The PoleCondition Martin J. Gander Trans0arent BCs
Transcription
The PoleCondition Martin J. Gander Trans0arent BCs
QSRUTWVYX[ZY\]^\-VYX_Za`cbedfVYg-`cb hjiklnm4oqpsrutvisowsxk yuz|{^}~7 z { yz}u~< < C -< ci|km ¡£¢pios¤ik¥§¦|¨|¨|© pª^m-ol«aªsk¬«®m l°¯²±´³¯sm4µ·¶ ³¯^i ¸ wc¹ x ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P RaXub R ` X]WVaX[Z´R TWV X[ZY\4]\%VYXub xW³£ªo¡Gm wxk l°¯xjµ ªwx¹kªs¹ xµ m-o "!# ¨%$'& )( $ ¨ * ¨ )( $'& * ¨ « m l°¯ ³£ªµ+ i³Gln¹ ¥q¡£¤,- ªklx£w m4o/. 021 3 ¨%$54 6!# ¨%$'& ¢iow ª¤sowx£w il m4o8u7 osm lC¥!r 9 & ,-ªskl ª@? :<; :>= ¡£¤ ¨ 4 ( A%oqªskwsxklªj¡@ª¹ Bxl°¯m ¡Ck ªs¹ xµ·ªojiq³£ªµ+s¤lxk ¢cln¯x + ox£x£w¡lªD x l°k¤so³!ilx£w m4o ( iklnm 7 ³m i¹ ª¤owi|k¥§³£ªow m lm ªo ox£x£w¡flªE xqm4µF ª¡@x£w isow (D 4|r ³£ªµ s¤lilnm ªois¹^wsªµ im-o ¢isow il (E io ¨ ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P TWVYXub] i¡@x£w c]\%VYX RaXub R ` X]VaX[Z´R TWVYX[ZY\4]\%V ªoqln¯xwsx£³£ªµ+ ª¡Gm lm ªoqª@? . 021 9 & isow B V :; ¨ /. 021 - kªs¹ xµ l°¯xWx ¤m i¹ xol³£ª¤ ¹ x£w ¨ 0 0 « m l°¯ := ( 4 ¡ 0 0 ªo ¯ªµ®ªxox£ª¤¡ ³£ªow m lm ªo¡®il ¨ D: ; ªoD: ; m-o/. 021 ªoD: = ªoD: = m-o iow 9 &ar m-o 9 ¨ Xub ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P []` \%V d Vag-` V VYg ]\%V X b ¯xx lxkm ªk kªs¹ xµ ¡®ik xqm4owx<xowxolYª@?Sl°¯xw^il<i iow ³!io x§kx!iw m4¹ ¥²¡@ª¹ Bx£w ¤¡Gm4o ª¤skm xk ¡@xkm xG¡m4o 9 ?ªkx iµ+s¹ xWªo «ax ¯i8B x ¨ isoqªkwcm4oik¥jw m [xkxolm+is¹^xc¤ilnm ªo¢ «®¯ª¡@x¡@ª¹¤lm ªo m¡ )( $ ! " $#&% ('/ " $#&% ¶m4o³£x ox£x£ws¡flª ¡<li°¥ ª¤sowsx£w?Cªsk (*)+-, ¢ «´xj¯i8Bx ¨jisow ¤¡Gm4o ln¯x/.Ym4km ³¯s¹ xGl w^il<i il (D 4 ªl<im-o )( $ 0 4<$ " $# %21 2= 3 ¯m ¡ m4µFs ¹m xG¡ 8 - %)( $ 5 4 76 = )( $ 5 4 76 = isow ¡Gm-µWm4¹ ik¹ ¥²ªo²l°¯x ¹ x2C? l ª¤owi|k¥ : ; ¢«´x ª lism4o )( $ 5 4 76 ; 8 9 )( $ 5 4 76 ; «´x ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P \%V d Vag-` ¡Gm-o jl°¯xG¡@x¡@ª¹4¤lnm ªo¡?ªkln¯xqm4olxkm ªk k ªs¹ xµ²¢c«´x xGl m4o/. 01 9 - ªoD: ; 0 = ' 8 0 = ' 8 9 - ªoD: = g-VSb \2X T ][`CX]` 8B 8? m-o xoxk2is¹m ¡ iq³£ªo8Bª¹¤lm ªo = , 9 > $ - '8 - «®¯xk xln¯x§m4o xk¡@xWlnk io Cªskµ « m l°¯ 8 ¥q³£ªo¡<lnk¤³Glm ªos¢ ³£ª^m4o³m wsxG¡ « m ln¯ m4o/. 021¢ iow ln¯x ³£ªow m lm ªo¡ ªlism-ox£w i|kxW³!i¹¹ x£w l°k io¡ i|kxol ª¤sowik¥ ³£ªow m lm ªo¡ u¡ r ¯x°¥ kx ¤m4k x i oªo@¹ ª³!i¹^³£ªo5 B ª¹4¤lnm ªo ª¤sow^ik¥§³£ªow m lm ªor E'- 9 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P R ª ³£ªo¡<l°k¤³GlYl°k io¡ i|kxol u¡£¢«ax ox£x£w lª 4|r ¶ ¹m lln¯xkªs¹ xµ m-olª m4olxkm ªkvisow x lxkm ªkvªoxG¡ ¦r ¶ ª¹ B x l°¯xWx lxkm ªk k ªs¹ xµ®¡£¢ m-o l°¯xWx isµFs¹ xn¥ !#"$ &% '( )* +,+-/.10!2435* 56 '( 5789 : <;=45>?! A@B)C9DE3CA 79 F36?56 GH9 I r ¡@x ln¯xx lxkm ªsk¡@ª¹4¤lnm ªo lª ª lism4o²ln¯x ª¤sowik¥ ³£ªowcm lnm ªo Cs ª k l°¯xqm-olxkm ªskªox¢ m4o²ln¯xx |iµ ¹ x °¥ F? + J:7,* F.B 7, 86LKM*?"N&)HCOQPSRUT4VCW :,X )*YG *Z> ![" 2B ± l ln¯xWw m ¡@³kxGlxj¹ x x¹¢ io + ³£ªµ s¹ xµ®xolµ ilnkm ³£xG¡ m wsxolnm ³!i¹ k ª³£x£w ¤skx ¹ x!iw¡flª ¶ ³¯¤sk ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P RaXub R ` X] W Va X[Z´R TWVYX[ZY\]^\-VYXub [x£³!i¤¡@xWª@?vln¯xm4koªo@¹ ª³!i¹oiln¤kx¢ ªoxWª@?2lxoqi,,k ª m-µ ilxG¡ u¡ iow ª lism-o¡ ± ¡@ªk s m4o ª¤sow^ik¥ uªow m lm ªo¡ ± u¡ ¥ ª¹ ¥oªµWm i¹sªskk2ilnm ªoi¹i,- kª ^m4µ ilnm ªo¡ ª@v? ln¯x¡¥µFª¹ ° o ¤m ¡<l°¢ hjiwi¢ais¹ xkos¢ ak¤sox!is¤ .Ymuh§xo!i¢ vil<i8? ¢ pi-¯xGl°¢ ¶£x2?Clx¹4¢ ¶u¯m ili r£r£r ¥ji,, k ª m-µ ilnm ªo§ª@S? l°¯xW³£ªo8B ª¹4¤lnm ªo ¬xkox¹ aism4kxk¢ ° ^¤-s m ³¯s¢¶ ³¯s¹m ³¯lx¢ut kx£xo sikw ¢¶l°k ism4o¢ ¶c³¯^i¸ wc¹ x r!r£r 7 x¹ w ¢ i¡£¬ni¬ª@B isow ¤¡Gm-o ¤iwck il°¤skx§k ¤s¹ xG¡ hjiG¥ ª ª@B ¢rr!r ª ³£ª¤-s ¹ x io m-o³£ªµWm-o 7 x¹ w . 0 ? kªµ 8 - 9 . 0 ªoD: = . 0 = ' «®¯m ³¯ ° ¥ ¹m4ox!ikm lC¥ m ¡Yx ¤m B i¹ xollª 8 , . 0 = ' 8 . 0 ªo : = = ' b \CX ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P X g]` ag-R]^\-VYX R]^\ `V < = = >? J B %OZQLI TJ J B C8DE GF6DH D 5 $[O 9 NVI N9OH 9CQ\D]C F6DX9CE ( &IOE &IQ 6FTJ\IDH B OE kCEOH : . 021 ]aml 5 B % > 9 > 9 J B 6H 6QROE(F6DHF SC(J GH X ;DTQUJ B 6QLIFVOENVDE JLIDHYD $J B C B N3J B NVIFVOWENVDE &J\IDHMIH O]NhH&DiC8DE SNhIHjJ B TWVYX[ZY\]^\-VYX O%PO&JLIDH^C_QLD`E 6X9N6a9b1JcN%O 3dO]CEeOF fJ\QROHZN ;DTQ\XgD GQRO dfVYg-` "!$#% &(')* ,+-/.1032425&6 8709 &IKJLIDHMI NO9P 6DTQ TJLIFVOWE(OH*OE J B 9 > @-A8B q] [ ` iEDW[ 6Q O O&J :;7 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P V o «´x V V Za` g®d 8? ¯iw Vag-` *ª¤skm xk l°k io¡ ?ªk µ i Clxki ¨ -¹ ¥sm-o i |i,s¹ i³£x l°k io¡ ?ªkµ m4o²ln¯x§k iw m+is¹4¢um-r+x|r jw m-kx£³Glm ªos¢ $ 0 4<$ 4<$ ¨ ± «ax ª l<im-o N Sxo³£x ¯m ¡f¡@ª¹¤lm ªo LN $ LN "N 0 ¯i¡ ª¹ xG¡ il 4<$ 4<$ 8 - ¢ iow i iklnm i¹ N N N ?k i³Glnm ªoqx %io¡Gm ªo m B xG¡ 4<$ 1 = # 3 4<$ 1 = # 3 " $# % " $# % $ ¦ 8 ¦ 8 LN LN LN ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P b \CX ][` dfVYg4` TWVYX[ZY\4]\%VYX ª¹ xW³£ªow m lnm ªoq¡<l<ilxG¡Yln¯ill°¯m ¡f¡@ª¹¤lnm ªo 4<$ 1 = # 3 4<$ 1 = # 3 " $# % " $# % $ ¦ 8 ¦ 8 ³!iso oªl ¯i8 B xWiso¥ ª¹ xG¡ m4o l°¯x km ¯l®¯is¹ v? ª@v? l°¯xW³£ªµFs ¹ x s¹ iox¢ @ ¨^¢iow ¡Gm-o³£x 8 ¨^¢cl°¯x ¡@x£³£ªow lxkµ ³!iso oªlC x kxG¡@xol°¢c« ¯m ³¯ µ®x!io¡ln¯il 0 4<$ 4<$ 8 ¨ «®¯m ³¯ m ¡fxc¤m B i¹ xollª 4<$ 0 4<$ 8 i¡Y«´xC? ª¤sow x2? ªkx²³¯ª¡Gm-o l°¯x ª¤owsx£w ¡@ª¹¤lm ªo m-o l°¯x§km ¯l Sª!« ln¯x LN LN LN ª¤lxkwsªµ ism4or LN ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P V \ V ] [` dfVYg-` TWVYX[ZY\]^\-VYX A%oqªskwsxklªj¡ilnm ¡ ?2¥ l°¯xª¹ x ³£ªow m lm ªo§i,,k ª m-µ ilx¹ ¥£¢[m l®m ¡ oiln¤k i¹lª x %isow l°¯xWwxG¡Gm4kx£w ioi¹ ¥lm ³ ? ¤o³Glnm ªo m-o l°¯x§km ¯l ¯i¹ ?s¹ isoxqm4olª i ª£«´xkS¡@xkm xG¡!r ª wsª§¡@ª¢ «´x u 7 k¡<l®µ i,q ln¯x km ¯l¯i¹ ? ¹ iox m4olª ln¯x ¤om l w m ¡£¬ ¤¡Gm-o ²ln¯xqh ª ¸ s m4¤¡ l°k io¡ ? ªkµ 4 C / ;; ; 4 Ru]\%V Xub N N N N N N N N N N N N ; N isow iow $ x N m4olª i N ª!«axk¡@xkm xG¡ -i ª¤l C N ¨r ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P YZ R]`Z [` X iklm ³¤s¹ i|k xj¤¡@x i ¦|¨|¨ b|b ª£«´xkS¡@xkm xG¡ ¶ ³¯^i¸ wc¹ x ? ±fo¡il Cªsk l°¯x 0 0 ¦ 4<; $ 06 ; $ ¡ilnm ¡ 7 xG¡i¹kx!iw¥²l°¯xj¹m4µWm l/ kª xklC¥ ª@? $ 3 / 4 4 N N N x£³!is¤¡@xl°¯xo i, ¹ i³£xl°k io¡ ?ªkµ®¡ O N N N ¹m4µ C? N $ 4<$ LN kx£³!i¹¹ln¯il ; == 4 = ; / 4 $ <4 $ i¡ªoxW³!io ¡@x kªµ ¹m4µ N N N N N N ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P _\CX ` ]` V§] [ ` R]\%VYX TWV` \%` X]b A%o¡@xklnm4o ln¯sm ¡ ª!«axk¡@xkm xG¡®±fo¡il m-olª§ln¯xWxc¤ilm ªo $ 0 4<$ 4<$ ¨ ¡ilnm ¡ 7 x£wE° ¥ $ iow ³£ªµ+i|km4o ³£ª^x§³m xol¡ ª@? ª£«´xk¡ @ª ? ¢ «´x u7 ow N LN "N N N P ,.- * W V R 0 ( /21 .,* - W?V 3 &+ *P & 9 ! 6.,* - W[V 3 RAT == = 9 >CB G >CBH RAT !"P) #%$ '& ( 4 5678 (59 ( :;' !" #%$ ( < '& P == = >?8@8 > (A9 >CB D >CBE ( < F > ?Cªsk ¦%$ r k¤so³!ilm-o jl°¯m ¡k x£³¤skkxo³£x k x¹ ilm ªo ?ªk . il ¢iow ¡@xGl7lnm4o . ¨?Cªsk «´xWªl<ism4oqi ¹m4ox!i|k ¡@ªµ®x ¡¥¡<lxµ ? ªkSln¯x§¤sos¬oª£«®o .C¢isow ioqi,-kª5 m-µ ilx kx¹ ilnm ªo xGlC«ax£xo 4<$ iow 0 4<$ r I O I b O O I b ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P ag4` SR ag-` \ ªsk ; b ln¯x N ¨^¢i¹¹^³£ªxj³m xol¡ lxkµ O k xµ ism4o¡ . ? ¨^¢iow ªos¹ ¥ql°¯x§k x¹ ilm ªo kªµ ; 4<$ 0 4<$ ¦ ; ¦ ¦ ; ism4o¡ lªj³¯ªª¡@x ln¯xx %io¡Gm ªoª^m-ol ; N A l®kxµ N N ¨ N N N ; N @ ¯xW¡ x£³Glnk¤µ «´xl°k¥§lª i,-kª5 m-µ ilxjm ¡ il C 8 ¢ $ $ r6A ?«´xW³¯ªª¡@x ; ¢ «´x xGl 4<$ ¨ 0 4<$ ¦ ª¤sk ?iµWm-¹m i|k[¡@x£³£ªow ªkwxk i°¥^¹ ªkv³£ªowcm lnm ªo ¦r N N ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P RagT Rub` .Ym ¡@³kxGlnm !ilnm ªo ª@? . m-o 9 Bikm+i,s ¹ xG¡ 9 V ` O ªo²l°¯x¡iµ x km w % % i¡ l°¯xjm4olxkm ªk ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P ( 4 ¨ ; ; " 0 0 0 - = ¦ ; @ ; = ¦ ; 5 " 0 ¨ rr r , . ¦ . = . @ ; . ¦ . = . ¨ q`X[` O O O O 3N MO "N O 3N O O O O MO S` Ru]V Sx£x£w lª \CXe][` R´gfT §` X[` l°¯x xoxk i¹kx£³¤skk xo³£x ¤sowsxk¡<l<iow Rub` k x¹ ilm ªo - . ¦ . = . ; . ¦ . = . ¨ «®¯m ³¯ m ¡fxc¤m B i¹ xollª ; . = ¦ 7; . 7; . = O N O O N O O N O O N 9 9 ;; ªk «®m l°¯ N ` «ax 8B "N + ¯i xl°¯x ¡Gm-µ ¹ x§kx£³¤k¡Gm ªo . . = ¢ «®m4¹4¹x lnk ¤so³!ilx£w ?ªk¡@ªµ®x = ¦ ¨ O «®¯m ³¯ . = ¦ `*O O O I `*O O b ¨ O ¨ ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P X]` ¯x ` ]|R]\%VYX V§] [ ` @? A%o¡@xklnm4o O A%o¡@xklnm4o = ¦ `*O °¥ `*O = `g-Ru]\%V X B O O O = ¦ ` ¦ `*O 4 = ¹ i¡<lxc¤ilm ªo m4olª§l°¯xl°¯m-kw ¨ ¦4 kx¹ ilnm ªo m xG¡ O ` = ¹ i¡<lxc¤ilm ªos¢ «´xWª lism4o O O isow ¦ m4olª§l°¯x¡@x£³£ªow ` xc¤ilnm ªoqª aln¯x§kx£³¤skk xo³£x ¹ i¡<ln¢l°k¤so³!ilx£w O `X ` ¦ ` 4 = = ¦4 O O ` = ¨ ¹ x!iws¡flª % O ?k2i³Glnm ªo§x %io¡Gm ªo^r m4ow ¤³Glnm ªo¢«ax²ª l<ism4o²iq³£ªolnm4o¤x£w ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P R ]^\-VYX RaXub \%V X ± l ln¯xx m4oom-o @ ª ?aln¯x§kx£³¤sk¡Gm ªos¢ «´xj¯i8Bx 4<$ 0 4<$ ; ¨ ¦ ; ¦ ; = ¦ 4<; $ isow ¤¡Gm4o = ¦ 4 = ; «ax ª l<im-o ? kªµUln¯x¡@x£³£ªow kx¹ ilnm ªo % ; ¦ 4<; $ ¦ + 4 = sRub RUTWVYX]^\2X [`Z ` ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P O "N N `*O O N O O ` O N «®¯m ³¯ « ¯xo -,kª ^m4µ 4 ; i N m4o¡@xklx£w ilnm ªo ¦ ` ¦ 7 m4olª§l°¯x uk¡<l 4 ¦ + ` ` = % ªox¢¹ x!iw¡flª§ln¯x J 4<$ - 0 4<$ ¨ "N X Rag ]^\ Rag `cb ag]b b % ( J B 3F6DHJLIH 4 I N3JLO D]C b ; N 7 ¦ ` 6HJ\DMJ B 6QRO&JLDZQ I N3J B ;QROFTJLIDH ¦ GEIXiIKJ $J B = ` J D]C 6HMIH J B 3F6DHJLIH 6HJ B F6DIH&F6I '( 6H &OWHJLE D N $J B 3EIXiIKJLIH6P 8 4<$ 0 4<$ ! > 5 > %'& )(*&+!,& -/. 1 0 > 9 > ;QROFTJLIDH ( hO]C&C_QLD IX3O&JLIDH SNf[/IkJ B $# SC "N 6QRO&J\DTQ % ( J B OHZNVIDH 0 4<$ < 4 $ - ¨ + "! 9 > ; ! 4 ¦ WC J B O6CC_QLD _IX3O&J\IDHjD 3IH /J B WC [ OHZNVIDHYI N 3D`JROIHYD 6Q N iJLQLOWH N Xi` DE(D N DZQLX /J B J\Q *H&FVO&J J /J B D J $J B MO&JN]J SC D]C b D]C b 6QLO*J\DTQ 6QRO&J\DTQ a b O ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P ]^\ _\ `c] [VZvb ±fo²ªlm-µWm £x£w ¶ ³¯!« ik jµ®xGl°¯ª^wj«®m l°¯qi²lC«´ª ¡£¤-wªµ ism4o wsx£³£ªµ+ª¡Gm lm ªoqiow§l°k io¡£µ m ¡<¡Gm ªoW³£ªow m lm ªo¡C i¡@x£wjªoWl°¯xª¹ x ³£ªow m lm ªo ¯i¡fln¯x ?Cª¹4¹ ª£«®m4o kª xklnm xG¡ TWVYXub` [`X ` b nV b IH - ( J B SC WC 6H 1OFTJ\DTQ 9 > OHZNVIDHfC8DIH_JN &OWHJLE O]N B PD D /J B SNiJLD ; 5 $J B 6H [/IkJ B 6E(b ! J B R - D &J D 3F6DH ! 5 5 6Q\EeO]C 6QKP 6H&F I N 4 B ! 5 $ 4 5 9 > ¶ ªq«®¯il«´xol «®k ªo ¯xkx¢ « ¯!¥ m ¡fln¯xkxjoª m4µFk ª@Bxµ®xol b 5 ! iJLQ H&FVO&J\IDHjE 6Q\D ` Z J B 3D 6QLEeO]C I N B /J B 6HjJ B 3F6DH 6QKP B IH SC 6H &OWHJLE D /J B iJLQ H&FVO&J\IDHjE 6E(b a 6HF OFTJLDZQ I N ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P RaXub \%VYX dfV \CX] \4b SV ]R´X] A ?«´x ³¯ªª¡@xWln¯xWx %iso¡Gm ªo ª^m4ol ; ¡£¤³¯qln¯il l°¯x¡ x£³Gl°k¤sµ i5?Clxkl°¯x h ª ¸ sm4¤¡ l°k io¡ ?ªkµUm ¡³£xolxkx£w i,ª¤l £xkª¢«ax ªl<ism4o N ' + - 5 ! 5 IkJ B D J D 6QLEeO]C J B GF6DH 6Q P [ B 1OFTJ\DTQ O]N PD B 6Q GO 6Q P Sb 6H&F 1OFTJ\DTQ O B %a b /J B GD 6Q\EeO]C WOE N B /J B 6HJ B I N O B SNiJLD 4 ¦ % / 4 ¦ % $ & $ 4 ! 5 4 ¦ % 4 ¦ F6DH 6H&F 6QLD I N ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P ubV v\2X VaX[Z´R ¨ 5$ 4 /!# ¨%$54 « m l°¯ l°k¤o³!ilm ªoqil 3 & 9 % # % " 1 2= 3 ¢ ¨^¢ ( ; 4 o l°¯x wsªµ ism-o (D b ¨ ¨ ¢ TWVYX[ZY\]^\-VYX NVIH 0.6 0.6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P u 1 0.8 u 1 0.8 ¢ 0.4 0.4 0.2 0.2 0 1 0 1 0.8 1 0.8 1 0.6 0.6 0.8 0.4 0.2 0 0.4 0.6 0.2 0 y x 0.4 −4 x 10 20 15 10 error y 0.8 0.4 0.6 5 0 −5 1 0.8 1 0.6 0.8 0.4 0.6 x V ubV v\CX jT Q ` ` g V§] [ ` −1 −1 10 10 error order 2 error order 2 −2 −2 10 error error 10 −3 10 −4 −4 10 10 −5 10 −3 10 0 10 b ; 4 1 10 3 10 ¨^¢ −5 2 10 10 0 10 N N b −1 ; 4 1 2 10 10 error order 2 error order 2 −2 −2 10 error 10 error 10 N N −1 10 −3 10 −4 −3 10 −4 10 10 −5 10 3 10 ¦^¢ 0 10 b 1 10 ¨^¢ N N ; −5 2 3 10 10 ¦ 10 0 10 b 1 10 ¦^¢ N N ; 2 10 I 3 10 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P R T T 9 \CX 6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.2 5 error u 4 3 2 0 1 0 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T 9 9 \CX 6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.08 5 0.06 4 error u 0.04 3 0.02 2 0 1 0 −0.02 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.03 0.025 5 0.02 0.015 error u 4 3 0.01 0.005 2 0 1 −0.005 0 −0.01 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P −3 x 10 6 14 12 5 10 4 8 error u 6 3 2 4 2 0 1 −2 0 −4 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P −3 x 10 6 20 5 15 4 error u 10 3 5 2 0 1 0 −5 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.04 0.035 5 0.03 4 0.025 error u 0.02 3 2 0.015 0.01 0.005 1 0 0 −0.005 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 6 0.06 5 0.05 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.04 4 error u 0.03 3 2 0.02 0.01 1 0 0 −0.01 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 6 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.08 0.07 5 0.06 4 0.05 error u 0.04 3 2 0.03 0.02 0.01 1 0 0 −0.01 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T T \CX 9 6 0.12 5 0.1 ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0.08 4 error u 0.06 3 2 0.04 0.02 1 0 0 −0.02 1 1 1 0.95 0.5 1 0.95 0.5 0.9 0.9 0.85 y 0 0.8 0.85 x y 0 0.8 x R T \CX V Q` ` g4b q]]Rv\CX[` Z ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P 0 10 −1 10 −2 10 0 2 4 6 8 10 12 14 16 18 TWVYX g ub \%V Xub T ¯xª¹ xW³£ªow m lm ªo B , wsxkm xWi ¡@ªk sm4o ª¤sowik¥q³£ªowcm lnm ªo¡ ¯x§kxG¡£¤s¹ lm-o ª xk2ilªk¡®ikx ciwxqi,-kª ^m4µ ilnm ªo¡ª@?aln¯x ¡¥µF ª¹ª@v? ln¯x .al ª xk ilªsk ªki ciwx i,- kª5 m-µ ilm ªo²ª@?vªkwxk ¦A ¦^¢¦A ¢ªox§¯i¡Ylª iwsw u A ¤sos¬oª£«®o¡6? ªkSx!i³¯ ª¤sowik¥ ª^m4ol ¯sm ¡ox°« i,- kª5 m4µ ilm ªo ¹ x!iws¡flª oªo ª@B xk¹ i,, m4o ª lnm4µ m £x£w ¶ ³¯!« ik µ®xGl°¯ª^w¡Y«®¯ª¡@xW³£ªo5B xk xo³£x wsx< xow¡ ik m l°k ikm4¹ ¥²«´x!is¬¹ ¥§ªoqln¯x§µ®xG¡£¯ ik2isµ xGlxk ` B m ¡ isoqis¹ lxk oilm x lª ]b ³£ªµ+¹ xGlnm ªo§ª@?aln¯xWisoi¹ ¥¡Gm ¡ x lxo¡Gm B xjo¤sµ xkm ³!i¹slxG¡<lnm4o X] ` V ! "#%$& ' ( )*+' , #-$ ./-/-0 , *+1' , #-$ 2 3( 14$& /1( 5$'762" ./-/( #84, 91' , #%$& .$140 :+& , & ; /' , 9<, = 5+>@?%*ABC1( = D@E FG H% I2J FG ?, 9</40 5K48149</-0 5 K( ( #4(0 5+L+50 M 1( :, $NO P
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