß5¼5¦¤» © f1 ´5´Q § ´Q5´ ¶ u«uyì¤5«uÂßQr5¼Q wþ xfv5jw}¡)h|Épxfv°lon5j jw{'qrp§"qr{'qrl Ý hz È e#tujw}wqrs p{bdfhrzlo|Up}Aj 8¹8¸ ½ ëÿì ¹` ¹`¬5 uª #«11 uÖ ¦ © f1 ¼5¼5Õ § r5ßQr Þ 5wt« ª t5 uUßQr5¼5¼ Ã htµg5qr{rjw})dfe#g5g5h}kl zh}÷f{rhpl¤qrxy og5hqrx#l Ã htug5e#l¤ploqrhx ¹`bo 5k § ä w5¬5t ± Í81uw6Pufß5¼5¦oÕ © f1 Õ5Ë5ß § Õ5ËQr Þ b ¨³f¬tuâï u±Â 5Ptë #u¬ ªuÐ) ßQrQt5Õ Ã htug5e#l¤jw})d#h{re#loqrhxhz ¯ qrxfjwp} ö { ytj"}wpqrs d Ý ikl¤jwtµi ä kut¬ § Ó 5 ± tf #b ª 6¡o0Uì ï°#²#ukff¸ Ð uttuoIßQrQt5Õ 27 ýqrloi ö }Aj È hlxfheuytnpzh} ¡ Q£¢µq ytqrl ö s sVef}wps Ýw ª b #Àot ½ ª ë ß5Ë5¦¤´ © f1ß5Ë Ö § ß5ËQt ï°#²#ukf ¶ 5ßQrQr5Ë Ã htug5eflojw} ö }wqrlon5tujwloqrs f · ª #«1¬u ½ Atouo1k#Ú ½ 15to¬5o ½ 1A#5Pº5#²#³ ¶ 5 ä 5¬ouwb5t#ìo#t» Ó uttu¨³ ½ 11utÍ ½ 0ë bk#5®c5o«5Pt0# ½ ob ª ½ ï°#1²f0ï°ut Ó 5ttóI5|b5t ª t5wu Þb ßQr5¼Qr ² pl¤}wq Û Ã htug5e#loploqrhxfi f´5t uobP Ì tìo#tt Ó #1ÔftUut6fuwo³ ä Au¨ Ð 56¬bkë 5w³f5t ï° 5t5« q #t5t f®Ut1ot ¶ #t5 ±Â 5t ä 5¬5ttÔf â k ßQr5¼Qr Ã hxfs }Ajwl¤j ² plon5jwtuploqrsVif½ b §gI u³ q u5P#0ë ½ f ßQt5´ Ó 6o ä 5Ô#5kßQr5¼5´ HP-15C ö vQ5pxfsVjkvi÷#efx#sVloqrhx#iÇupx#vQ"hh { ¸ ±±± ßQr5¼5Õ 99dflopxfv5p}wv ¥¤ §¦ §¨§ zh}ýqrxfp} Ý ÷f{rhploqrxy ¤g5hqrx#l ö }wqrlon5tujwloqrs f¸ ±Ï±± ¦oßQr5¼ Ö © q wtou¹8¸gï ä ½ ´5´5¦o´ © 1 r § ´ Öt ®U5t5t I ßQr5Õ5´ Ôö d#ef}5j Ý øÏz}w}Ah} ö xfp{ Ý iwq'i 8¸¤to#w5¬# ä kbu¨t¯Õ5ß5wó##´5 1 ß5´5ßQ § ß5´5»Qr¦gxi¡1²f i¡5t5 ù ##¨5 © 0¾# ¬ Ó b65t ½ «¬uk5

dFH`d ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ ®U5t5t I ßQr5¼Qt
Ã p{rs ef{rpl¤q'xuy ö }Ajwpcpxfv ö xyt{rj¥hrzcp È jwjkv5{rjo{rq { j ~ }wqrpxuyt{rj 8t11²#6¨tu «5t¨w1 ®U5t5t I ßQr5¼5Õ ýÏ}kpx#sVn Ã efloibzh} Ã htug5{rj Û {rjwtujwx#lop} Ý ÷fe#xfs l¤qrhxfi f · Ì P¹8o5¬hb ot ½ wÀ»Ñ1«u 5 ½ t5³f¨6¨º5fu ²f³îë ì ¶uä u5t ½ ¸¤ü uU¦¤µP# Ð «tft5« ± t#65t © ª t5¬umÕ5 â Íro#kutuw¨6¬³ ä ku¨0¾ ßù #wÔ ®U5t5t I ßQr5¼5¼ ut1²#tuuo1ku#u5À¹8»ë k#¨³#¬u¨0ë bto5 ó¥ ª ½ ®U5t5t I 5t ª b#tut0ì¤k#« ÌÏ ßQr5¼5´ ~ n5j È jwp}øu}wlon5hythx#p{'qrl Ý hrzSd Ý x#l¤p Û dfjktupx#l¤q'sVifpx#v ©¢uqrpytxfhiwl¤q'sVi)qrx È e#tujw}wqrs p{õ}Ahyt}wptµtuqrxuy»x5qr}Ahx#tujwxfloi f · Ì t q u5¬#tt Ð u ui¾«uw5 ª #«1¬5o# ½ t ä A#fw5«t5tf15fu¨º5 u ²#³ì ® q u1 ß5Ë5» § ßß Ö 0¾#wo § Ó b5P ½ «öuk5 ®U5t5t I 5t Ð 5t ±Â ßQr5¼ Ötgö xfhtup{'qrjwi)qrxlon5j ý² ö sV}wqrlonõps { pytj ä A# Õ5¬ ¸ ±±± ¹`³#1b1 b ª #«1¬u ½ wot«uo°¦ouw²f5t5f¸¤t# © f1 »5´5´ § »5»5ß ®U tft5t Ð w5 I 5P q ¬tu ¶ uttë ßQr5Õ5¼ ~ n5j Ã õ}AhAyt}wptutuqrxy ¯ pxuytefpytj ä kuto § Ó 5 ± tf ## ª ¬o¨0Uì ®UAttu¬ q 5t®c¨tr ßQr5¼5Õ ö }wqrlon5tujwloqrsrzh} fjksVloh}aõ}AhsVjwiwiwh}wi ä kb ¼5oÑ¸ ±±± ¹`³#«1#¨61# ª #«1¬u ½ wot«uoS¦ ª bbf¸¤o5³ © f1 ´ Ö t § ´QtQr ®Ut1ot ¶ #t5 ±Â fßQr5¼5ß ~ n5j ö }klphrz Ã htug5eflojw}õ}AhAyt}wptutuq'xuy h{'eftuj o¹8u#t ± o#t ½ ¨# §gI uu³ q u5tf0ë ½ ®U1¨t` I `5t|ë 5tÔfu¬ I ßQr5¼Qt ~ n5j ö }wq'l¤n5tµjwl¤q'sÂhzclon5j ¢uq ytqrlop{ Ã htug5eflojw}ô ö È jw| ö g5g5}Ahps n f¹`¸ ½ ë q uf6 ´5¼5¦oß © f1ß § »Qt ë 5¬165 ¶ I 5P®c# tuk1 äf ßQr5¼ Öt ÷#qrxfqrlojõ}AjwsVqriwq'hx ploqrhxfp{ ö }wqrlon5tujwloqrscôfdf{'piwn È e#t·"jw})d Ý iwlojwtui f¸ ±±±Ì w5t b ª b11 uª § »Q5¦oß © f1» § ß5¼ ¾u¨#tQï ßQrQr5ß d Ý iwlojwtuiõ}AhAyt}wptutuq'xuy#ÅÑqrlon²¢hv5ef{rp ä kP¬ § Ó 5 ± P# b# ª ¬o¨0©ì q uü¬0ì¤bt Þb 5t®Ut1ot ¶ #t5 ±Â ßQr5Õ Öt 95pv5q'xuyl¤n5j ¡¢µ}wq zlÀqrx÷f{'hpl¤q'xuy og5hq'xfl ö v5v5qrloqrhx f¸¤to# «5¬6b ä kbu¨¨PîÏoouw»5¦o» © f¼Q § ¼5Õ ¹`¬uw²fut¿5 ä 5 Ó ßQr5ÕQ ÷f{'hploqrxuy õhq'xfl Ã htµg5efloploqrhx ä kuto § Ó 5 ± tf ## ª ¬o¨0©ì ¹ 5wo¿55tu¬ ± 5w ±) 5t ½ uÍ8b1#1#¨ ½ öuWï ßQr5Õ Öt ~ n5j)d#q ytx F¯ hytp}kqrl¤n5t È e#t·"jw})d Ý iwlojwt f¸ ±±±Ì w5t uª b11 uª § ´Q5¦oß5´ © f1 ß5´5»5¼ § ß5´Q5´

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 dFHx¦ I 5¬Pu¬0ì
¹ fßQr5Õ5ß gö e#xfq Òjwv¯p{ yth}wqrlon5tÎzh})jw{rjwtujwx#l¤p} Ý zefx#sVloqrhxfi ä k#uutfb ot ½ Þ ¸ ä ¹81 tìo#t ª b11o ª #t »5¼51 »5ÕQr § »5¼ Öt SUTWV ç àV Þ b æ é SiTWV ç àV Þ Ì t6U¬6b#t¬5PU #Àot«#k¬utt511kb#o¬P5 uA#«oouokb ot¬uÍ8 !#"()x69wy)¿%$'& ¯ jkl Ë { gQpxfviwjwlÀt±° β q ß fpxfvpikiweftuj)lon5plkmhpl¤q'xuy og5hqrx#lphg5jw}wpl¤q'hx#i)p}Aj j Û psVlo{ Ý }Ahe#xfv5jwvQ ~ n5jwx ¦ t ⊗ Û © ¦ t ⊗ Û Û ©pqri)j Û psVlo{ Ý jwefp{Pl¤h Û }khe#xfv5jwvloh gv { iwq ytx#q Âs px#lpv5q ytqrloik² h}Aj)g5}AjwsVq'iwjw{ Ý Û qri)}Ahefx#v5jwv" Ý lop { qrxylon5j)iwq ytx#q ÂsVpxfvhrz Û qrtupytqrx#qrxuypm}wpv5q Û g5hqrxfl8ë#iwl1{rj zl1hz)l¤n5j { {'jwpiwl1iwq ytx#q Âs px#lv5q ytqrl¤i78px#vµ}Ahefx#v5qrxuylohmpx qrx#l¤j¡ytjk}o (&wy6960) Ì th1A##À²#ku5ÔfU1to 5üü1utt# Potumbmt#À¬P #«1¬5o## t Û Ä β q Û Ê Û t55 w³ § #1À#)t# ½ ¨1«µ¬tukµmt¾5 w³b ¸oÏmt5wu¨mo¾¨5 Û ¾ot5m5P¬ufu Ì tu¬P#«11o5obb t Û Ä Û Ê β q Û b#ÔfUÔfot5Ú BB ²b² BB ²b² Ê BB ²b² BB ²b² ¿ ¿ BB ¿ ¿ ²b² BB ²b² Pk Û P5U²fu15woobPÎto 15 ¬ uÌ P #Au{°#o5A «5 Ô#u b 5tottf#ku g¯§ {°f¬5k«5wÔfu a Ì b11o t ⊗ Û ok# t Û tbumkbttÑ#¡Ïotµ #ku {Uf¬m¦¤¬P°#tum«5 Ô#u o b © t ⊗ Û Ä t Û § Û «#¦ β q ©ÀÊ β¹ ¦¤»5´ ©

dFH§H ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ Ì thf51hb } Uß¯ .bb...b
Ufk5oum¬t5 5tË#¬Pu 6¨ ë bk 1ku6³ } Äß a.bb...b kb1PUo pSÊ ß5 } ÄË#otu ¨ ¦o»5» © ¾uÍ8Pb«1¬ t ⊗ Û § Û Ä t Û § Û «#¦ β q ©1Ê«} β q § Û Ä β q ¦ Û Ê«}w©1§ Û «#¦ β q © Ì t 1¬1k²f6 ¨t ot#«1¬5obb t ⊗ Û § Û kb1Pufot5À¨f¦ t ⊗ Û © Û Ì tµo#Pµ β q ¦ Û Ê}w© tuA B cotµfÏot5Ïku¨1ocokb 5t r oot uI#ku)Ï# b BB ²b² BB ² Ð Ë Ë BB Ë Ë § ²b² BB ²b² ¿ ¿ BB ¿ ¿ 21Ë Ë BB Ë Ë ¸¤ .bb...b ¤ otu } ÄË5f1²#ow5¬P51¨u²## A ¤kb ¬tÏ# «5 Ôf B b²#otµ6¡oukutµmk#1tu11b5t¨Ñ¬PµtuÏu¡¤Ïmot5Ïot kbtu¬oukutuf156Uotobtu P β q Û ¸¤ .bb...b h otu } Äß5`5tßµ)1²#ow5¬uokb B ²#u5¨S#¬PS²##wk 8¯5#56otSku¨1 β q Û Þ t56³#t¨uÂotu5¨ .bb...b Ä ¸o } ÄËotu B cufuP Z cb 5t¬tu¡oukutµcA#1tu11bft β q Û ¹`«5 ³ tu } Äß5 B c# Z ufut¬tÏ¡oukutUk#1tuÏ tf¨5f5ot¬oukut β q Û uÌ ¨««5 ¿5 ¦ t ⊗ Û © Û Ä β q Û ¦¤»Q © ª b²#tPuf5¬#t¦o»5´ © 5t¦o»Q © #6fu¦ t ⊗ Û ©À§ ¦ t ⊗ Û Û © Ä Û § Û «#¦ β q ©pÊ ρ⋅β q Ì tku¨16I#À o#wt¬PUb11o5¬# r00...00 + aa...aabb...bb 1 2 -- - 1 2 -- - 1 2 -- - 1 2 -- -

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 dFHxR - bb...bb
aa...aA00...00 Ì tµw1µo#Òb«1¬6t } `uf5¬#¦o»5» © `c¬tµ5«µ5m¬P°w1µo#Òk#1tP a... ab...b ¬ gh§ {15u Ì t1c#«1¬P t Û § ¦ t Û § Û © «Áb5ot § 1#t 5 ¬tu¬U1ku¨6b)uÍ85o³Ñuf15o¾A#1t6t Û ¬ g§ {U15u¨bot°5 P Û Ê β¹ Û #uUPbÀ5 w³#1 I tu Û Ê β q Û bu5 w³#1¬¬tu t Û Ä β q Û Ê Û b#ÔfUÔfot5Ú aa...aabb...bb + aa...aabb...bb zz...zZbb...bb Ì t¨ t ⊗ Û Ä t Û § Û «#¦ β q ©pÊ| β q Pk | Ä § £3 3 ¤ βà ´5²#1ÀotuÍr5 51# | 6U1P«1#wo5t ¾uÍro t ⊗ Û § Û Ä β q Û § Û «#¦ β q ©pÊ| β q ¸o 1¬1k BB ²#² BB ²b²bË Ë BB Ë Ë § ²b² BB ²b² Ê ¿ ¿ BB ¿ 21²b² BB ²b² á q btt#um¦ t ⊗ Û © Û Ä β q Û Ê| β q §Ï} β q tuk } Äß .bb...b h b .bb...b Ä 5t " j Äß ~ Þ P5³`¦ t ⊗ Û ©Ï§ ¦ t ⊗ Û Í © Ä t Û § Û «bÏ¦ β q © Ê| β¹ § ¦ β q Û Ê| β q §p} β q © Ä Û § Û «#¦ β q ©pÊ} β q ½ PbP5f5t } ÄßuÍ85o³ Pkb1Pt a...ab...b o g§ {°15utbuUk#1tP uÌ t Ì tu#kußQ1kbu515u å æ ç è ñ ô ñ ô ï è é ô ù í ñ ü î ñ ö ú õ é ô ö õ ï é ö é ó î ï é ÿ î ó ó ¢ õ ù ï æ ÷ ñ ï ñ õ ö î ð î ó ù í é ö ï ñ ô ö é é é ü õ ó ï è é ô þ é ÿ ñ î ð ÿ î ô é ú è é ó é î ñ ö ú õ é ô é ö é ó î ï é ÿ î ó ó ¢ õ ù ï æ ý ý æ « æ 4 õ ù ö ñ ö ñ ¤ é ô β q 5 6 7 β q ý 8 β q õ ö ð ¢ ñ ü Ë β q 5 6 7 β q Ì û é é þ ô ï è é ü õ ó í õ ü β q 5 9 n n @ A 9 1 2 -- - 1 2 -- -

dFHx ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ fhgpirqtsuirvxwytgtUUvvtAqtqtsrvUt ¹`111#¨ot5 Ub11oÎ1¨6t¬t¤b6 t5##w6¬t ~
n5jwx¾lon5js htug5e#l¤jkvµiweft d¯qrijwefp{#lohcdi° Σ Û £ ¨oß δ£ P Òø ¨gÈ εB P Σ ¢ Û £ ¢ 8|cn5jw}Aj ¢ δ£ ¢ ≤ ´ ε ehsuqtqgf Þ wIku5t otuwk#)u¬5¬¤bmot¨1o#w1 Σ Û ¦ ut ¸¤t¬k#1 i l Ä Ûkl i ¦ Ä¦¤ß Ê δ¦ © ¦ i ¦ ngl Ê Û ¦ © Ì tu«¬Pµb11ou¨1 i Ê t6 ¨#Àouw«fu5b t65 Û ¦ «¬u²#³5uÍ81ku¨¨6b tbt δ£ r Ì tuuÍr5Ï#u¡ÙutIb Ûkl c¦oß Ê δs © ¦oß Ê δ © … ¦oß Ê δÊ © #5t ²#³kP1«²#u tf#¬tubu¬Ù1utÏ# Ûks «1Ï²fµ¦¤ß Ê δ © ¦oß Ê δ © … ¦¤ß Ê δÏ © #5t ¨# uÌ P1kb#À# Ì tubku¼¯ktU56btuÍ85o³¬t5«Pu¨f#t³¬P #u¡ÙutÀ# Ûkl «#k#«15¬u ¸¤Ïo5 i j Ä s j ÄË¯5t Ýkq ° Û q s q ngl ° ¦ Û q ps q ngl © ¦oß η q © i q ° i q ncl ⊕ Ýkq ° ¦ i q n l Ýkq © ¦¤ß σ q © s q ° ¦ i q i £CED á © Ýkq ° ÅE¦ i q Ài q ncl © ¦oß γ C ©À§ Ýkq Ç ¦¤ß δ q © S = X [1]; C = 0; for j = 2 to N { Y = X [j] - C; T = S + Y; C = (T - S) - Y; S = T; } Σ j 1 = N x j

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 dFHxu Pk5¬tÆï°kuÔuo¬u U5k²##1Pu²#³
ε ½ otb#¬th#¬Ù1utÀb Ûkl i qÏUot16¬5¬uÍ81k¨##À6t¬uku¨of¬1 tbIo²#u5u)o #«1¬otbu¬Ù1utÀb Ûkl i q §ps qÏ5t s q I tu|{SÄß5 l Ä ¦o l ¦¤ß Ê γ l ©p§ ³ l © ¦oß Ê δl © Ä ³ l ¦o¦¤ß Ê l © ¦¤ß Ê γl ©p§ ß © ¦oß Ê δl © Ä Í l ¦o l Ê γl Ê l γl © ¦¤ß Ê δl © ¦oß Ê l © l § l Ä Í l ÅE¦¤ß Ê l ©À§ ¦o l Ê γl Ê l γl © ¦oß Ê δl ©oÇ ¦oß Ê l © Ä Í l ÅEß § γl § l l § l γl § l γl § l γl δl Ç ¦oß Ê l © ª 5totu#u¡ÙutoUb Ûkl «otu¨uuÍ81Au¨#tUÃq5t d q1k1uofu³#otu Ã l Ä ´ ε Ê â ¦ ε s © d l Ä Ê ηl § γl Ê ε s Ê â ¦ ε © Ì fuI¬t#utu 5P¤b «1o# d q5tÃq6fuÍr15PotuuÙtobtU# i q15t s q¡ ftb t51ouwUPf#fP Û ¦ 6¬ q h ß¯¬f i q Ä Å i q ncl Ê Ýkq © ¦¤ß Ê σ q © Ä Å i q ncl Ê ¦ Û q §ps q ngl © ¦oß Ê η q ©¤Ç ¦oß Ê σ q © Ä ÅE¦ i q ngl §ps q ngl ©p§ η q s q ncl Ç ¦oß Ê σ q © s q Ä F i q §Ài q ncl£G ¦oß Ê γ q ©p§ Ýkq Ç ¦oß Ê δ q © Ä Å HF¤¦o¦ i q nl §Ïs q nl ©Ï§ η q s q ngl © ¦¤ß Ê σ q ©Ï§Ïi q nl£G ¦¤ß Ê γ q ©ÏÊs q nl ¦oß Ê η q ©oÇ ¦oß Ê δ q © Ä Å HF¤¦ i q ngl §Ïs q nl © σ q § η q6 q n l ¦¤ß Ê σ q ©Ï§Ïs q nl£G ¦oß Ê γ q ©ÏÊ q ngl ¦¤ß Ê η q ©oÇ ¦¤ß Ê δ q © Ä ÅE¦ i q ngl §ps q ngl © σ q6¦oß Ê γ q ©p§Às q ncl ¦ γ q Ê η q6¦ σ q Ê γ q Ê σ C γ CV©o©oÇ ¦¤ß Ê δ CV© i £Cc§Às IC Ä ¦o¦ i q ngl §ps q ngl ©p§ η q s q ncl © ¦¤ß Ê σ q © § ÅE¦ i q ncl §Às q ncl © σ q6¦¤ß Ê γ q ©À§ps q ncl ¦ γ q Ê η q6¦ σ q Ê γ q Ê σ q γ q ©oÇ ¦oß Ê δ q ©

dFHx} ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ Ä ¦ i q ncl
§ps q ncl © ¦o¦¤ß Ê σ CV©À§ σ C ¦oß Ê γ C © ¦oß Ê δ C ©o© Ês q ngl ¦ § η q6¦oß Ê σ q ©ÀÊ ¦ γ q Ê η q6¦ σ q Ê γ q Ê σ q γ q ©¤© ¦oß Ê δ q ©o© Ä ¦ i ncl §ps q ncl © ¦¤ß § σ q6¦ γ q Ê δ q Ê γ q δ q ©o© Ês q ngl § [η q Ê γ q Ê η q6¦ γ q Ê σ q γ q ©ÀÊ ¦ γ q Ê η C ¦ σ CcÊ γ CmÊ σ C γ CV©o© δ C Ç ¹`t d q5tÎÃqÏ5A#t³²#uPb11ou1obku ε s ¬tuo#w«5U5 ²#«1Ùuo Ãq Ä ¦ σ q Ê â ¦ ε s ©o©¤d q ncl Ê ¦ § γ q Ê â ¦ ε s ©o© Ãq ncl d q Ä ¦o¦¤ß Ê ´ ε s Ê â ¦ ε ©o©od q ncl Ê ¦¤´ ε Ê Ο¦ ε ~ ©o© Ãaq ncl u¨totu¨¤b «165Uffu Ã s Ä σs Ê â ¦ ε s © d s Ä ß Ê ηl § γl Ê ß5Ë ε s Ê â ¦ ε © 5t6fPuw51Àu5¨³otuÔ²#³P1o#ot5 Ãaq Ä σ CcÊ â ¦ ε s © d q Ä ß Ê ηl § γl Ê ¦gkq Ê ´ © ε s Ê â ¦ ε © Þ t5³ t5I 5t¬u6cotu#u¡ÙutIb Ûkl i q Ì fuÏotc51u#u Û Ê r l Ä Ë5fuÀ51otïµAuuÔu¬ouw 6¬¨1²f¨ 1oU# xÊ ßuf15Ë5f5tb11o i Ê r l Ì tu i Ê r l Ä i Ê §ps Ê f5totbu¬Ù16PI# Ûkl 6 i Ê u¨ot5ot#u¬Ù1uPI i Ê r l t d Ê Äß Ê ηl § γl Ê ¦ xÊ ´ © ε s Ä¦oß Ê ´ ε Ê Ο¦ x ε ~ ©o© XYd òcòaVâàV æUä Vâh ãYÞßçeæ êQP2RSRSRUTV P ÞXW fcV Þ V æ `cb¥`cd çeæ h Y ` a b c Ì ² ¸ ² ³ ² ² ¸ ± ± ± Ì ² ³ ¶ ï ² ²

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 dFHx Ì th1Auut151umt5Ut
ot5ÀÁb5ot § 1#tÀ5w6¬tu¬°«1À²# «16ut¬uÎ5kuo1³ft1k## 5««u U5³u1ut#oU1k#1uwouU¤b ot#wkuotu¨5tÎ5w5³#Àotum1kbfw5 ¸o15wo5¬fot¸ ±±± ö5t5kkuf1kuU¯5kuo1«1u«uto5obPf5tÀ1#¨¨²fo wo1uo1 1kbfw5)ot5 b Ô¾b kuo³¾5tu6fu¥51 5oSku1o)bP³¾#¨³föu«)ot5 #to# oot¬5P5A hÌ tku5um«ftÀ²#ou«1ou¬bt1ot5À¨ 1kbfw5Ut#1²f1b ¬5²fo5¸ ±Ï±± ¨³föu« ¸¤tuuf1#wo5²#b¤ 5A #1«²#u5üo wo1¬PAu5wÔ¯b15fßQr Ö · I tuÑµ1A#fw5 Òbu ²#u uu «5ttu5t²fbo111#wÀ¸ ±±± 5 ¬P«u¬6fotup5t³ touwÏ5¬ku¨À¡¤ Ï1¨À²f²#u5¨#Àbo 5k²f1#ft#Àokb ¬oukPu5 ¬tu¬Eº ukok uto#wo1t5ou³f¬P¸ ±±Ï± ö5t5k#uUPbÀf15w5P¬uot5À¬P¨5«A## 5 61umuto51k1oU#561#to#wt¨³föu« ë #¨À1k## 5« 5o15³Ñ1kb°¬oukP1ku¨16¬)#¬oukut1³#¬u)o#¥u5wuo³¾#ku5¨#t Þ bmbPuf«#¨À1kbfw5UPf#fotbtuwbbpt1²#uw²#u uuÏ«5 5t²ft5 ³¤b «5o¨f5tÎ¬P¸ ±±Ï± ö5t5k#u¨ rÀ#«1uou³¨1uo³¬P 51 5³ 6¬ t6«¨1bPfu ¨#tc«1Ï²#u1uwo#wu °Þ #Â5t#¬Pob5t³ 1kbfw5Uüu«uto5 ³¤1t¬#t¨11u²#³³#¬u ²# 5w³f5tot ö5t5k#u¨ rÀö³otu¨¤1to#t5I5 â À#1 üfbIA## 5« Ôft ot5À¬tuou5o1k²fu³##tot#1bp¬P¸ ±Ï±± ö5t5k ë 5t³1A#fw5«uw«5³t#Iku5¿5ot5Àufu1kbfw5 ¬P5Iübt³ot t1uwo#w5¬c5tb 5obtc1ku¨w²fu²#³otµ¸ ±±± ö5t5k5b11o ¬oukPIku¨¬#¬oukutÀ¨³fö ¸oo5ofot51ot#wbpotö5t5k toutuo5 ¬oukutÀ«1u«uto5obPUob²#o5¡¤kutÀku¨1o hÌ tu toutÀUuutÀotuÙtobbpot¬u v5jwiwloqrxfpl¤q'hx ot¸ ±±± Õ Ö ö5t5kÂÚ · ½ u¬P5¬#5³²fu¬PmuÍ81o³u¨ft5ou²#³ot1u)# «16o³¨11u²#³ot¨³föu¦¤o#)uÍ85«1ufP¬u «u5oAu1oU ¨²#uÍrAu¨#tU#m5k#1ut¬o#m1A#uAu © ¹`b5tf15fuU15¬P ku¨1obpto «u5o5615o#tU6uöt5o#tU²fu³##tot1up'#toA# ¾#tu¬P6¨fotö5t5kuÙtuUotku¨Àbp5#1uw5obouwU#Àot5 uöt5o# rmo#w55t¬tµ#1uw5Pf5u º ¦o¸ ±±± Õ Ö § ßQr5¼ Ö b Õ © ¸¤#¬tu #k¨fot¸ ±±± ö5t5kAufk¬P5Iu5AuÏ²fA#1tub kuo³oot 1kub«#Iotuuöt5o#«to t P²#u15u#²#1Iotuö5t5k#u t#ÀAuf16Aot5Àot1ku¨6bbp¬t5Àu¬P5¬#²fu¬u «tu²f³1üpr 1kbfw5 ÒÌ P1¨r¬oukutt³#¬u«Ò5³uup¬Pupku¨1oÒohuöt5o#t o ¬oukPIAu¨#t¨51t¬P¨5«A## 5o1kb¬oukutÀAu1o ¦obo«u 5«5¬65³¨ © fuuot#1#¬Pb¨³föu«51#to# oot ö5t5k

d§R§ ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ ¹ùw5PbIotµuÍ85«1uc«¬tu1Auu6t151uÂu1ut#«¨#«µÔft ufµ# ot 5³Áb5ot §
1#tÀ5 ¬tu¬°A#1tÏ ¸¤#Aumoku³#uÍr51u ¨5otuüfA## 5«m«¨À²#5²f¬1kuIP 1kbfw5 61²# touw1kuoÏf5t15wo5¬f#5¸ ±±± ¨³föu« t5À¬tAu¨##Àot uöt5o##Àu55wot«uo°b1u 5¬#5³²# ½ 5¨¬thb#1t#ot ¸ ±±± ¬5t5k9ruÙ1to## v5jwiwloqrxfploqrhx tuwtu¬tA## 5«pr5²#o³ oÔft P 1kbfw5 61²#P¬u 1Auou ª #tüfP¬³füuw5bp¬t uÍr51um#6fu5²##fu tu«16ut¬u5m5115kut¬6³Ñ1b o5²#°A## 5«m tf § ufu15P#15fuf«5³t# #wÔb Au¬³#¸ ±±± ³#¬uUot5Àt#w«56³ uu)ku¨¬¬u¬P5¬#t o6¡oukutÀ1ku¨6b¬t5¬P1A#fw5«u uÍr¬ â ¬Pu)uÍ85«1u«5³ #wÔff²#À1A#t¬t5Àotu³ #wÔ«5³ ²#u³fbP¬P5fu 5#1k## 5««upr5²#o³ ¸¤otüo#t 5ö³uÍr6öt«1u«uP¬5obPU#À¸ ±±± Õ Ö 5wot«uo ²#5ub«¬tµAu¨#tcbI¬tµu¬P5¬#¤b «5omotu³ÑPb «5³1 I µ¬Pu kuf ¨#«uÍ85«1uokb ¬th151u)o¨t ¬t5ÀuuwPku¨¬ Ï)1kubot5«1A## 5Í81uoU551uIoÑb11o kbtku¨¬ ufu¬t#1fÀ1kb5²#³# Au tuotuÍ81uou1ku¨6bUü I 5¨kufÀbPbp¬P1kb#oot151um¬öw5oottouuo151u¬o#w kuf1ku¬b1 o1tuÍ81uou1kubufu tu6I#u¨'Àt55o #1m1A#fw5« Ì PuüÍ85«1u¨t ot5Àu¨16¬5¬t5Àot¸ ±±± ¬5P5A 1ku¨ ²#uf¬P¡¤uAutuUÀ5 5«#t¬oukPI6«1u«uto5obt5 1kufutÏm¤kb ¬P1b ¬5²fubu¬Ù1utÏt1 5P¨#o 5k t#¨µ²fP5f# S55w5ou³¯1kÏ ÂÌ hÏu#¨¨#o 5kurotut «¨PÙ1 ¨PAu5o 1kbfw5«t5tf15fuU5PutA#t«uP¬¬P5I6«À¬Pf5 5²#o³¬P¸ ±±Ï± ö5t5k1u «oU5P5 1k## 5««u UouÍr1kuötÁb5ot § 1#t ü5t¬U1b t6otum1kbfw5Uu1ut Àtu3wwy)x832×§$&$©$ed'fg& ×5¿ 13E7)x¿C)8v25B925476983Â ª 1 AutÀ1u«uto5¬#tbp¸ ±±± Õ Ö 5 ot«uoS5²#ÏfuÎt¬ fA#11ötf1tu²#³otufko t6otu³¨11b À¡¤uAutÀÁ#5¬t § 1#tÀo#w«5ot5k 5A Û lojwx#v5jwvQg"piwjwv ³#¬u¨fuÍ8u«1Ùu²#³ot¸¤t¬u Í8¼Qt°o5«³¯#P1kbu¨¨#wr1kbo16#¨11b t¤b5ÑuÍroutu#1²#6UAu¨# o#w5I²f1ÀbP³15wo51111#wÀo#m¨tf5tÎb²#1kub¥Úu¬P³1k#f t¬w1¬#t¬#5#)¬#k5otf5t#1²f1ku¨#tfbPfu ¬t # § ot § Á³¬#)¤A#otuÍ8¬uPu#1²#¤b «5of5totu³1kb¨5 «#uU¦¤t#Iotu¤51 © totku¨1obp5wot«uo°b1u 5¬#t5k kbtu¬¾¨6t#°#¥Ïb1²f°1k¨#uu¬Pb1fotu³Ñ5A°Ôfu11Auföuwm uÍroutuÏb1²fo#w5 ¦ë bobk#t5¼5Ë5Ë5Ë u1A#u¨#wA#1Pk1oUo

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§Re ²##¬¬P1k¨#5Pw5P##Àottfbm#1²#6o#w«5ootu¨bu
¸¤t¬utÍr¼Qt5t#«15o²#°1k#u¨#wmkbtku¨¬mo¾¬tµ1ku¨6bbÏot°t#6 #)#1²fo#w«5o²#1Àku¬5¬P¨5« 5tf5UotuÍ8¬uPu#1²#¤b «5 © dfq'xuyt{rj v5he"{'j ³#¬u¨8t1P¯b q ¸¤¹ ª 1k#u¨#w¨`1A#f6So1¨111# P¤b ¨P#5P#1²#Au¨#o# «5¬²#IP111#wÀ¤bm5¸ ±±± § #«15t uÍroutuÏb1²f1kubo#w«5 ¦ Ì t¸ Ð ë ä1â I ± q 5ktouo1k1kbu #t³¾15wo5f111#w1o#¥¨tfS1kubt`²#11o#¥¬t°11wbSb¬PÂuobt 5¨¤³6I5t#6 à #1²#h¨³fö © Ì üt #«1¬5ob«ftÀ²#ut5¬ouAuP¬³#5uÍ8oPu § ²#5u ¨³föuot5b¨tf à b²#³#¬u«#tüm ª uwbbp¬PuÍr5« ok#15f´5ßßtÚ Ó uk¬P#tö5toU» Ë¯5tÕ Ë5ktouw1kuou5UÏb1²f1kubÁ1#5ot § 1#tÀt1²#uwf5totuÍ81k¨#» Ë à Õ ËtPuwoUot double Ï5¬o³f1 â t#6 à #1²#h¨³föfotuÍr1ku¨# 61²#u515ou#1²f1ku¨6b ¨Pot5PÂ¬P«#¨Pu¡ÙutP¤b «5Poh¨ )Ì t1 q #²f5¨#P«¬Pf5 » Ë à Õ Ëkb1Pu#wkuo³¬#1²f1k¨# ¸o¬PtuÍrÀP¬thÍ81ku¨b » Ë à Õ Ë 5#5²#u515oÎ#1²f1k¨#tf5P#À#1 ötku¨ 6t²#µuf15to¾otµ51(i¡¨Ï5¨¨ftuo q b¾otµ1kbfw5 t1 t · ± f15º 5Í81uou â 5uÍ8¬uPu § ²#5u¨³föu«fuuotb#¬thÍ81ku¨b» Ë à Õ ËP5Uo³# double fotf1#out 1²f#«1¬uku#¬umuÍ8oPu#1²# o#w5¬f5Pot1otuo5À«bÏ 1²fkb1PuouÍ8¬uPu#1²# 1kub I tuotku¨¬P515¨ftuoot5w5²f q ft uu¬f «5³otu²#öbku«u«#w³#5tt q Uu5ku double fotu51 ²#k#1tuo#1²#1ku6¨# ¸¤ottuÍ8ÀtufotuÍ81Au¨#» Ë à Õ Ë«5³ 5f5²#µuf515ouuÍroutu1ku#³#uPÑku¨1Ï¬P5¬ouwcokb ot #1²#u1ku6¨#«f561uöbku q #5¨totu1kbfw5¬1 t · ¾bIuf15º â I#1wub#otuÂb¬buc5Au1#¨¨²fubo#ÏÚotµb1uÂ#1uµoÑ¬#A 5totUkbtot51#ÀotuÍ81k¨#» Ë à Õ Ë¬tu#tP²#uo#k int main() { double q; q = 3.0/7.0; if (q == 3.0/7.0) printf("Equal\n"); else printf("Not Equal\n"); return 0; }

d§Rd ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ #«15 t o q bmÀ#1Ôfuu
q ku#¬umuÍ8oPu1kub ot#1Àö#wt ½ #1o«¿5Pb1um«ftÀ515ootuÍrAu¨# » Ë à Õ Ë5À#«1o«uf1ukt51Ïb1²f1kubbm1ukt51uÍ8¬utu #1²#Au¨# ¦ I o#tÍ8¼Qt#«16of¬P1kbfw5 1wP¬ · ± f15º tu #«16 ¬«#1o«¿55o#«5t · ¾b ± f15º Pu«#«1uo#Âu²##ft © Þ t5³f¨#«b«uwo#)uÍ8¬uPu § ²#5u¨³föu«51ob«5o5³t5tfot kbtt1kub«#¬51¨#1uw5o#t1A#1tk1oU6ku#¬u Uo kbtotbku¨1o¬¨P##mb²#1ku#tf5²#uÀ1#¨²#³ o u 5t# hÌ t1f#otu¨³#¬u« 5 rÀ1k6À¬P²#ut5bm#Àot1kbfw5 ¨1³²#³ku5P¬cbAµbu5t51³#P²f5¨1tÏ ö5tPbI¸ ±±± Õ Ö 5 ¬tu¬ ¾u¬Pm5 51ött5k 5kbmot#«1umbpo5t o1k#fh5¸ ±±± Õ Ö b15tÀutA#t«uP Ü ott5k 5AP5UuukÎ #wkuo³kbtuku¨1Àou5u¬P5¬#tf5ÀAuf16Auo#f5tot #«16mt55¨#Pu¨#«P¬u «u5oAu1oUouöt5obtot5À5A ²#u³fbP¬P1üpr#t¬k#f5ÀU5 uÎ¬ (&4F2 )B9E7E7ÂÆ478AÀt69¿ 13u325B92§476983ÂÆ698A$&%(2§)83D()xDgh5¼©B9Â)xD Ø(½3Â2§)¿CÂ ª bPfuto#t5 6¨#«56t¬5PUot5ÀuÍroutu § ²f5ü³#¬uU«¨À1kbÏ1 ku¨1oÒot5t5k5tu5t5Ò51 5¬urPt#tbk51 5¬¬P5Ñ¬t#üuku# ¨P# à #1²#³#¬u¨ftot¤b «um5 5³fU1k#f5Àu5I5« 1kub5t#oou«#k¬t5¬P5¬ou Ì w51uÍ85«1u¨15¬P ª 1kbfw5 5²##f5 u5Ubk¨²#o1kbfw5U²f5ü#otuÍ85«1u 1ü²f6 ¨t ¬t5Àot b UP5f5À²#u Ú ¨#«5115kuto³ 1#wo5²#1k## 5«¨ t5ktuub ¬5²f5kbUtf à #1²f¨³föu«¨ uu)P#wkuÀk1oU#uÍ8oPu § ²#5u¨³föu«1kuü³²#u51¨h¬t #«16m5tÎt5k 5k#tÄob5bP5³1A#f6«#AAu¨#ot5 ot1kbfw5 uÍr1u¬ ª 1 AutÀ1kbfw5«t65t#5#uU5Ô#I¬Ù11Io#)1kbfw5o1uo³¬P 1kubÀÍ81uo ½ Uot¬6b · 5t#5#u5t ª #«1u ¨º#5#´5ßQ «uto#t¨b5t³1kbfw5«t5tf15fum# rÏ¨1uo³¬P5u5#1wkutµb 5uÍ81ku¨¨#6Ô# 10.0*x ot¨5#toÍ8À¨t#1Îuf55¬o¬th¨5« 51 ¹`b5tf15#uf¨5 ½ 5 kPÁ1utu¬PUku¨1uÀ²#³ 5w5obPU5«#t¬oukutÀ5wot«uo1w6bm¬ot¸ ±±± ¬5P5A ë #k kuuto³#5tf15fc6Ô# ½ ¾¹`¸ ª t5fu²#uu6tÁ1utu²#³¬5tÏ5A § #to#wt uÍroutu § ²#5ü³#¬u ¸¤o5ofot ½ ¾¹8¸ ª ¬5P5AÎuÍr1o³56 U #«16mou515oÁb5ot § 1#tÀuÍr1ku¨b¬1ku¨6b um¬t5¬t5 t#w5³5¨¨#5ou ¬¬o³#1 ½ ku¨¬fot51#ÀotuÍ81ku¨¨# 10.0*x «5³5w³ 5³fUot5Àu1ut#5w6o³bpo5¬#wrÚ Potum¬P

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§R§¦ uÍrAu¨#)««u65¬u³¾5¨ftu¬¾µ5w65²#°b¥515 )5)u¨²#uÍr1wu¨¨#
¯5k#umuÍr1ku¨b Ü tuotum¬thÍ81ku¨b15wo5¬ub15wb Ü Potum¬PuÍr1ku¨b15¨üÎ5U55Af1PIooto#tf5t6À¨# PotuÒotµ5Af1«utÏm15ü²f³Ñf5°#Ò²#³ÑkuoukP Ü ¬tµwkut1ku¨6b «# Ü otufubpb1o«6¿55¬#5 P¬tA## 5 5Ub1u Ü ot 1kub«bu5tuÍ81Au¨#«uf515o#««uotbü²#³otu#«16 tu ot1kbfw5 5#«1u Ü 5t# 5tf15fö5t5k5APbÀutok³o²#5«¤bmotf5f5 uU#ÀuÍ81ku¨¨# uf55¬# )± Íroutu § ²f5ü«¨³föÂw1«bPu¬Ù16P¬³ tuuÍr1ku¨#tÂ5A uf55¬uÎuÍ8¬utuÎ1Au¨#ku#6ö tutuu)1#¨²#uf³fuÀf5ot5 «¨À²#¬#AuÎ5A¬#kuotP5wk u¨À1ku¨6bAuf16Au hª bt¬ 5tt 5tf15f¬kuf1kot5 10.0*x uf55¬o¬P¨5«51uuw³ tuk b «1#¨h1uw¤b «5t1ut5o³#ot#¨³#¬u« ut¤b ¬t5¬u³f5 P otb¨³föu«¬uf5615¬ 10.0*x ¡¤uAuto³³#P¬5¬5³uf15ut #touÍroU6«1#üU¯1ut5o³#À6¬ #1k## 5««u U#À51 5ot1«u 5 ¨#o 5k²#³1kufutototu¤A#ku³ftb¬P¨³fto5Í¯#Àotum1kbfw5Uo uÍrAuUotumtoPuü5t¬ ¶ ku511kbf 5«UÏP#ot5¨1«1o#ot5À#6fuuÍ81ku¨¨#5 5³f uf55¬u¬ot5«f561 pi q u51ot5fb ¬P1kuütou Ì tu#Auo# #«1¬PP¦oß Ê Û © wooutuk Þ #wow5Ú â 5uÍ8¬uPu § ²#5u¨³föu«fb1um«5³uf55¬otuÍ81k¨# 1.0 + x ¬th¬tk6tuÍ8oPu1kub5tb«5Aotku¨ o 1.0 I tuotu5«Í81ku¨bU15ü¬¬t#o1PobotuÍrotu t uof¬P#«1um«5³ö#Aof5«ub ³A#1tP6Io¨tf 1kub hÌ t¨f x t#I¨«5¬t5 1.0 + x k#1to 1.0 uÍ8¬utu 1kub²#1À¨51ut#1fot5 1.0 + x kbtUo 1.0 ¨tf1kubP otuÑotf5Auo1 tu²#³ log1p(x) 6#²#¿5ukhPöu5b x r5totku5¬ uwk# 1²fbt pqÆ 5¬Pu)5kfu)ot5 Ö ε ¹85w³f111#ötku¨Àbpot uÍrAu¨#¬tÍ8¬tuft1Potkubwkut#Àot1²#uÍ81ku¨¨# real function log1p(x) real x if (1.0 + x .eq. 1.0) then log1p = x else log1p = log(1.0 + x) * x / ((1.0 + x) - 1.0) endif return

d§RFH ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ 1.0 + x `mu515ouÍroutu1kub
¸¤¬P55üb x )«5t²#1Pb f1o¨51ut#1fot5 1.0 + x kbtUo 1.0 ¨tf1kubPf¬P¬P 51Auo1 tu²#³ log1p(x) 5ÑuÍ8uuotUb kt51²#³¯tu5w³¯5Ò«1Ñ5 x #5t5f5«otµku5¬µu kbÂ551A#5«#t °Þ bÂh#tku¬uuÍr51ubo5Ôf x o²#´ sr ~ut Ê ´ sr tpv f x Uot«5u¨Àtf1k¨#P1«²#um¨¬t5 1.0 + x kbtc1oÑ¬tutuÍrÏ5kfÂt«²#u¬#ß Ê ´ wr ~g§ SÌ tu log(1.0 + x) 511kbÍ8«5ou³´ sr ~g§ Ð u5ötut#«t5o#)¬PuÍr1ku¨botÍro t6Uu515oÎuÍ8¬utÏÎ1ku¨#tf6I#«1¬uÎÍ85o³5tÏ6fu x ¨ log1p(x) kuo1wPU511k#Ír5¬u³´ sr ~§ tUPu5w³ h5U5kf5Uot uÍr5Àf5 ¦ Ì PU5¬5³t511ut o5Àu5¨À#tb«u I tu¬t 1kuutb#«16u²#³ot¹8 I b Ôf¹8t# ª #«1u ¥ ´ ß Þ #wow5Õ5Õ #«16mo#mÍr¼Qt¯¨³föU¨tot -O #1o«¿55obÁ5#fotfutuw5oub #«1¬u 1.0 + x uÍ85o³5Òu ²#u ½ ÂSku¨1or¬PU¤1t¬#Ñu Ò¿5uk o# log1p(1.0e-10) 5t 1.19209E-07 ob log1p(5.97e-8) © Þ bm¬P5## ¬t # Ì P#kuo #wÔ# Au¬³fotuÍr1ku¨# 1.0 + x «¨P²fu515o«otS¨5« 5³¾u5¬6«6P511u5w Ü ¬tS5fb ¬P 5o5f# uÍroutu § ²#5ü³#¬u«UbP³ tu 1.0 + x Uuf55¬uoÑuÍ8oPub²# 1kub#tt¬5P5tÎ¬¨P##mb²#1ku#5Pbotu hâ #1 üft log U¯#utu SP¬ t¨6Soto# Þ #wow5Pf#«16mb16 uf55¬UotUÍ81ku¨b 1.0 + x ¾uÍ8¬utu1kub¾otk#1#Pb¬5#«1¬P oU#f5wot6ot¨51k¨#tf²f1Àufuto³ 5tt#À5¨¨«ot5À¬P #«16 Ï¾¨ ¦ â tµ55¨¾5#P°u«5ÒuÍ85«1°6tf#f6tÑu1u § uÙ1Puo1Pob ¸¤ot5À5ü#«1um#1Îö1Ôfuu¬P5k#1utÀ uÍroutuAu¨#u¬Pb#¬t¤1t¬#kuo1wPUtf1ku¨6bAu¨¬ ²#I¤ p5t³uÍr¬P Þ #wow5#«1u Uotü¬tu © I «ftÀotuAu¤bk 5o¬u«1Àout¨1kot5 1.0 + x uf55¬ubP¨öuto³²#³5¨#t6tIo 5w5²# uto# ¬1P5¬u³5 u5kUot5tf5 5²# real U«5³¬#²#U¤b6 ²#³b1u)ot5À¨²#öo1ouU51Ôfu1Àkuföu)uÍroutu1ku6¨# o#mbt51u5w5tbp¬Pf5 5²#5P51ö#AuÎ«ub ³6¨P# 1kubo#)5t#otu ¸¤t¬u5 #1Îtuu¬u5Ah¬t5w65²# 6¬ o³#1ot5À#wku¨btootuÍ8¬uPu1ku¨6bob «5 ¹`¬5P5A Þ#â q Ì q ½ Õ5Õ#uUPbÀ1kb 5³¬¬P¨f5P t6 Þ #wo 5»r5Ë#¡¤ ¬t SELECTED_REAL_KIND «ut5P¨ o#Âu¨w²ft5w#1co# «5¬b6ÏbucPb uÍro³kfk«1u«uto5obPUot5Àuf55¬uÍ81ku¨¨#tuÍ8¬utÏ 1kub¬5 5w65²#uUo²#u5k oot5À1k¨# ¸o¨t#wofotuk Âthb ¬5²f 5³¯¬ o¬t6Ò1kbf 5 Ñ¬5P5A Þ #wow5¬t5PÂ#5w5touu o1kuPIotuÍ81ku¨¨# 1.0 + x okb ²#utu515ou 5³¬P5 t55ouU#1m1kb#

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§R§R Ì tuk5k#¬tumuÍr51uUot5À5«5¤1to##uÍ8¬utu
§ ²#5üÎ¨³föu« ufu tu51²#uÍ81k¨#ö#AuÎ5tot1kb1Pu¬ot5« 1kub hÌ t51 v5he"{'j¤}Ahefx#v5qrxuy ¸¤otuo5À1k¨#buf5 uÍroutu § ²#5ü³#¬u to5³kb1Pu5Au1À¬uÍ8¬utÏÎb²# 1kub ¸¤Ï¬P5ku¨16motuö#ko¾#1²f°1ku¨#tbcA#1tÏ5f5 Ì tSb«²ft5ob#1otu¨S k#1ttfÂ5³#u«µ51S¬t51Â¬oukutPot5 P5 bt5²#uuÑ#²#¬56tu²#³¯kbtt¯otÙ1 ¨Pku¨1Pb k¬³¯ohÏb1²f 1kub hÌ t5t511u tuotku¨À5UkbtuouÍroutu#1²#6 1kubU · t5 5³5üº5f f6I6uÍ85o³P5 5³²f uu Ïb1²f 1kubt«²#u ¨f¨¬Puu#tA#1t6tUÏouwtu²#³¬tkbt § ou § o § ufuk ¸¤1ot)ü#tA#1tÏt¾A#1tÏÂ¬t°¨5«S6Au¬#5)¬PSÙ1 ö8ot tuk#1ttuwkb tuÍrut5Ï1tÏ¬tµ5¨Ï15 ¦g¾bouòt#1ftòt5 #1²# § A#1t6tbt³5¬ouob²#1kub#«11o5obP â t51kb ot5À¬t1«¡¤uAutuf1kbofbmf1#outÀbp g5§ ²#Àt«²#u ¨f#)ot ¨f5ASA#bP#1 g5§ ²#PP1«²#uorA#1PuÙ1 ¨Po ²f¬Â5t«¬tu¬ g ²#oÒffuÂot ¨5U515Ò6P¬PUku¨ uAkbtui¡¨t#tUo g ²#¬Â1kbu ≥ ´ g°Ê ´ Ì t¨fuÍ8¬uPu#1²#1ku6¨# uPb1fot5À¨6t#h1Au¨# #«1¬5o#tU# rÀ¨¡oum#1²# § A#1t6t © ¹`b«56##wot«ot5Àu1ut#b AuIk#1tP5¤5 oÏb1²f § kbtt ¸¤¤5ofufub5fb ¬P«Uot5À# rÀAuf16Ab kÀA#1tÏt 5t b Ô#wkuo³bf5 uo³#À5tPuUot5Àb'Àbt¤b o¸ ±Ï±± Õ Ö 5o5 o#1²# § A#1t6t hÌ tbIüo1#Àotu¨5kot1#wo5²# 5fb ¬P«obm1u o#w«P«5¬u1o11kub5woto°«utobPu #15fß5¼Qt Þ bmuÍ85«1ufot1kbu1ku¨w²fÎ Ì tubkut¤bm¨¬ot Á#5¬t § 1#tÀt«²#umtot#5t 15woUÏbu¨' b Ô#wkuo³ #1²# § A#1t6t5wot«uo8Úuo ³o¨1Àot#1²#61Au¨#P1«²#um´ sx ~ Ê » × ´ ~pyE ßto 15woUu5 6¬5IbI´Qt¯²#o I Puu5b1u 5¬#6 kbtu# Au¬³oÏb1²f1kubt¬tht#bkum15wÀ´ sx ~ Ê ´ ~pv 5P¬P bkuÂ15 Ïc´ ~py ß5b²f1 tu«5«b1u 5¬#«ckbtuÙ1 ¨ÏoÑuÍroutu #1²#Au¨#5tÎ¬tu¬#1²#61Au¨#tfot1k#u1k1k#1utf #Aum15 I#À´ sx ~ Ê ´ ~p 5t¯ bkum15wÀ# ´ ~pyE ß hÌ t5oou)t«²#u #1uU´5Õ¯²#o¨fo¨f15k5 rI²f#«11ouuÍ85o³#1²#61Au¨# â À#1 üf bö1²f1#¨¨²fob11o¬P¨f15k#Àott1²#u)6 uÍroutuÏb1²f1kubt²#1Àotku¨1ot5fb ot bt#t#um²# 1#wo5²#o¨tf à b²#¨³föu« ½ ¨#f5ou)¬u1¬t1o11kub «¬165ob5## ¬t5¨1«uot5I5P5wo5P1kboct5u²#uu#«11ou Ïb1²f1kub hÓ 5t6t«Í8¬A#pb²#5tÎÍ8outu#1²# 5w5²#u#wkuo³ #1Î«5Ôf¬th«1u«uto5ob¨ftÙ5to³«#A uÍrP¨

d§R§ ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ Ôf 1b ¬5²f5fb ¬P«obm5P1o11kubt1²#uw ku1Auut¬u5m5w
5³#m##1²f°1kubt«²#u )5¤5#1²# § kb1Pt 5 ¬tu¬ Ì P5fb ¬P«¬³f156³ku³bouttf¨5m¬®U5t5 r ¨««5o#o#w1 ½ ¬Pto#w51uÍ815t5obbp¬P¨1«5obo#w« ffu#5#´5»Qr1f#u¬f s 5t y 5kÁ#5¬t § 1#tÀ5w5²fu ¬ ¡ s ¡ ≥ ¡ y ¡¥5P b11ofÚ otu«#¨Ï5wot«uo e Au# muÍr5¬³¬tµA#1t#¬u A#Òot5#wku #«1¬P thÌ tottfbu r #wÔ#1²f § kbtu5 ¬tu¬ t uuÚ s Ä´ sx ~ Ê ßu5t y Äß à ´ ´ srx t b¬tu s + y kb1P)Ùw¬¾´ sx ~ Ê » à ´ uÍ8oPu#1²#Au¨#tf5tÎ¬t6U51kbt¬´ sx ~ Ê ´#1²f 1kub²#³¬PA#1t § ¬6 § o § ufuw1 Ü ¬tUottuÀkbttuwk#)6 #«1¬P t Âß à ´ Ê ´ srx t t¨ rPku1kuüto5²#SuÍr5¬³¾#1²#S1Au¨# 5t¨Ñ5 r²f°b11ouÍ85o³Ñ²#³¬P°uÍ81Au¨#¨t 5²fb Ó uk°5f5t #1Î²#1#¨¨6²#oku#motkbt#¡p kbm²#³#«1¬Pot1 uÍroutu#1²f1ku¨#tr²f1P¬tu°1kbfw5 #1t5¬uÍro #wÔ¯¬ ku1otÙt51#1o11oU²f5Ôo#1²f1ku¨6bt5t#1²# § A#1tÏt#1 5¬Á1À¬t1kbu¨f¬# Þ bm¬PUku5¨#tf5ot#1f1#wo5²#A## 5«o# ¨15ot«16¬1h1Au¨#5 ¬P«u¬6S²#³otuü¬t# #wÔ# Au¬³5t u¡Ùuto³# f5 u¬³#À5tPfotu³b r # Ô55 oü# uÍroutu § ²#5ü³#¬u Þ t5³f¨#«5fb ot«¬P5I5ÀÙwI#PI51u5)ou1ut#b Au kbttÑ«5³¾6o5 #wÔÑ#wkuo³ ob²# § kbtt ¸o¬tuS5ü8ot #¨À#Àb1P o#1²# § A#1t6tuUt#Àot1u«uto5¬#²f1Àot Ù15¬6b¬t5ot°5f#wot #wÔf)5m5uwoü Ì ¾1¬ 5¬u °1kb°ot o# t5w5tÀ# Ì tubkuÕtÚ fhgpirqtsuirv zct pxfvux¾p}Aj¥qrx#l¤j6ytjw}wi¥}Ajwg5}Ajwiwjwxflop"{'j¥qrx 99¤ §¦ v5he"{'j¥g5}AjwsVqriwq'hx¾|Uq'l¤n ¢ t ¢ ´ sx ~ pxfvxn5pi)l¤n5j)iwg5jws qrp{zh}wtxî° ´ ¦ ´ £ fl¤n5jkx ¨ t x P ⊗ x°t3g5}Ah5qrv5jwv"hlon mhpl¤q'xuy og5hq'xflPhg5jw}wpl¤q'hx#ipp}Ajpjkqrl¤n5jk}p}Ahefxfv5jkv°sVh}w}AjwsVlo{ Ý loh)v5he"{rjpg5}Ajws qriwqrhxh}À}Ahefxfv5jwv Â}wiwlÀloh°j Û lojwxfv5jwvv5he0"{rj)g5}AjwsVq'iwqrhxpx#v¯lon5jwxlohSv5he"{rj)g5}Ajws qriwqrhx0 t = s + y; e = (s - t) + y;

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§R§u ehsuqtqgf ½
¨1« otb#¨m¬P5 t h Ë u Ä t x ¹`5t²#³Ñ1 uwm#Ï b 5bPüÒ5uff5utÏüo¬PÑ P´ sx ~ ≤ t ¤´ sx § 5tÔ# ¨µo# b ot5À²#bo t 5P 5kP¬ufuw t#ü5À¨ftÙ15tÀ²#oU#11³ot1t6¬ 15¦o f161¦ tu© Ä1¦ © Äß © Ð uo#k¨5tf 5¨1u t ¤´ sx ~ f5oo ¨5tf t Â5utouf ½ 6¨#8²#u5¨SotS¨5u«f5uÂ# t 5t ¨5oö³ t à ´¯¤ ¤´ t ¬th#wku¨1#tt51# x «¨Àt5bthbp ¤b « u1utP# tb t # U65AfuÚu6 ¤ t fotuuuto³ß¤ x ¤´5f5P ¨P x )µ¨ b ¯1 u Â# b x Ä«ß Ê ´ sr C o#¥¨#«v{ Ü «5 ³` h t otu«ß à ´h¤ x ¤ß5#¨ x Äß à ´ Ê ´ srC á u¡ ¦ ½ x cotu1 #I 1 uwU#I # otµbu¨Ï1#¨²#µ51ub x oÑ#tµ x Äß Ê ´ srx ~ °Ð u5¨ t à ¦oß Ê ´ srx ~ © ct 5k#umot5ottuÍ8À¨«5um#1²#h1Au¨#t«²#umuöt5 t 5 r t5 Ä t © u j ut#¬otk#1ttu A#mb«1¬6t föt5 Ä t à xÊj f5tÎ¬P #«1¬u51 ⊗ x ²#ot¦o#thbm © kb1Puf5b tÊx#j ª bPümÙ1 ¨Àot5 tu5Á1#5ot § 1b6tÀb1u 5¬#kbtu #wkuo³o#1²f1ku¨6b ¸o¬PU5üh¡ j ¡Ò¤ß à ´ ¸¤ x P5Uoto#w ß à ´ Ê ´ sr£CE á u f¬P xfj Ä xf t U5tou#um«1o1#p´ sr£CE á u 5P ¡ x#j ¡Ò¤ß à Ê ´ sr£C ~ ¡mÌ t)«1uÂot5Ñ¡ x#j ¡ ≤ ß à q u5fot51¬tS¬oukutS²#u uu t 5t otPuÍr15Afuku1kut¬5²fSt1«²fuÂßµ5t«otS¡oukutS²#u uu t 5P«¬t tuÍ8P«5uku1kuüto5²#St1²#u)u¬P¥ßµ t h ´ sx ~ b¥ß à ´µ t Ä«´ sx ~ mÌ t¨ 5 ¡ xfj ¡ ≤ ß à 5 tÊxfj kb1P¬ t ¦ ± fu t Ä´ sx ~ 5t xfj Ä ß à 5¬t 1kb Pkb1Po t ²#³otukbt § ou § o § ufu«w1 © ¹85w6³# x t5cot o#w ß Ê ´ sr C botu xfj 6m5«toufÂ1o1µ#I´ sr C 5t ¡ x#j ¡¤ß à ´ Ê ´ sr£C á u Ü ot «16u¡ x#j ¡ ≤ ß à ´ I 5'Àt5 t Ä´ sx ~ ot5¨h²#u51 t öw¬³ fAu5ou)ot5 f¨ t ¬ou U¤A#otu5AuÀAu1kuüto5²#Pu#t²fb U²f³ ±ß Ì t¨f5 ¡ x#j ¡ ≤ ß à ´5f5#5 tÊx#j 1k#1to t ¦ ± u¡ xfj ¡ÒÄß à ´5fot 1kb 61kb1Po t ²f³otk#1t § ¬u § ¬ § uuw1²f51 t Uuu © Ì t6Ub«u¬u¬th1A##Ào#m#wkuo³A#1tÏÎ5wot«uo ¸¤#1²# § A#1tÏt5wot«uoI5³ö1t511u¬P5 ¬tb kuo³ kbtu«fbout1¦ouuot#1f6 5)5o15³ÑA#1tu © `¨ ¡ j ¡À¤«ß à ´µ5 5²##f ¸¤otm5ü °5511u5t¬¾ot°5kf1«utombÏ¬P°1k#1m15w5fw51 1kbu °btu¥ot°o51ot5 ⊗ x ²#°kbtu UÌ ¯5#1to# ot¨8t#¬S¬P5P¬P¸ ±±± ö5t5kkuf1kuÂot5P5uÍroutu«b²#Sob «5P5w ³ 5u5¨9tQu¨ftÙ15t²#o¨`¨¾ot51ot°t«²#u t ± ß à ú5t t ± ß à u5k°uÍr5¬³ ku1Auut¬5²fuÍ8¬uPu#1²#1ku6¨# Ì t1¨ x t5¬Po#w ß à ´ Ê ´ sr£C á u #¨Ñ¬t5¡ xfj ¡ ≤ ß à 5botuk#1tt t Êxfj ¬ÑuÍ8¬utub²#µ1ku# «¨À1A#1Au1À¬t5À¬ouwokb t ²#³5À«#¨Àß à 5f5t5Ut#ou5²##fu

d§R§} ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ otUf5 6Pkbt¬ t 6«#1²fu1kub
¹`«65w³# x t5Uotu¤b ß Ê ´ sr C föt5«¡ xfj ¡ ≤ ß à ´5¬tuA#1t6t tÊxfj ouÍ8oPu#1²#Au¨# «¨À1kbkuÏot5À¬ouwok# t ²#³5À«#¨Àß à ´5f5totf561 6 kbto t 6#1²#Au¨# ¦ q u51ot5 t h ´ sx ~ ¬PU5¨ © Þ t5³ 5kuoÀo#tüm5uU t UPbÀot#wkuo³kbtu f1#¬6PI1o#1²#6 § kb1Pt ¸¤otu¨5 t5f¢¡ j ¡¥¤ß à ´ Ê ´ wr£ á u ot b ¨À5u Pk v 6Uott«²#umbpuÍrow¯²#oU¬PuÍroutu#1²#6 o#w5 ¦ ½ 1uÍ8ötuÍroutu § ²f5ü³#¬uU111#wÀ5uÍ8¬uPu#1²# o#w5 ouÍr5¬6³îtQ¯¨ftÙ15tÀ²#o Ü ¤bmotob «5o v ÄttQ Ö »¯Äßß © Ð u5¨#1²#6 § kb1PtbP³A#1uU5t#wkuo³kb1Puku¨1 tu otUu#tk#1t6tÒuouw«Pu²f³¬PUk#1t § ou § ¬ § uu¾w16 «¨t²#5 ufutofu Ì t1 x t5oto#w ß à ´ Ê ´ wr£Cg á u fotu xfj Ä x# t 5 tou#u1o1Sb1´ sr C 85t ¡ xfj ¡I¤«¦oß à ´ Ê ´ wr£C á u © ¦¤ß à ´ Ê ´ sr á u © Ä«ß à Ê ´ srC ~ Ê ´ wr£ ~ Ê ´ srC ~ ¬ ¸¤{ ≤ v #¬tU«1u¡ x#j ¡ ≤ ß à ¸o{ h v ut5f ¡ xfj ¡ ≤ ß à Ê ´ sr£ ~ ¬ ¸¤uotum5ü¬tÙwÀA#1tÏtbp¬P1A#1 ufum ku¨1Àot5À¡¤uwokb t ²#³5À«#¨Àß à 55t²#³1kuf6b15k#«uto¨fot übtk#1tP kb1P¬ t ¹`«5 ³86 x P5)¬PS¤b ß Ê ´ wr C òtu xfj 5«touf)«¬16µ#I´ srC á u #5P ¡ xfj ¡¤ß à ´ Ê ´ sr£C á u Ê ´ sr£ á u Ê ´ sr£CE á u¡ ¸¤{ ≤ v fot«1u ¡ xfj ¡ ≤ ß à ´ ¸oa{ h v t5¢¡ xfj ¡ ≤ ß à ´ Ê ´ sr á u¬ ¸¤ uotu)5üfotÙwIk#1tP#Àot1k#1 1uumk1Àot5À¡ou ok# t ²#³¾5P«#¨1ß à ´585t«5f5²#³¾1AubÂ5k#«uto¨8otSü#t«k#1tt 61kb1Po t Ì tµ1kutÑ1k#bÏ¨t mot5ot°1k#156t1Ò#1²f § kbttÑ#t³Ñ otfbout1bur5Puu¬PPr6Pkb1PÂootb kPku¨ mÌ tS1kb#15 ¨P Uot5ÀuÍroutt#1mAu5bPt¬t1ot1#¨²#o³bp#1²# § kbtt5²#t5utftufu¤bm1k## 5 ¬bP³ Á#5¬P § bt #1uw5obP Þ #)bkb«5ou1kbfw5fÀ«5³²#1b¨²f¬ ¨³föu«5o5³5#1tÏobÒotµu¡ouom##1²f § kb1Ptf`t#o¾«utob«#k fPuw51b²#t5o#tU#ÀÏb1²f5tuÍ8oPu#1²#Au¨#b«1¬5o#t (&wy6xG3wxB9¿C¿C478(G3B98(G3u3BxG3)ÆØvu(1(1369w2p)69w$&%(25)x83D()xD(©wy)'047Âx47698 Ì th1AuutuÍr51uUtbt#À²#o5Ô#u¬¨#fu¨Àot5ÀuÍroutu 1kub g5jw})iwj UP5w«¤1 ë 5t³1k## 5«U5²fPuÙ1Àokb uÍroutu 1kub tu¬tA## 5«m5²#o1¨hIuuofu³ ut¤b ¬t5¬u³ wkutÀ1kbfw5«t5tf15fuUt#À1kb¨¡ÙutÀ«u5to#m 1kbfw5«u)o1uo³ tu5PP uÍ8oPu1Au¨#¨t#1²fuü Ì t5¬ t5À¨11b ÀUPuÏÏ btum¬t 5³fU t «ft 5PIo«5t5fot1bpuÍroutu1ku#

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§R§ ¸¤1#wo5²#h1A#fw5 ¬t5À1uU#1²f1ku¨6b5UoUPbt5
b Ô#P 1kubt¬tuk5kÙ1 5³f «6#t 5tÀ¬#tokb1ot¨#À u 1kubÚ ß mª b1o1A#1¬Po5öu¨Àbuf¨tuÍ8¬utuÎ1k¨# tuk 1#¨¨6²#µbuÍ8¬utu § ²#5ü¨³föu« ª u5w6³Ñ«#¨P1«u 5t¨#o 5k°#u t#ÀAuf16Abk#Àot5 ¬tu¬°ot5ot5À¬PAu5ouwkbm5 #1uw5obc²fb1Pu²#³¬P · «5Ptuu1¨#tº I Pu«5oh««ub ³5k ö#Au6#1²#Au¨#tfot«5ttu1#1¨5³¬5Ôfuo²#ot 5k#uIku5ofk#1t#¬Àuwk#)¬P5IAu¨#tftotP11À5¬¯5A ¦o #P¬³# kbP#³ © 5¨«uoht5f²#uu¾k#1tu tu¾otu³ ukUuP¬uku 5totuk1o 6PÔ# ü²#ukbtu tu«otu³5Au¬#k µÌ t1¨ t #«1¬P¯¨#«S#1¬tStouwu5¬Sku¨1oÂuÍ8¬uPuAu¨#«5³¯³fu «#k51w5oku¨16¬fuÍ8¬uPu1ku¨6bUt#Àu¨P¬5 ¸¤otU5ü «ftI1k¤uÂot5I¬tub1uÂüuÍroutu1ku6¨#«bP³ tu« PPb 511ku5²f³ ot1kbf 55tÎ1¨#1²f1ku¨6bbotu ¨ ´ u¨¤b «5 Ïmot5#1²fÀ6Iku5¨#t5²#³o5À5t 6utb#P #¬tu Aub À¬¨#«uottu ¹8#«b11o5¬#t5²#1uw¤b «u «#ku5¨³ PuÍroutu1ku6¨#5f55²fuf²#Iotu³55¨²# 5 wu«b1Pb²#1kub oÑ#t³¯¨#« t5Pfk5oupu¬o#w Âª #tü #«1¬P¾ot ± 16u5t#w #ufu¬#¥##1²#6S1ku#t1«²fuw cÐ ³ #«1¬Potf15kuU#Àotuu«uto5t5115ot¬tum15 ¸ ±±± Õ Ö uÍ8¬uPu#1²#¤b «5 oo umuÍr1#tutÀ 5t#u 5 ow5³5f#1ku«5o1k1tÏ Á #)#fu Á o#)uob U#p1w5¬5 utf¬P uâ uÍroutu § ²f5ü³#¬u«#otUU¬to5öu¨ 5³o#«1¬uot t#w â ¨P# à #1²#³#¬u¨f5Í8outu#1²#hob «5 #1Ît5f o²fu«15ou¨#o 5k¦opbt uk¨11b ¬u5À5 © f5t1 u«165¬# b²#1¨ um¬P5¨6«1³1tb²#1kubP o¬6t¬tuÍ8u1o#Á5#oÏouwt tu¬tum1PuwÁ #)#fu Á #wkuf5tÎÀbfku1u5ot¬th#«11o5¬6b ouÍ81À5t ¾#o ot5toh¨11b totÒ¨bPuÍ8¬utu1ku#t5°5tf15f«1t1kb²#bo 5tÏ5ob«bÏot u¨Ï5f55²fµo#w«5Ïot5cAu5bt5²f³Ñ¤5öb¨Ñ¬t5Ï 1kbfw5 5t##¨ P«u¬Pbo15tutkbtut¬515w5ouw ot5tP5o¬t1kubÑ5Pw5tfb15Ño#w5¬rhot5tot1kbfw5 5 o³ot5À¬t u¨À¤5¨Ào#w«5À Ïutb#¦o ¬t5ÀÀt5 Ï 5t#ot5b²# © » u¨¤b «5 Ïmot5#1²fufuÀÀt5¬²fu«15ou¨#o 5A Þ bm«#k#«15¬u1k## 5«¬P5ot ± 1u5t# uÍ85«1ufot 1kbfw5«up5³¨1³ ¨Ñoh5bottuu¬ ¬ huwbPÒb1¬t

d§§ ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ 1kbfw5 5t6töu5ku³#uÍ8oPu1kubufupÀ¨ ½ f5t
ot5tf15f«1À1A#f6utfk#t«uto5115 5«uouw¨¬P5Iot1k## 5 5u¬u «tot 5tf5t1ku##Àot ÏÀ5f55²fo#w5 m¶ # rÀ1¨h um1kub Ü A#1tÎk1oUb k¬6³o¬P1k¨##Àot #1²#¤b «5of5²#uÀ1#¨²#³ ouÍ8oPuw5P# hÞ #)A## 5«¬P5I5k «#¨Àu5¨6³ woo¬u1utÎb#wkuo³A#1Pu#1²#1ku6¨# 5 ¬tu¬ft1tb«hbp¬thÍ85«16«uto#tu5²##5tf15f «¨Ï1kbµ 5³Ñ¤bÂotµ1kbfw5«uÒoÑt5¬µot5ÏuÍroutuAu# «¨Àt#I²f1üufuot#1ftouwu5¬ku¨¬«5³²#h#«11ou ku#¬u ¬ ÏmuÍ81bPPI 5tf¬P5#1²# ¦o¸¤to «u5oku¨1o #«1¬u6ot 5³5¬61t1mb²# § k#1ttp¬Pu³tuwÁ Pöbkuo«u«#w³ÀÚup¬PAu1Àbp55 ot«uoS#1uw5o#kb1Pu Ùw¨Ï¬ Ö »u#PÙ15t²#o¨botukb1Pu5f5o¾o Â¨ftÙ5t²f¬ tu À«1À²#ut#w5¿5ufotÙ1P51ku¨1À«5³¬oumokb P5 bt5 ²#uu#²#o5tu²f³Ñk#1tt3i¬1¨Ï#tµo¾hutb «5¿5uP1«²f Sâ I#1 ü otU¤b #ÀÏb1²f § kbtttft³1tÔfu³¬5¬ouÀ5t³1 5o5 1kbfw5 5 ü³ © Öt q btku¨16¬#wkuo³¬²##oot1kub5t 5t#hbp¬thb²# o#w5 hÌ töwÀuto#Au«utÀbp#1²#Au¨# b16²#bIüo o#Ò1kbf 5«mot5ouüotuÒt«uw5t¨#o 5k°#Ò¬P°5 ¬P«u¬6U¬uÏtu5 ot«oÂb1²##¬¬tSw5tf5tAu¨#Ñ#Pot#1²#6o#w«5 ¹`1Ñ5kuo oÀ1A#fw5«outo²f¬Ù1À¬ ob ¬5²f 5³ Ü otu³²fu#« ufu«#A6¡Ù1À¦o5Puwk#)1k#t © tuotu³«1À1#³««³ ¨²#kb¬tu5t#otu)o ÔfUoo#AAu1oUo²#kbtuo15wo15 o#w5 hÌ t1f1k## 5««um1t5uÍ8¬utu § ²#5üÎ¨³föuoÏu# kb²f1¨À¨#o 5kot5À«1À²#1#wo5²#o51¸ ±±± Õ Ö 6«1u«uto5obt #1Îf1Ôf³#«o511ku5o²#ut5²#h¬u«165¬ot5 ¬P«u¬°# ¨P# à #1²#³#¬u ¬t#1ÀuÍ8¬ 5bkt5 ³u¡o# ¾wkutÀ5tf15#h¨111# ¬51Ùf#Àotu¨hb1o#t ¸oo5ofo 5tf15fu t55¬ou«1ouof¬P1kbfw5«u)ot5²f¬³o#t¬k#1ot1b uÍroutuAu¨#5À5 hâ tPbo5²#uÍ8u1o# ª rQífot5ou¨Àkufb¬ ot ª 5tf15fu t6t 6otÙ1P51ö5#u#Àö5t5k¿55o# Ôf¬P1 kut ª ö5t5k ª rQí 5 5«16ut¬5o#o515o uÍrAu¨#tU6o# «5 um¬P5ot5Àtb «5³5¨#5ou ootumo³#1u ²# ª rQí Au#«utU1¨PbtbÀ#t³otAuuuÍ81k¨#515o#«ot# Ì th¬tkuku#««utuÎ«u¬Pb5kt5 5ouw¿5uÎ²#³¬thÍ8outÀ¬ P uÍrAu¨#tU5k · 1k#«#¬uºo umo#w5¬f5tot1u«uto5¬#6 ut#1 5#uÎ¬utoo³ t«u¬PbÎIü²#³uÙ1t6t¬tAu1k#u¨b

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§e «5A FLT_EVAL_METHOD
Úu FLT_EVAL_METHOD Ë5fu5uÍr1ku¨b uf55¬uÎ¤b «5À¬P5Ib ku¨1#tUooUo³#1 Ü FLT_EVAL_METHOD ß5 float uÍ81k¨#t5AA#«#¬uÎ¬oto#w5Iot5À#wku¨bto doubleÜ 5t6 FLT_EVAL_METHOD ´5 float 5P double uÍ81Au¨#t5A1k#«#¬u oot¤b «5À¬P5Ib ku¨1#tUo long double ¦ ½ «1uP¬5o# 5 uoü FLT_EVAL_METHOD o ß¯¬t65o¬t5ÀotuÍ81Au¨# uf55¬#u¬t#tuouwt5²# © ª rQí 5¨kuf1kuUot5À¬P <math.h> tu5umÙ1ÏÙtoto³#1u float_t 5t double_t t65k5Àu5¨À5 Ïu5 float 5t double #ku¨1u¬u³#5t5AtoPuo«5o¬t¬³f1u 1u¬u515o float 5t double uÍr1ku¨bP Þ #)uÍ85«1uf FLT_EVAL_METHOD ´5f²##o float_t 5t double_t 5k long double Þ t5³ ª rQí Auf16Au¬P5Iot <float.h> tu5u)ÙuÙ1P1ku1A#u¨# «5A#¬t5À1uo³ot 5t#5P1kubbp¬Po#w«5o#wku¨1#tto u5Á1#5ot § 1#tÀ¬³f1 Ì th#«²#t5o##À¤5o1kuAuf16Au#mkb«utu²f³ ª rQí 111#woUb #IotuÙ1ub1o#tc6öu5²##fu²#ÏPbI5 µÞ #ÂuÍ85«1u#À5««16ut¬5o# «51ot long double o³#1o5uÍ8¬utÏÎb²#¤b «5À5tuÙtu FLT_EVAL_METHOD o²#´5fot1k## 5««um5Au5bt5²f³5¨¨«ot5 uÍroutuAu¨#k65¬u³o5öf¨1k## 5«Ô#h¬t ± 1Ï5t#w uÍr515¨1³1¨6t¬u «u5of5 5²#ubpo³#1 long double ¦ob double_t © hâ ¬thbotumt5tfot5««16ut¬5o#«¨ÀÔfu 5t#t³f«bUuÍ81ku¨¨#tuÍ8¬uPu1ku¨6bu tuotu³5A¬#ku «u«# ³¦¤ tuotb«u)1¨À11Á#5¬t § 1#tÀku#¬u © f5t «Ï¬#Aotuku¨16¬U#ÀuÍr1ku¨btU5¨ftu¬f5 5²#uUu5ku double o #tuwÀ¬PuoÏb1²f1kubufupotu³bt5²#uuÔfu1À ku#¬u ÂÌ t1¨8tu¬Pot double t#pot double_t o³f15Ñ²f#«1uo 1kboto5¬uÀ#b1 kutÀÍ8outu § ²#5ut5k 5k Ôf ª rQíA#fÏ¨#1o#t¬¨#«#Àot1k#²#u«öw5ou²#³ot uÍr51uU6otUuob²#1Àt#À5 ½ ª rQí uwbbp¬P log1p o1Pob f15w5P¬uuo #wÔ#wkuo³p¬tuÍ81ku¨¨6b 1.0 + x U5¨ftuo 5w5²#¦o#I5P³¬³f1 © 5Pot5If5 5²#1ü¬PA#1#Pb ½ 1#wo5²#ufu¡Ù1ut ª rQí1kbfw5o#)¨1ooth#1²#1Au¨#«t«²#u)totf«5t 15 ¬ t uofUbk¬Ù11VÚut 5 ¨1À5Àot#wkuÀ1#ö#5tÎ5f# #1²# § A#1t6tÑ °5tt##5w5touµ¬t5 double uÍ81k¨#tm5k°k#1tu #wkuo³o#1²f1ku¨6b di â t¨#1o#¬öt double_t o³#1h¬ 1uw¤b ot1o¬tb²#1ku##¨P# à #1²#³#¬uU5tÎ uÍroutuAu¨##uÍ8¬utu § ²#5üÎ¨³föu«¨f¬t5Àu¬Pm5öt 5 ¬tu¬ t²fµ#wkuo³Ñkbtu SÌ P#ku ßQ¨5³fmot5 µ5«¨1Ï5Ï5t³

d§d ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ ²#I1b¬6b«1kbu uÔ#P otu1Au¨#«bIotu1tu ³#P5w6¬tu¬b5P ot
FLT_EVAL_METHOD 5tÎPfk#t«uto5115 5«u¬um«5A#¨t#1Î#1 otÒt¤b «5ob ÒÌ to# t¯ow5f«utP¨P ÒbPUb¨²#1u«uto5¬#Ú â À#1 üfoÙ1tÎ¬PUb¬#tfot1k## 5««um«¨ÀÔ#P ¬P5 double uÍrAu¨#tU5³²#uf55¬uÎuÍ8¬utuÎ1Au¨#tfot5À¬Put¨tb²# § kbtt1kb²fu55öt5fb oto«5oto#tf5tot5ÀuÍ8¬utu 1kub«5³²#1ut¬u5Î5#kt¬ Ì tubkußQ ½ «#A#²##1 ¨#1o#¨1³¬¨1uo³ot5À5uÍ81k¨#²fA#1Pu#wkuo³¬ #1²#Au¨# â uÍ8¬uPu § ²#5u¨³föu«¨¬t6U«uku³kfkP5t#6t otkb1Pt1ku¨6b«bÏ²#1À1Pob o1t5ou³ ª rQí #uUPbÀ1kb 1#wo5²# 5³o¬P ¦ ± 5w³ 5ooU#Àot Þ #5¬P § ä b6t ª ± o¨fot #wÔft#«utÀ¬P5I1uÙuott5P#u¬²#h«5oot ª ¬5P5k¬ ¨11b ÀÁb5ot § 1#tofAub«utuot5À«16ut¬5o#tb¨³fö ¬ kbtt1kub«#uU1k#f fegetprec 5t fesetprec ¤1to#t¬ fÀ5tÎüÀ¬Pkb1Pt1ku¨#tf5P5bf#1Uo¬P fegetround 5t fesetround oto#tUot5ÀfuI5PüÀotkb1Ptkuob Ì t6 ku#««ut5o# 5ku«#fu²f¤bkotP5t#u uk«5h¬ot ª rQí 5o © ª b6tuto5³ ª rQí'511kb5o¨11b ¬P1# ¬5²#6¬³5«#t¨³föu« o ¬oukPI6t¬ufu)5 ot«uoS515²#ou¨1ff¬²foou 5³o¨11b ¬oukPIÁ#5¬6t § 1bPI5kt¬u¬Au ± 5 ª rQí «1uP¬5o#¨11u5 #include <math.h> #include <float.h> #if (FLT_EVAL_METHOD==2) #define PWR2 LDBL_MANT_DIG - (DBL_MANT_DIG/2) #elif ((FLT_EVAL_METHOD==1) || (FLT_EVAL_METHOD==0)) #define PWR2 DBL_MANT_DIG - (DBL_MANT_DIG/2) #else #error FLT_EVAL_METHOD unknown! #endif ... double x, xh, xl; double_t m; m = scalbn(1.0, PWR2) + 1.0; // 2**PWR2 + 1 xh = (m * x) - ((m * x) - x); xl = x - xh;

&Ú§ÓgÕ!#"$¦%'&)(10 2)34QÕ6$¦%8759A@B$¦CQÕ6@EDÕ7QÚ0 45FEGIHPC50 QSRUTV0 4QÕW5FE0gÓgÕ6@EC¨X`Y6ab0 @ECQÕ Rc%¦@ÕÖÚ2)$Õ6@E9 d§§¦ <inttypes.h> tu5uÙ16¬t5PuÙtuÒot#¨P¬ufupo³f1uÒot«u«uto5¬#
¨11b o¨bt5«u5#ktÑo¾otuÒ¿5um5tu¬Ù1ut³ÀÚo#ÒuÍr5«u int32_t )5tofu¥o³f1°uÍr5¬³Ñ»5ú²#o u int_fast16_t )ot°«1uut¬5o# r o5¬uIP¬ufu)o³f15Àu5¨ÀßQt²#o uf5t intmax_t ¬t u¨ÀP¬ufu o³#1S¨11b ¬u )â t5«5ftµ¨5¨tu«¤bÁ#5¬P § bt1¬³f1u5Úo# uÍr51u float53_t bt5«hÁb5ot § 1#tÀ¬³f1 ¬uÍr5¬³ Ö »²f 1kub²#1À1#¨²#³ um 5t#u float_fast24_t #1P5«ot «16ut¬5o# rU¤5öu¨À¬³f1 ¬5I65À´Q²#À1kubPf5t floatmax_t #1Ît5«ot u¨Àk5bP5²#³o5Io³f1¨11b ¬u hÌ t¤5¨À¬³f1u#1 5 b1uw#uÍroutu § ²#5ü³#¬uUo#utu 5¬ot¤5öu¨À1b¨²f #°1²pi¬bt6³¾o¾¬P°#töw5P1ot5ot°f561u)#P5«u5w5²fu)«¨t# 511u5m¬t5tf5Uku¨ÀbpAuföum¨16t hÌ tuÍ85 6¬¬³f1u b 5¨b«uwbuÍroutu § ²f5ü³#¬u«ouÀ¬tk#1tPAu¨# «#ok#1tÎ¬ot¨1uÙ1uAu¨#tf5 P umw5tf¨²pi¡uÀ¬ot ¨5SbPöw5t UÞ t56³ double_t #1t5«°u¬³f1 o²##¬ot°1k¨# 5t 5tfbpot¸ ±±± Õ Ö #1²fo#w«5of1k#fP¬ À#1²fuf55¬# Ì ##uotu outA#tP¬515w5«uoum«5k#Ut5u5bktf³f¨1 ¨tu« bku5³¨11b I561Ù1#1¬#tu¨w²#uÎ5²##f5tÎ5 1kbfw5«uwo6t5ou5¨6³5P1t5²#f1#1¨³otÁb5ot § 1#t ü5t¬Uotum1kbfw5Ukuf1k ë 1¨À5tf15f¨11b Ào#)uÍ8¬uPu1ku¨6b²#¨b«5ou 'i â ¨P# à #1²#S¨³föu«¨8o#1#PotSÙ1#1¬#tÂ¬u«5²##fS#tu85t«¬tuk ÒPhtuuoh¬oukut¬5oo5t5tuÍ85 oÑo³#1u Ò± Í8outu § ²#5u¨³föu«¨ t uo81bS¡ÙPt#u5Ú¬tu³¾¨11b 1tu¬Pu1kÏb1²fS1ku¨6bPb 11kuÍ8¬utuAu¨#Ñb«1¬5o#Ñ5Â¬Ùuto³¯5Ò°«6Íro1kb1¬P br5P ¬oukP11kbfw5)5o#¥¬oukP1«Í8¬k ë bku#fu¬`¬t°PbS# tuo 1uÍroutu1ku#¨Pb16t#I²fuoÀob1u w6¬u ¨ t5A#oou o1¬uÎ²#³²#utt«5 Ô#¦o5tÎ¨#«u¬uUob6b¬ ftÀ²#³t1«u 55t5³fö © oku#5kÁb5ot § 1#tt5wot«uoÂ5 · ttukut¬6³htuÍ85oºS5totuAu¤bkUtu¬Pu uü fPPbm515²fbpot1kuo5²#o³bptou#um5woto ¸¤töu5fot Pb«1À²#1kuütouo1k## 5««u ¨f5Potu³ 1kuf1k5tf15fu 515²f#ÀuÍ81Au¨totumüuo# Àt68v'0E7u3Â4F68 Ì thobkufbPku«5 Ô#5kt#IP¬utuÎ¬15w5fuÍroutu § ²#5ü³#¬u ²#IouÍr1#üfu 51o55u¨fotÙwÀ²#ut¬P5I561¸ ±±± Õ Ö ³#¬u««¨ uuuP¬5fk1oÂ¤b¥otS¨5S1A#fw5 I °t5fS¤b1üb¡oukutu ²#u uuuÍ8¬uPu § ²#5u¨³föu«5tt#6 à #1²#h¨³fö¨f²f1À¬Pk5k

d§FH ×)452)$¦%¦@E9ÔÓ F¨(10 2)34QÕÖÓgÕ6@B0 CÒØp45@EG$ owotuÒÏ¡oukutc5«#t¨³föu« ot«5#Ïotuüo5«u SÞ bÂuÍ85«1u ¨#«tf
à #1²f¨³föu«1A#f6¨P#6tökob¬1o1³ t1²#uwm5t5uotk o·i¡11#t°Ù1P5A#1t6t UÌ P)#1uw5obP`5u ze#iwjwvtuef{rloqrg5{ Ý ¤pv5v f551öt5«1k## 5oA#1¬ouAutÀku¨¬ 5k#¨¡¤uAutÀt#6 à #1²#h¨³fö¨f5PfÔfuÍroutu1ku#tfÀ5 ufu51¬P¨5«h1A#fw5 ¬1kb¬oukutÀAu1oU#ot5«¨³föu u1utP# tu¬Pu)5t tuÀUü ¦ ½ o1u«16¬1³ § 5Î55¨ o#1ot1¬ot1k#u¨b Ì tu#kut5f5ot#1fÀ5²#1ut# § 1#wo5²# 5³¯¬1uw¤b «¬UAu¨#Ñ«16¬165ob ¬Pbtottuuo# ¨16¬ot © ± fu¬Pb#ot°¸ ±±± ¬5t5k r5to15¬µ¨5b 5obt Àtufu ¬Puu¨#to#w«rÚuottouwÏ5¬1k#1IuuAuÎ¬ · uöt5o#tº²#u³fbP¬P1üpr#t¬k#1ot5À ut#1foPbÎÀÍ85o³ 5totÙt511 Uk#1tub k¬6³oÙ1ÀoUtfbm#1²#61Au¨# uöt5o# Ì thu¬P5I¸ ±±± Õ Ö 1ku¨ ²#u1kuü³otku¨1Àfu1kbfw5 «¨ uu)t#tu¬Puu¨511u5t ë 5t³1A#fw5«uwÔ#h¬²#uu¬P5Iotu³ 51tu ö5tot²#uP5f#p#1°1kbfw5 5t1k#fot5P #wÔ¯#wkuo³ ot#11kuoukPSo¯¬PS#«1u¥ot51#«1u)1#otSb11ouot51w1t) ¸¤«5t³ 5³f¨f¨11#wotot²#uuÀ b o t6uf#51obm¬PuÏftuwU# #«1¬u³#¬u«Â5P1k## 5««P¯5tf15fu uto# ¬1P5¬u³ tuP#«u oÁ#5¬P § 1bPI5 ot«uofotfb5Uwo156³1b¨²f¬5tuf hÌ t 51ot#wÒ#Pot¸ ±±± ö5t5kÒÔft ot5¬r5P¬tu³¯ rP5¬ou«1P¬5tu ½ U¯Au1ofu¨1otu5 ³1tuw¨5#to#w5to¦o«#¨À# © ¬t¸ ±±± Õ Ö ö5t5kotkb1ft#1P¬tb«1¬ut1¬ ³r1A## 5« Ò#P1#wo5²#¨#o 5k «¨À#tot1o#1 ot1kuo5²#Á#5¬P § btÀ5 ¬P«u¬6 ¸¤À1k## 5««u U5kouÍ81#À¬thou5¬Aubp¸ ±±± Õ Ö 5fotu³ tuu 1kbfw5«t5tf15fuUot5À«5ÔfÁb5ot § 1#tÀ5 ¬tu¬°1kuo5²# hª rQí «1k#fu1kÏo5²#o³¬¨#«Ïfku5À¬tuÍ81utbpAufwP 1kbfw5«uwo wo«¬61fu ¨#tbp¬P6)1k## 5«¨bthobmu5 FLT_EVAL_METHOD I Potumo1o1k5tf15fu 1t#btöu5o5 1kbfw5«uwo wotf1kbfw5 o³#P¬5Í¾ot5À1t5«²f#b¨³ uÍrAuüUotuÍ8¬uPIo tÀu1ut#¸ ±±± Õ Ö u«5toku«5t¬²# üu h± Ír6ötuÍroutu § ²f5ü³#¬u«otk5ouot5À1kb1uÀ²#³o1¬6t1 o5¨1¬P5Iotb1um5tott5k 5k5Ôft ²#uo¬umot5ot 1kbfw5«ut °#«1¬5o#Ñ¨t#1«²#1uw¤b «u#Ñ°ffu¨³föu ÂÌ P5 5¨1«1o#¬Pü#tÎo55³ÀÚuot51 5³kuf1ku¯#«11ouku¨1 u1utt#I#ot5tP¬t5À1k#1uÀ²#1À#t³#ot#t61¨#t¬P5 6f²fS okb o`5t«#otS1k## 5««u¬`¬PSb«u¬85totSP5A 5k 5À²#u¨À#t³¬P1kbfw5«u)5Ôft t5À¬t#¨h#t¨#tU5³²#

What Every Computer Scientist Should Know Abaou...

What Every Computer Scientist Should Know Abaout Floating-Point Arithmetic

More Decks by Paulo Bordoni

Other Decks in Education

Featured

Transcript