On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs

(1)

HAL Id: inria-00073949

https://hal.inria.fr/inria-00073949

Submitted on 24 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs

Anne Aubry, Philippe Chartier

To cite this version:

Anne Aubry, Philippe Chartier. On Improving the Convergence of Radau IIA Methods Applied to Index 2 DAEs. [Research Report] RR-2744, INRIA. 1995. �inria-00073949�

(2)

ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

On improving the convergence of Radau IIA methods applied to index 2 DAEs

A. Aubry , P. Chartier

N˚ 2744

De´cembre 1995

PROGRAMME 6

(3)

(4)

$6798:+<;=/

>@?A>CBADFEHGJI*K?L%M*NOE3PHQSRTEVU

WYX[Z]\OX3^`__abc d^TegfHhieYj[flkmanpoqksrFtOhua]vA_ZuwxaHeskyj[^`oqkmZ]nzao4emZ]\`kyflkmaHe{nihu_JxaX|kgtOhua

W}X[Z]~qao4%ey^]wOkmn

%^`u,Z]X[owiaCX[afluaX3f[uan.p]CczxafHa_uXlaO]c`V^`\]aj

V`]4ikyjV^VaXuX[ajqanio3j^znuajkm_iemazo[af[unikytOhuao[Zzkm_uX[Z`]ao[ua2Z]X3wiaXVaV^OkmZ]huXZ` ¡¢hunu\]a£

¤'huolo3^_ao[uZuwuj4%uan¥^`uieskmaw¦olZkmnVwiaH§¨%©j«ª¬o%kyjo[uan¦jquZ`%nuZ`0o[ikyj'f^`nVakmnVfHZ]X[VZOX3^`o[aw®k¯nio[Z@^

_Z]X[a'aH°®flkmanpo']aX3jkmZ]nzZ` Yo[ua.fHZwia4%±'wia]aHemZ]Vawzi²®©ªV³'^kmX[aX¡^nVwz´ªiµ=^`nunuaXª

¶¸·p¹º¼»½AT¾]0wOks¿*aX[anioqky^em£À^em\]auX3^kyf2jq²ujqo[a_@j'Z` kmnVwiaH§ÁÂv*4hunu\]a£À¤'huo[o3^¸_ao[uZuwujv^]wu^`h¥¬q¬ÀÃ_aoluZwujv

XlZpZ]olawo[XlaajvVjkm_iesks Ä²Okmnu\¨^]jljqhu_uoqkmZ]nVjv,fHZ]_VZijkmoqkmZ]n*ª

ÅSÆÇ

È|É¼ÊË

Ç

ÈÌÍsÉ¼ÎÄÏÐ

ÑÀÒÔÓÕAÒSÖ×ÄÕAØqÙÙØ¼ÚqÛ]Ü*ÝlÞ4ßàÚáØâ,Ø[ÝlàãåäæØqà]Ûuç3èHé[êë4ÕAØqÙÙØ¼ÚYÜAØqáØqì`Û]íuÕAÖYÓÜ*î

Unite´ de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) Te´le´phone : (33) 99 84 71 00 – Te´le´copie : (33) 99 84 71 71

(5)

& $4/ .$4/ $ #"%$"'&)(*(,+ "%21 &44/ " $4/;J8 + $ 43

$&.6

·u·@Cdao4^`X[oqkyflemauXuxajqanpolahunuanuZOhu]aHese¯aolaf[unikytOhua¡jkm_iema,Z]huX%^`_JxaHeskmZ]X[aX}ey^fHZOnp]aXl\]anVfHa¡wiaj4fHZ]_£

,Zpj[^`nio[aj^em\]auXqkyt]huaj}emZ]X3jlt]håhunua'_Jxao[uZwia'wia%^]wu^`hwia%oq²iVa'¬¼¬0ajqo4^`uieskgtOh4xaa^hunjq²ujqoa_a%wOk¿xaX[anpo|k¯aHe

^Te¯\YxauXqkytOhua®wkmnVwOkyfHa¨wiahi§*ªYWZ]huXCey^¦_Jxao[uZuwia@wia¦%^]wu^h=¬q¬ÀwåZ]XHwiX[a®Âvj[^z_kyjqa2anhupXla@X[aftOhikmaX[o@wia

j¼k¯_ie¯aj_ZuwOksrFf^`oqkmZ]nVj'wih¨fHZuwia.%^]wu^`hV2wxa]aHemZ]u4xaV^X©%ªu³'^kmX[aXao´ªiµ=^`nunuaXª%Z]hVje¯ajaH§uiekyflkmo[Z]nVj'ao

uXuxajqanio[Z]nVj}emaj^_ aHeskmZ]X3^`o|k¯ZOnVjZOuo[anphuaj'jqhuXwOks¿xx aXlanpo3juX[ZOiea_ajª

½,º[·= jq²j|oa_aj^em\YxauXlZ]£ÀwOks¿¡xaX[anioqkmaHegj4wkmnVwOkyfHaÂvp_Jxao[uZuwiaj4wia.4hunu\]a£À¤'huo[o3^Âvp_Jxao[uZuwiaj4wia.%^]wu^`h

¬q¬vA^`X[uX[aj«vfHZOnVwOk¯o|k¯ZOnVjjkm_ieskrFf^o[XqkyfHajv*fHZ]_,ZpjkmoqkmZ]n*ª

(6)

<

(A* *C$2&}3

¬n X[afHanio²]a^X3jvYwOks¿ aXlanpoqkyê.êm\]auX3^Tkgf¸at]hVôqkmZ]nVj=¼¡©j>V^]aVaan0jqo[hVwOkmaw!p² T^`X|k¯ZOhVj@^`huoluZ]X3j=¼j|aa

?

³%µ0u@Àv

?

³BA*C]D@Àv

?FE

d4Wu5@G>3v^`nVwo[uaHkmXk¯_VZ]X[oH^`nVfHa%^]f'HinuZ`4emawi\]awi²o[ua%wia]aHemZ]u_anioYZ` VjqVaflksrFf'jqZ`em]aX3j

j|hVf[@^OjY%JIKIA2 ÔX[Z]_

?

WaoC]b@uZ]X4%±'' ÔX[Z]_

?

³%µ0u@Àª`'najq,aflkg^Teem²km_VZ]Xlo3^`niofleg^Oj[jYZ` ¡©jY^`Xqkyjkmnu\

kmnzuX3^]fHoqkyfHa^X[a¡j|a_km£ÔaH§ieskyflkyoj|²jqola_@jZ` {olua' ÄZ]Xl_

='LM>ONQP$R

=TS'>VU WX=

P

=TSY>1Z1[\=TS'>'>Z

] U ^=

P

=TSY>'>1Z

S`_

?

Sa1ZYS)b@cZ

uaX[aJ^-dWeCkyj^]j[jqhu_aw¦olZ@,aZ VZOhunVwiawzkmnp]aXHjqaCk¯n ^nuaHkm\]iVZ]huXluZpZuw¥Z` 4o[uajqZe¯huo|k¯ZOn Z` ='IK>3ªA³aX[a]vA4a

^X[a4kmnpolaX[ajqo[awCkmnZ]uoH^kmnik¯nu\^nphu_aX|kgf^Te ^ùuX[Z§ikm_@^òqkmZ]nolZ='IK>Y^]ffHhuX3^ò[a¢ ÄZ]XVZ]olo[uawOks¿*aX[anioqkyê^`nVwo[ua

^Te¯\OauX3^kyffHZ]_,Z]nuanpoHjªue¯oluZ]hu\]¦jqZ]_aZ` o[ua'kywia^]juX[ajqanio[awkmno[ikyj%V^VaXêyjqZ2^ùiem²@o[Z_Z]X[a'\]anuaX3ê

¢hunu\]a£À¤'huo[o3^2_aoluZwujv,4a.4ksese ÄZufHhVj'Z]n¨^]wu^`h¥¬q¬À#_ao[uZwuj«vV\`km]an¥o[V^o%o[ua²¦4aX[aChVjqawzo[Zuhikseyw¥o[ua

fHZuwia%4%±'uª`4uaHkmXfHZ]nVjqo[XlhVfHoqkmZ]n^]j*4aHeseF^]jjqZ]_aZ` Vo[uaHkmXYuX[Z],aX[oqkmaj^X[auXqkmafV²X[af^esemawCkmnI]afHoqkmZ]n¦]ªg]ª

µ uan¦^`uiem²Okmnu\®^j|£Àjqo3^`\]a%^]wu^h¦¬q¬À _aoluZw2o[Z='LM>3vio[uaZ]X3wiaXHjZ }fHZ]ni]aX[\OanVfHa.^`X[aX[ajq,afHoqkm]aHem²¦ghJ

ÔZ]X%olua

P

£ÀfHZ]_,Z]nuanio¡^`nVwigC ÔZ]X%oluaj[£ÀfHZ]_VZOnuanpok=¼j|aa

?

³lA*C]-@>3ªu¬n j|Z]_aj¼k¯olhV^`oqkmZ]nVjvA%uaXla.\]ao[o|k¯nu\¦^n

ÔffHhuX3^ò[aTêmhua%Z` [_@^²Vakm_,Z]X[o3^`nioM=Ôkmn_af[V^`nikyfj ÔZ]Xk¯nVj|o3^`nVfHaD>3v`Z]nua}kyje¯aw o[ZhVj|a^wOks¿*aX[anpoY^ùuX[Zi^]f[*ª

´'anuaX3^esem²¦jqVa^-H]kmnu\Vv,o[uaZ]X3wiaXX[awihVfHoqkmZ]n¦uuanuZO_anuZ]n¦_2^²2VaCZ`]aX3fHZ]_a'i²@o[ua' ÔZ`esemZ`4kmnu\@o[aflunikyt]huajnm

o1pGq

^CrVX3jqo4VZij[jkmiksekmoq² fHZ]nVjkyjqo3j¢kmn¨^`uiem²Okmnu\@o[ua%^]wu^`hz¬q¬À _ao[uZuwolZoluakmnVwiaH§¦Z]nua ÔZ]X[_hiey^`oqkmZ]ni= Lr>Z`

='L:>3v

= L:>ON

P R U WX=

P

Z[>1Z

] U ^-dD=

P

>)Ws=

P

Z1[\>1t

I`k¯nVfHa.^]wu^`h¬q¬À0_ao[uZwuj4^`uieskmawolZkmnVwiaH§®ZOnua¡¡©jYaH§uikmik¯o¢ Ähiese*Z]X3wiaX4Z` fHZ]ni]aX[\]anVfHa ÔZ]X

P

^`nVw

[=qjqaa4uaZ]Xla_u;`£[

?

³BAC]D@>Hv`o[ua%Z]X3wiaXZ` fHZ]ni]aX[\]anVfHakyjYnuZ` gvhz¡^Tegj|Z ÔZ]X"[uªi³%Z`4a]aXv]j|Z`em]kmnu\

= Lr>¢f^nVafHZ]nVjkywiaX3^ìem²z_Z]X[a.fHZpj|oqem²mVÔj^_2^ò[o[aX4Z` Y À^]fHovio[ikyjX[at]hikmX[aj'olZaTêmhV^òla.o[uakw]^]fHZ]iky^`n

Z` Fo[ua¢ ÄhunVfHoqkmZ]n xJ=

P

Z1[>vUy=)Ws=

P

Z1[\>1Z^d5=

P

>)Ws=

P

Z1[>Y> ^òYaÔf[jqo[az=ÀZ]XY%uanua]aXo[ua%fHZ]ni]aX[\]anVfHa%XH^ò[a4Z`

o[ua a%o[Z]n2kmo[aXH^`oqkmZ]n¦\Oao3j4o[ZiZjq_@^ese>3vÂk¯nVj|o[a^]wZ {o[ua' ÔhunVfHoqkmZ]n{xJ=

P

Z1[>rU|=)Ws=

P

Z[>1Z}^=

P

>>ªu'nuZ]o[uaX

wiX3^%V^]fYHkyjYo[VôYo[ua%nihu_aXqkyfê j|ZèmhuoqkmZ]nkyjnuZ]o ÄZOX3fHaw^`ni²emZ]nu\]aXo[Z'eskmaZ]no[ua%fHZ]nVjqo[X3^Tk¯nio_2^`niks ÄZèyw

^=

P

>rU

] ª

o1ppGq

^%j|afHZ]nVw'kywia^%fHZ]nVj¼kgj|o3jAkmnfHZ]_uhuoqkmnu\'o[ua:[`£ÀfHZ]_,Z]nuanio*p²jqZèmOkmnu\o[ua4^]wuwOkmoqkmZ]nVêpatOhV^òqkmZ]n9^dD=

P

>WX=

P

Z[>3v

k ªa]ª]kmn2uX[Z]~qafHoqkmnu\.olua%nphu_aXqkyf^e jqZe¯huo|k¯ZOn@Z]no[ua¡j|Z]£Àf^esemaw~Hikywuwian¦fHZ]nVj|o[X3^kmnio1uª]4uafHZ]X[X[aj|VZ]nVwOkmnu\

nphu_aX|kgf^Te}j[flua_anuZ`0X[a^Owuj

='L>

] U ^=

P

G>1Z

P

U

P

JK`

's

WX=

P

$

Z1[n

>Z U ZZgZ

P

U P

J"

Ws=

P

Z1[5$T>Z

] U ^-d$=

P

>)Ws=

P

Z[5

>1t

4uaZ]X3wiaX4Z` {fHZOnp]aXl\]anVfHa' ÄZOXolua

P

£ÀfHZ]_,Z]nuanpo¢kgjjqoqksese{g`h ¡^nVw@f^`n,a.jquZ`%n2o[ZVanuZ`0g`h=

^eyjqZz ÄZ]XoluaO[`£ÀfHZ]_,Z]nuanioi²¥hVjkmnu\¥o[ua@¬_ieskyflk¯ouhunVfHoqkmZ]nJ4uaZ]X[a_zª³ZT4a]aXvAo[ikyjo[aflunikyt]hua2kyj

Z]nVfHa¨^`\i^kmn fHZ]_uhuoH^òqkmZ]nVêsem² _ZOX[a¦wia_@^`nVwOkmnu\Jo[V^`nJolua¦Z]Xqkm\`kmnVêZ]nuamj|Zèm]kmnu\ o[ua¦nua<km_ieskgflkmo

V^`X[oZ` l='L >%X[atOhikmX[aj^`n ^]ffHhuX3^òlaa`êmhV^òqkmZ]n Z` :^ d = P

>^`oa^]fl jqo[a*v*^`nVw nuZ]oZ]niem²]v^]j'uXla]kmZ]hVjem²

_anpo|k¯ZOnuawAvu%uanua]aX4o[uafHZ]ni]aX[\]anVfHaX3ô[a%Z` olua a%o[Z]nkmo[aX3^ò|k¯ZOn@,afHZ]_aj4o[ZiZj|_@êseÀªu¬Ào%f^`n2Va

nua]aX[o[uaHemaj[jnuZ]o[aw o[V^òkmn®^'V^X3êsemaHe aniOk¯XlZ]nu_aniov]o[ikyj^]wuwOkmoqkmZ]nVê fHZijqoY4Z]hieyw,ajqV^]wiZ`4awi².o[ua

hVjqa.Z` ^jqafHZ]nVw¸uX[ZfHajljqZ]Xª

¬no[ikyj4V^`,aXviauXlajqanpo%^Co[ikmX3w^ùuXlZp^]fl@%ikyfl¨wiZiaj4nuZ]o4X[atOhikmX[a.^`n¦^`nVêm²ioqkyfêA ÄZOX[_ Z` X^d.^`nVw2%uZpj|a

fHZO_uhuo3^òqkmZ]nV^Te¢fHZpjqokyj.VÔjkyfêse¯² atOhVê%o[Z¦%VôkmoCkgj ÔZ]X.olua®jqoH^`nVwu^`X3wz ÔZ]X[_hiey^òqkmZ]n*ª¬ÀoCkyjCV^]jqawÁZ]n o[ua

(7)

ZOVjqaX[TôqkmZ]n=olV^ò.o[ua2aX[X[ZOX3jk¯n o[ua [`£ÀfHZO_VZ]nuanio^`X[a2aj[jqanioqkyêsem²=Z` ^2emZf^TenV^ò[huX[a]v^òema^]j|ohu=olZ¦o[ua

ZOX3wiaXZ` ¢fHZOnp]aXl\]anVfHaZ` o[ua

P

£ÀfHZ]_,Z]nuanpoª*j^fHZ]nVjqatOhuanVfHa]v*_@^DHOkmnu\[2_ZOX[a^]ffHhuX3^`o[aCkyj.^2_2^`o[o[aX'Z`

XlafHZTOaXqkmnu\ o[ua¨j¼k¯\OnikrFf^npoo[aXl_@jo[V^ò2^ùVa^`X k¯n o[ua¥jqZ]£Àfêsemaw ~ E

£ÀjqaXqkmajZ` ¡olua¦aX[X[ZOX1uªY4ikyjkyj_@^]wia

,Zpj[jkmiemai²¨fHZ]nVj¼kgwiaX|k¯nu\¥o[uafHZ]_VZpjkmoqkmZ]nÁZ` o[uaV^]jkyf_aoluZwz4kmo[zkmo3jqaHes %Z`]aX¡j|a]aX3^ejqo[aVjªA¬o.f^`n¥Va

nuZOo[awo[V^ò4jkm_ksey^`XkgwiaÔj4aX[ahVjqawi²@ªWª¤ªudV^`no[ZwiaêA4kmo[2o[ua%Z]X3wiaXX[awihVfHoqkmZ]nZ` Y´^`hVj[j_ao[uZuwuj

uan¦^`uiesk¯awzo[Z2fHaX[o3^kmn¨j|oqks¿ uXlZ]iema_@jl=¼jqaa

?E

d;@>3ª

¢uanua0ZOX3wiaX%fHZ]nVwOkmoqkmZ]nVj44kseseVa.wiaolaX[_kmnuawkmn{I]afHoqkmZ]n¦uªOµ ikemaCo[ua²@f^`n,a.wiaXqkm]awkmn¦^j|o[X3^km\]ioq ÔZ]X[£

%^`XHw_@^`nunuaX4 ÔX[Z]_o[ua4Z]X'HZ` {©ªu³^kmX[aXvVd'ªAAhuikyfl¨^`nVw ªp¢Zfluak=

?

³lA*C]-@>3v]olua²®^`Xla¡jqo|kese*X[aHey^`oqkm]aHem²

hunKHinuZ`%n j¼k¯nVfHaÁo[ua² ^`X[aznuZ]o®jl^òqkyjrVaw p²J_ZpjqoflegÔj[jkyfê4hunu\]a£À¤'huolo3^ _ao[uZuwuj2=qjqaa êyjqZ

?FE

d4;@Z]X

?

d%d]n@>3ªY¬o4ksese.o[uan ,a¨jquZ`%nJo[V^o@jqZ]_aZ ¡o[uZijqa¨fHZ]nVwOkmoqkmZ]nVj^X[a¦^]fHo[hV^esem²JX[awihunVwu^`nio^`nVwJ_@^² Va

ZO_kmo[o[awAª*4ikyjkyj^zfHX[hVflkyê¢ÔjqVafHoCZ` %o[ua_ao[uZuwAvj¼k¯nVfHa2kmoêse¯Z`jo[uafHZ]nVjqo[X[hVfHo|k¯ZOn Z` 4 ÄZOX[_hiegÔj'4kmo[ ^

_2^`nV^`\]a^`iema'ema]aHeYZ` {fHZ]_ie¯aH§ikmoq²]ª

¬niI]afHoqkmZ]n ;uvio[ua'km_ieskmawz_ZwOksrFf^ò|k¯ZOnVj%o[ZCo[ua.fHZuwia.4¡±'^`X[a'eskyjqo[awAªkmnVêsem²]vVnihu_aXqkyfêX[ajqhiemo3j%^`Xla

uXlajqanpolaw2o[V^`o4ksesemhVjqo[X3^`ola.o[ua.^]wi`^`nio3^`\]aj4Z {o[ikyj4nua o[aflunikyt]hua]ª

!"#!$%'&(*)+-,,.0/1"2)

^]wu^`h¬q¬À_ao[uZuwuj%f^`nVawiaHrVnuawzp²2o[ua.g` .tOhV^]wiX3^`o[huXla' ÄZ]Xl_hiey^]j

3465

798

=TS'>;:$S0<

' 8

=>= >1Z

U ZYZ1gZ

3

7 8

=TSY>;:$S <

8

=>= >1t

V^`X[oqkyfHhiey^`X4_ao[uZuw?4kseeAVaflV^`X3^OfHo[aXqkA@awi².olua%o[Xqkmiema.=>B.Z

ZC=>u%uaX[a=

>T

'

EDEDED

kgj^.gGF.g%_@^o[Xqks§ v

U = Z'Z

>H ^2g«£ÀwOk¯_anVjkmZ]nV^e%]afHo[Z]X.^nVwI=kU =>=

ZZ=

>H ^2g£ÀwOkm_anVj¼k¯ZOnV^e%]afHo[Z]XªA¬n o[ua@j|at]huaHeÀv

4a4ksese* ÔhuX[o[uaX[_Z]X[ahVjqaCo[uanuZ]o3^`o|k¯ZOnVj JlUy=3$Z'Z->H _KL ^`nVwM=ONlUy=;=ON

Z'ZC=ON

>H ÔZ]X^Teek¯nio[a\]aXHj PAª

QSR6Q!R6Q T

½U «*`

p

½VU ½SW V¾*YXZX =·i[F½*¾

¢uaHk¯XfHZiaH°®flkmanpo3j'^`X[a'hunikyt]huaHem² wiao[aX[_ k¯nuaw¸p²2o[ua ÄZ`esemZ`4kmnu\®fHZOnVwOk¯o|k¯ZOnVj5m

Oª =

Z'ZC=

^`Xlao[uaZ]X3wiaX[aw$@aXlZpj4Z` Yo[ua^]wu^`hXqkm\]io%VZe¯²inuZ]_ky^e

\ =>]>rU

:

>^

:_]

`^

ba ] >^

=;] hJD> >c t

Âª

ZZ

j[^oqkyj Ä²dk=gD>1m e

P2_(f

]

ZZ1ghJgZ

H = N U

=>PJ D>

t

;Âª%4uafHZiaH°®flkmanpoHj

Z` Yo[ua_@^`o[X|k§hB j[^`oqkyj Ô²bij=)gD>m

e

P2_(f

]

ZZ1ghJgZBj=

N U

=;PJ ->

= N t

¡j4oluak=Y^`X[a.^ese{wOkyjqoqkmnVfHovB kgjnuZ]nu£Àjkmnu\]hiey^`Xª

QSR6Q!Rml n

½=· *·_W o½*o·u

p

·u

µÁa4ksese¢XlaH ÄaXCuaX[a@olZ¦o[ua@^OwuwOk¯o|k¯ZOnV^ejkm_iesk Ô²Ok¯nu\J^]j[j|hu_uoqkmZ]nVjp2=#q>4kmnio[X[ZuwihVfHawAvY^]j'dj=sr>.^`nVwIij=;t3>%Z`

uXla]kmZ]hVj'jqhuVjqafHoqkmZ]n*vÂp² wuª¯dª

E

huoHf[uaXJ=¼jqaa

?

³µ0u@AV^`\]a`>3ª

(8)

Oª%'hua¨olZ=o[ua fHZ]nVwOkmoqkmZ]nVjMik=)g->3vdj=)gD>@^`nVw9=

U ]v4%^]wu^h ¬q¬ _ao[uZwuj2^`X[a jqo|k2² ^]ffHhuX3^`o[a =ÔkÀªåa]ª

2U

)Z

U)ZZ1g->3ª=q4ua¥]afHo[Z]X3j

^`nVw = Z'Z

c

>H ^`Xla¨jqZ`emhuoqkmZ]n Z` o[ua j[^_a dX3^`_aX

jq²j|o[a_2>3ª

Âª%4ua²¦^`X[a.fHZ`esemZuf^`oqkmZ]n¦_ao[uZwuj =¼4uaZ]X[a_ £TC

?³ %µ0u@ÀviV^`\Oa.u>3ª

;Âª dk=¼glhJD>}kgj'j[^`o|kgj¼rVaw=)AAa__@^`£[

?

³µ0u@ÀviV^`\]a.;\>3ª

,ª p2=)gh D>kyj'j[^`oqkyjrVaw =)AAa__@^`£Ô

?

³%µ0u@ÀvÂV^`\]aC>Hª

%uo|k¯_2^e{fHZ]ni]aX[\]anVfHaCX[ajqhiemo3jV^]aVaan¥Z]uo3^Tk¯nuaw® ÄZOX%o[uZpjqaC_ao[uZuwuj%i²2wuªsdª

E

huo3fluaX'Z]n¦o[uaZOnua.V^`nVw

=q4uaZ]X[a_£T;Z`

?

³%µ0u@>^`nVwz©%ªV³'^kmX[aXv dªAAhuikyfl=^`nVw ªV4Zuf[uaCZ]n¦oluaZ]o[uaX%V^`nVw =¼4uaZ]Xla_@j;`£[]v

O£Ô ^nVw]£ÀbZ

?

³lA*C]D@G>3ªu4ua²¦^`X[a.fHZ`esemafHo[aw¸kmn¨^`iema@]ª

kmnVwiaH§

]

=© > kmnVwiaH§= kmnVwiaH§¨

P

>^

`^

[ £

`^

îe¯a2m%uoqkm_@êY\èmZ]Vê*aX[X[ZOX%ajqoqkm_@^òlaj ÄZOXoluajq£ÀjqoH^`\]a'_ao[uZuw¦%^]wu^h¬¼¬

, S* "2 V)Sh% ! %'"2OV/1 "

µ uan¦^`uiem²Okmnu\¦^jq£Àjqo3^\]a¡^]wu^`h¦¬q¬ _ao[uZuw? U|=;B.Z

ZC=>*o[Zo[uajq²ujqo[a_ ='L:>Hvp4aZ]uoH^kmn

] U ^=

P

c>1Z =3D>

P

$ U P

}

Ws=

P

Z1[ $

>1Z

U Z'Z1g$Z =¼>

P

D

U P

.

Ws=

P

Z1[nc>1Z =);>

[n

U

'X

[5

Z =ÀK>

uaX[a

^`X[a¥o[ua fHZpaH°flk¯anio3j¸Z` .o[ua _@^o[Xqks§YB ^

ª4¬n =ÀK>3vM[ `^`nikyjquaj,af^`hVjqa LJ= >U ] ª`AAaohVj

nuZ`ÃXlaieg^OfHa¦o[uaz]afHo[Z]X2=

H*B

^

>H p² ^n ^]w~¼hVj|o3^`iema¦]afHolZ]X U =

Z'Z

>H kmny=À>3ª

E ² wiZk¯nu\JjqZVv

4a¥wiaHrVnua=^ nua_ao[uZuw ?ª4¬okyj¸a^]jksem² j|aan olV^`oo[ua¥Z]X3wiaX2Z` fHZOnp]aXl\]anVfHa¥Z` .o[ua

P

£ÀfHZO_VZ]nuanio

Xla_@^kmnVj%hunVflV^`nu\]awAª,j4 ÔZ]X%o[uak[`£ÀfHZ]_,Z]nuaniovVoluaey^]fYHZ` ^]ffHhu_hiey^`oqkmZ]n¦_2^DH]ajo[ua.aX[XlZ]X3j4uhuX[aHem²¸e¯Zuf^eÀª

¢ua¦fHZ]npOaX[\]anVfHa¦,aV^]kmZ]huX2Z` o[ua [`£ÀfHZ]_,Z]nuaniokyjo[phVj anpoqkmX[aHem² wiao[aXl_kmnuaw i² o[ua ÔZ`esemZT¢k¯nu\ Z]X3wiaX

fHZOnVwOk¯o|k¯ZOnVj4 ÔX[Z]_¢uaZ]X[a_ C`£Àb^`nVwC`£TCCZ`

?

³%µ0u@Àª

C½*o½F

p p

½VU

Q"!

#d2!#$#e'J/0!/ Y,. )n,TK1/r5

P

!#k[ \$3, }.! %$&%

'.%!2k# )(X+*r.!

# P

U-,k=)/.

> 0 e

S"_Mp B)1DdZ2=TSY>43 rXZ5v=TSY>

H76=TSY>rU Z

#D[JU-,j=)8

> 0 e 9

_Mp B)1DeDZ23=

9

>:3<;Z5v=

9

> H 6=

9

>vU Z

*v p B)1Dd p B)1De !$ ) ,5,O 5,=6 !i$ 1!/5%3!$#>5rZ2{,5!/0! kn,j!-,1,5!#?*r

+ 5,

a]aX[oluaHe¯ajljv@V^]j¸nuZ]o2anuZ]hu\] fHZ]_VZ]nuanio3j2o[Z ^esemZ` ÔZ]X^`n Z]X3wiaXZ` fHZ]npOaX[\]anVfHa \]Xla^`o[aX2o[V^`nyg

=A U = H B ^ >H kyj2o[uazZ]uoqkm_@^e]afHo[ZOX5>3ª³anVfHa]v4o[Z \]ao®j|hi°®flkmanpo2 ÔX[aawiZ]_¦v4a¦nuaaw o[ZJfHZ]nVjkywiaXo[ua

fHZO_VZpj¼k¯o|k¯ZOn ?>BZ` ?Z`]aXDC jqo[aVj«ªA¡jT^`X|kg^ie¯a2jqo[aVjlK[v

U ZZC ^`X[afHZ]nVjkywiaX[awAvA4a^eyjqZV^]a.o[Z

kmnio[X[ZuwihVfHa.o[uaXH^`oqkmZpj@EnXU GFr

B

Kqv

U Z'ZHCYª_?>Bkyj%flV^`X3^]fHo[aX|k@aw¸p²2o[uaCo[Xqkmiema=GIZHJZLKr>%uaX[a

M

I#kyj4olua.iemZfYHi_@^`o[Xqks§i=;B

>T

}

EDEDEDB@4kmo[ g F gCiemZfYHuj%Z` o[ua ÔZ]X[_

NO

ØqØQPSRTVU&WXY[Z\ÝáØ]`Ùä_^Sä_ZHÙ`ZY^aØqÚÔØYÙbZ^Sä_Z3ÙÚAc

(9)

B0 U En BJZ

B

U E J H Z ks

Z

B

U ] Z ks

t

M

JJkyj4olua.iemZfYHi]afHo[Z]X=>dc>T

EDEDEDB24kmo[iemZuf'Huj%Z {o[ua' ÔZ]X[_ drU-En

ª

M

K¨kyjo[uaiemZfYHi]afHo[Z]X=>i`T>c

EDEDEDB4kmo[iemZuf'HujZ` {olua' ÄZ]Xl_ i`XU E56=r

a ^

N

E N c JOª

¢aieg^Oflk¯nu\

a J H I ^ c H

i²®^`n¦^Ow~¼hVjqoH^`iema]afHo[Z]X@ Z`¿*aX3jQChF gwia\]Xlaaj4Z` * ÔX[aawiZ]_¦vk ªauZ]VaH Ôhiese¯²zanuZ]hu\]

ÔZ]XDC o[Z k¯nVfHXla^]jqao[uaZ]X3wiaX'Z` fHZ]ni]aX[\]anVfHaCZ` {olua+[`£ÀfHZ]_VZ]nuanioªu¬o%kyjZ]huX^km_nuZT o[Z@jquZ`0uZ`0o[Z

fHZOnVjqo[X[hVfHoD ^`nVwuZ`0o[ZCkm_iema_anioo[uanua0_aoluZw ?Bª

7 0/,* &}3-' 4Y*'

¬n.o[ikyjjqafHoqkmZ]n*vo[uaaH¿*afHoqkm]a4fHZ]nVjqo[XlhVfHoqkmZ]n.Z` ÁkyjwiajlfHXqkmVawAª`¬ÀoYjquZ]hieywCVaa_uV^]jkA@awCo[V^`okmo3j*fHZ]_VZ]nuanio3j

wia,anVw Z]n E Z'ZHE

v*o[ihVj ÄZ]XHflk¯nu\¥Z]nua2aT^Te¯hVôqkmZ]nJVaXjqo[a*ªY³ZàOaXvAo[uajqa^]wuwOkmoqkmZ]nVê¡fHZO_uhuo3^òqkmZ]nVj

,afHZ]_aCnua\èsk¯\k¯iema@Ôjj|ZpZ]n¦Ôj4o[uawOkm_anVj¼k¯ZOn¦Z` Yo[ua.jq²ujqo[a_ =YIK>Ykyj4ey^`X[\OaanuZ]hu\]*ª

h)!( )6"

¢ua.fHZ]nVwOkmoqkmZ]nVj ÄZOXZOX3wiaXjP^`X[aanihu_aX3^o[aw,aHe¯Z`0olZ]\]ao[uaX%¢k¯olo[ua^]jljqZflky^`olaw2o[X[aajª

M

PU g]v3g 0$m=;i

>YkyjX[at]hikmX[aw¸4kmo[

=>i

>

H U Z = ] >

e

P2_(fiZZ1ghJg HK N U Z

?? N

Z5Z

@d@eDt =>P3>

M

PU g` ]vKg Km=;i

>^nVwi=;i

>^X[aX[atOhik¯Xlaw4kmo[

=;i

>

H K U Z

??

Z5'Z

@d @e Z =)gD>

H I ^ K U g Z

?

ZZ

@et =)g` D>

AAao

Va I K h

K

ªu¬ =)gD>kyjjl^`oqkyjrVawAv,o[uani=;i

>}kgj¢at]hikmT^Te¯anioo[Z

H I ^ U ] t

M

PU g`JuvKg ;Km=;i

>3v =;i

>^`nVw =;i

>^`X[aXlat]hikmX[aw¸4kmo[

=>i

>

H7K

U Z

??

Z5)Z

@d5@eZ =g` \>

H I ^ K U g=Z

?

ZZ

@e Z =g`;\>

H I K U

Z ???

Z5Z

@d@d5@eZ =g`=>

H

K:tSI ^ K U g $Z

?? Z ?

Z55Z

@e5@d5@eDZ =g` \>

H I ^

=K:tSI K >VU

Z ? Z ?

Z55Z

@d@eDt =g` b\>

AAao

Va I K h

K

`ªi¬ =)g`=>}kgj'j[^`o|kgj¼rVawAvuo[uan{=>i

>Ykyj%atOhikmT^emanioo[Z

aØYábZ^*Ú^ÀÝ[ÙáÚYZH\âØAZ3Þ4ßZ3ÙØqÙ&ÂäæÚÔØ4ß \ZáàAÂc

(10)

H7I

^ U ]

=)g ;>1Z H

U ]

=)g K>1Z

H K:tSI

^ U ]

=)g =>1Z H I ^

=K:t > U ]

=)g =b>1t

M

PU g` ;uvKg !m=;i

>3v =;i

>3v=;i

>^`nVwi=;i

>4^`X[aX[atOhikmX[aw4kmo[

=;i

>

H K U $Z

??

Z5)Z

@d5@eDZ =)g J>

H7I

^ K U g ;KZ

?

ZZ

@eZ =)g C>

H I K U

Z ???

Z5Z

@d @d @e Z =)g =>

H K:tSI

^ K U g KZ

?? Z ?

Z555'Z

@e@d@eDZ =)g ] >

H7I

^

=K:tSI K > U

Z ? Z ?

Z55Z

@d@eDZ =)g ]D>

H I ^

=K:tSI K > U

Z ? Z ??

Z555'Z

@d @d @e Z =)g >

H I ^

=K:tSI=K:tSI

^ K

>'>VU

Z ? Z ? Z ?

Z555Z

@e@d5@eZ =)g ;>

H7I

^

=K tSI K > U

Z ? Z Z ?

Z55Z

@d5@eZ =)g K>

H I K U

Z ????

Z55'Z

@d @d @d @e Z =)g >

H I=K tI ^ K > U

Z ??? Z ?

Z555Z

@e@d5@d@eZ =)g b>

H

K:tSI K U

Z ?? Z ?

Z555'Z

@d @d @e Z =)g `>

H K:tSI

^

=K:tSI K > U

Z ?? Z ? Z ?

Z55Z

@d@e1@d5@eZ =)g C>

H K tSI

^ K U g Z

?? Z Z ?

Z555Z

@e@d5@et =)g >

AAao

Va I K h

K

ªi¬ =)g`J>}kgj'j[^`o|kgj¼rVawAvuo[uan{=>i

>Ykyj%atOhikmT^emanioo[Z

H I ^ U ]

=)g` C>1Z H U ]

=g` \>1Z

H

K:tSI ^ U ] =)g` ] >1Z H I ^

=K:t > U ] =g` OD>1Z

H I ^ =K tI

> U ]

=)g` >1Z H I ^

=K:tSI =K:tSI

^

>'>VU

]

=g` ;>1Z

H7I

^ =Kt > U ]

=)g` K>1Z HI

U ]

=g` >1Z

H I =K:tSI ^ > U ] =)g` b>1Z H

K:t U ] =g` >1Z

HK:tSI

^

=K:t > U ] =)g` C>1Z H7K tSI

^ U ] =g` >1t

¢uajqa¡fHZOnVwOk¯o|k¯ZOnVj%^`X[a%ZOuo3^kmnuaw2p²2WYX[Z],ZpjkmoqkmZ]n¥^`nVwi²hVjkmnu\@jkm_iesks Ä²Okmnu\®^Oj[jqhu_uoqkmZ]nVj=Ôkmokyj}k¯_VZ]X[oH^`npo

olZnuZOo[ao[V^`o%olua.fHZ]_VZijkmo[a._aoluZw$?>B@j[^`o|kgj¼rVajjdj=¼gh D>3vik=gD>^`nVwhpz=)ghJD>'>3ª

#/1G Y++

¬nZ]X3wiaX%olZey^ò[aXjkm_iesks Ô²¦o[uaatOhV^òqkmZ]nVj¢ ÄZOXD.vi4anuZ` jqo3ô[a¡j|Z]_aV^]jkyfX[ajqhiemo3jª

}· ¨ Q"! Xx Y B ^

B ^

*r

UYJ

H

: xJ=#K+h

B ^

>rU ]

C½F½SW3

E ² wiaHrVnikmoqkmZ]n*v4a¦V^]ax U B ^ J H B ^

ª¡jo[ua¥_ao[uZuw ? U =>B.Z

ZC=>%kyjjqo|k2² ^]ffHhuX3^`o[a]v

x U_J H

%uaX[a U B ^

JÁ^`nVw9J

UV=

]

ZZ

]

ZD> H ª%³%anVfHa]vlxJ=K h B ^

>U J H

=#K h1J

H B ^

>iU

_J H h J H

JZJ H U_J H

h J H U ] ª

}· ¨

l !

v=

>T

}

EDEDEDB Yk.g Fig5/0,+kk! 4 I ^

:

(11)

i U 5 B ^ Z

^ U h 5 x`Z

U ] Z

U

!#

U

hJt

C½F½SW3

E

²¦wiaHrVnik¯o|k¯ZOn*vI I ^

U%vikÀªåa]ª

e Z _ fpZZHC gZ E5sB i

^

N

E N N U-#

Kt

A a__@^Ckyjo[uana^]jksem²uX[Z`]aw2i²kmnVwihVfHoqkmZ]n¥Z]n

ª

}· ¨ ! XU

a

H

Z'Z HB

c H

'JDC FOg#+K,5$!/3 I K h

K

!#

9

('J

g#k, !$/1 jB = h

D

=

*ri_K )(s 5 .!$//X ?3 glh _ X{,B O

e

_(fiZZHC gZ X U

N P

E N

g ^ N

9 N t

C½F½SW3AAao¦,aemaj[jo[V^`nZ]Xat]hV^eo[Zgh uªKuZ]X^Tee

_(fpZ)ZC gpvi4aV^]asU<1L3,vu%uaX[a

1 U ^

N

E N

=E

N = g

N

JD>

Z

L U E5mB{=AE5s=MgnJD>

h

=GEn=MgnJD>

t

L32f^`nVaCaH§V^`nVwiawz^]j ÔZ`esemZT'j

L3 U N 7

P E N

g ^ N

Bj=ONlh

D

N 7

P

EN g

D

^ N

=ON

U N 7

P E N

g ^ N

9 N h

g

D

Jt

%¢k¯nu\2o[Z ik=gD>3vi4aV^]a

9 N U ]

ÔZ]X%^esekmnio[a\]aXjPg]vuuanVfHa

L U

N P

E N

g ^ N

9 N h

g

D

Jt

I`km_ksey^`Xqem²]v71@f^`n,a.aH§uV^`nVwiaw¦^]j

1kU

^

N

E N

7 E N g ^

N = t

Z`.vdk=¼$gh ->km_ieskmajolV^`o

e _ f ]

ZZ3$g`h=_gZ =

UYJ

H = U

JZ

^nVw^]j+3 gh uvu4aZ]uo3^Tk¯n

1 U

^

N

7 E

N g ^

N J

U

a ^

N

=GE

N g N >

h{g

N c J

U

a ^

N

g

D

N h{g

D

N c J

U

g

D

J$t