• Aucun résultat trouvé

Arnoldi-Faber Method for Large Non Hermitian Eigenvalue Problems

N/A
N/A
Protected

Academic year: 2021

Partager "Arnoldi-Faber Method for Large Non Hermitian Eigenvalue Problems"

Copied!
20
0
0

Texte intégral

(1)

HAL Id: inria-00073688

https://hal.inria.fr/inria-00073688

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.

Arnoldi-Faber Method for Large Non Hermitian Eigenvalue Problems

Vincent Heuveline, Miloud Sadkane

To cite this version:

Vincent Heuveline, Miloud Sadkane. Arnoldi-Faber Method for Large Non Hermitian Eigenvalue Problems. [Research Report] RR-3007, INRIA. 1996. �inria-00073688�

(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

Arnoldi-Faber method for large non hermitian eigenvalue problems

Vincent Heuveline and Miloud Sadkane

N ˚ 3007

Octobre 1996

TH `EME 4

(3)
(4)

+

#,-!%.

/,021#3 45176984;:&<=4;>?021&4@1#ACBC0?>2DE:#ACFG@HA7IE@1&4

JKML

NONQPRTS;UVOXWY[Z]\^U`_ abN-\_dc\^U`OUVefZ]\^UV_da

g

Neih;e^\

L

NONej_dOcYVNk;Ne

l&m

_onN\Qp'Y[Z

g

UVa

q

Z]ccH_

m\ g N m

Nj K N mj K

Nras;tdudu vRTwQj\i_]x

m

NXyoz zd{|R}yo{~cEZ€dNe

X‚#ƒ-„o…]†H‡]„ ˆŠ‰

Nbc

m

_dcH_de^N,Z

m

N-ei\fZ

m

\^N g p m

a‹_dY

g

UŒe'ON-\

K _

U`\

K

Z]xHN

m

cH_dYVh;a_ OU[Z]YVe'Z]a g‘g

UVe^jWeieUV\ieW‹eiN’?_

m

j_ OcW\^UVa\

K N m

UV

K

\iO_de^\“N-U` Na;”dZ]YVWNe_]’'Y[Z

m

dNXa‹_da K N m

OUV\iU[Z€a‘OZ]\

m

UVjNeo•

‰

NXUVY`YVWe^\

m

Z]\iN –

Ž

U`\

K \K N K

N-Y`c—_€’

ei_ ONc

m

Z€j\iUVjoZ€YE\^Nei\c

m

_ xYVNOe*–]\

K

NQxEN-aN˜E\_dx\Z]UVaN g ’m

_dO™\

K N 

Z]xHN

m

Z€jjNYVN

m

Z]\^U`_ aXx;hbj_ OcEZ

m

UVaQ\

K

UVe#ON\

K _ g

Ž

UV\

K \K

N|š

K

Nx;h5e

K

N”xEZ€eiN

g

Z€jjNYVN

m

Z]\^U`_ aG•

›œ;HžfŸ H…d¡&ƒ]ˆ¢!m

h;YV_]”e^cEZ]jN –xYV_;j£p

m

a_dY

g

U–



Z€xEN

m

cH_dYVh;a_dOU¤Z€Y¥e*–dS;j K Ž Z

mi¦§

š

Km

UVe^\i_]¨=NY

©VªX«

¬­®*¯

«

¬°&±`­²2³´

Unit´e de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) T´el´ephone : (33) 99 84 71 00 – T´el´ecopie : (33) 99 84 71 71

(5)

+ +

+

#

+

#'. $%&!. , "# &-

œƒ œ ˆ

_dW‹ebc

m

_dcH_de^_daeWaNON-\

K _ g N g

N\h;cENp

m

a‹_dY

g U m N g N-O,Z

m^m

NN Z*” Nj

g

N-ebcH_dYVh;a_dONe g N 

Z]xHN

m

N\

g

UVeij-W\i_ ae!ei_daW\^U`YVUVefZ€\iUV_dacH_dW

m

joZ€YVjWYVN

m

YVNe!”dZ]YVNW

m

e'c

m

_dc

m

Ne

g

NcYVWe'

m

Z€a

g

NecHZ

m

\^UVNe m NNYVYVNe

g

NO,Z€\

m

UVjN-e

a_ a K N m

OUV\iUVNaaN-e

g

N

m

Z]a

g

N\fZ€U`YVYVNd•

_dWe\iNe^\i_ aeXY[ZON\

K _ g

NX_dx‹\iNa;WNeiW m g

N-ebc

m

_dxY

L

NONec

m

Z]\^U;WN-e;WU

O_ a;\

m

N-a5\efZe^WcN m UV_

m

UV\NcEZ mm

Z]ccH_

m \ L

ZYŒZ]j-jNYN m

Z€\iUV_dacEZ m g

NeQcE_ Y`h;a_ ONe

g

K

Nx;h;e

K

N”=•

C H„ ƒ ž ‡ œˆ

Ne^cEZ]j-Ne g N

¢“m

h;Y`_]”=–;p

m

a_ Y

g

U=cEZ

m

xYV_;jd–;cH_dYVh;a_dON g N 

Z]xHN

m

–S j K Ž Z m^¦§

š

K‹m

UVei\^_]¨=NY

(6)

JK

NXp

m

a_ Y g U§ š K

Nx;h;e

K

ON-\

K _

g

z;–#yP–yov–=t‹yd–#y j_da‹ei\iUV\^W\iNe_da‹Nb_€’ \ K NXO_

m

NbZ€\i\

m

Z]j\^UV”dNZ]j-jNYVN

m

Z€\iUV_da

\iN-j

K

aU5W‹Ne’?_

m

j-_dOc‹W\iUVa!\

K N m

UV

K \§

O_ ei\#NUV Na;”dZ]YVWNe_]’GY[Z

m

dN!O,Z€\

m

U`j-Neo•

JK

N!U

g

NoZ~xEN

K

U`a g \K

U`e\^Nj

K

aU;WN!UVe

\i_ei\Z m \ Ž

U`\

K \ K

N!cH_dcWY[Z m p m

a_ Y

g

U=ON\

K _ g

y

Ž K

U`j

K

Z]c‹c

m

_]k;UVO,Z€\iNe#\

K

N!_dW\^N

m

O_de^\ cEZ

m

\_]’&\

K

N!eicHNj\

m

WO_€’

\ K N OZ]\

m

UVk

Z€a g \ K

Na

m

Ne^\fZ

m

\#\

K

N#p

m

a_dY

g

U;c

m

_5j-Neie*–5Z ’?\iN

m

Ze^O,Z]YVY]a;WOXxHN

m

_€’U`\^N

m

Z€\iUV_da‹eo–]We^UVa'cH_dYVh;a_dOU[Z]Y¥e

\ K

Z]\'Z]OcYVU’?h\

K

N“j-_dOcH_da‹Na;\ie_]’#\

K N m N ;WU m N g

N-U` Na g Um

Nj\iUV_ ae

Ž K

UVYVN

g

WOcUVa~\

K

_ eiN!UVa\

K

NWa

Ž

Z€a;\iN

g

_da‹Neo•

JK

NZ]YV _

m

UV\

K

Oe

g

UVeijWe^eiN

g

UVa‘\

K

NZ]xH_]”dNONa;\^U`_ aN g m

N’2N

m

Na‹jNeZ]e^eiWONb\

K

Z]\~\

K

NXWa

Ž

Z]a;\iN

g

NUVdNa;”dZ]YVWN-e

YVUVNUVa‘Z]a N-Y`YVUVce^Nbe^h5OON\

m

UVj

Ž

UV\

K‘m

NeicHNj\\^_\

K N m

NoZ€Y#Z]k;UVe'U’ \K NbOZ]\

m

UVkUVe

m

NoZ]Y#Z]e!UVa

z;–&y-PE–&yov;–=ty _ m Z€a

NYVYVUVceiN!_dx‹Y`U;WNQU`a\

K

NdN-aN

m

Z€YGj_ OcYVNkXa‹_da K N m

OU`\^U[Z]aj*Z]eiNZ]eUVa

y i•

JK

NZ]eie^WOc‹\iUV_da\

K

Z€\\

K

NW‹a

Ž

Z]a;\^N

g

NUVdN-a5”dZ€YVWNeYVUVN“UVa Z]aNYVYVUVceiN~U`ea‹_d\

Ž

UV\

K

_ W\

g m Z Ž

xEZ]j£;e'e^U`a‹jNd–=Z]eQcE_ U`a;\^N

g

_dW\x;h SZdZ g

t u;–cG•

t z i–o•V•V•&\

K N

j K

_dUVjN!_€’ NYVYVUVceiNe'Z€eNajYV_ eiUVa

m

N-dUV_daeUVaš

K

Nx;h;e

K

Z€jjNYVN

m

Z]\iUV_ a,OZ*hbxHN!_]”dN

m

YVh

m

Ne^\

m

UVj\^UV”dNbZ€a

g

U`a‹N¨MN-j\iUV”dN

U’ \ K Ne

K

Z]cHN“_€’#\

K

N!j_ a5” Nk

K

WYVY=_€’ \ K N!Wa

Ž

Z€a;\iN

g

NUVdN-a5”dZ€YVWNexHNoZ

m

e'YVUV\i\iYVN

m

Ne^NOXxY[Z]aj-N

Ž

UV\

K

Z€aNYVYVUVceiNE•'N

c m _dcH_deiN

g \ K

N!We^N_]’&\

K

N~Y`N*Z]ei\e ;WEZ

m

Ne'cH_dYVh;a_dOU[Z]Y¥e_dae^_dON'cH_dYVh;d_daHZ]Y

g

_dOZ]UVae

z i•

JK

UVecEZ]cHN

m

UVej_daj-N

m

aN g Ž

UV\

K \K

N!j_dOcW\Z]\iUV_ aX_]’ Z|’?N

Ž m

UV

K \ §

O_de^\ N-U` Na;”dZ]YVWNeZ]a g \ K

N!j_

m^m

Ne^cE_ a

g

UVa

NUVdN-a5” Nj\^_

m

e_]’ Z|Y¤Z

m

 Nj_dOcYVNk!a_da K N m

OUV\^U¤Z€abOZ]\

m

UVkb•&p'eU`a

z – Ž Nj_dae^\

m

Wj\'Z!cH_dYVh;d_ ab\

K

Z]\#j-_da;\fZ€U`a‹e

\ K NXWa

Ž

Z€a5\^N

g

NUVdN-a5”dZ€YVWNebZ€cc

m

_]k;UVO,Z€\iN

g

x;hp

m

a‹_dY

g

UŒe“c

m

_;jNe^eo–&xW\~\

K

NXj_ OcW‹\fZ]\^UV_da _€’ \K

NcH_dYVh;a_dOU[Z]Y¥e

UVe g U¨MN

m

Na;\o•

‰

N“W‹eiN\

K N 

Z]xHN

m

cH_dYVh;a_dOU[Z]YVe

v \K

Z]\OU`a‹U`OU

¦

NQ\

K N m

N U`_ aNajYV_de^N

g

x;h\

K

N~cE_ Y`h; _daG•

JK

N-eiN

cH_dYVh;a_dOU¤Z€YVe Z m N!\

K

Na We^N

g

’?_

mm

Nei\Z

m

\^UVa\

K

N!p

m

a‹_dY

g

U=UV\iN

m

Z]\^U`_ aG•

JK

UVe

Ž _ m £ U`e~U`a‹eicU mN g

x5h\

K

NXj_da;\

m

UVxW\iUV_ a‘_€’ S;\Z

m

£dN,Z€a g

Z m

5Z

t Ž K _ WeiN

g 

Z]xHN

m

cH_dYVh;a_ OU[Z]YVeUVa

\ K

Nj_da;\^Nk;\_]’#YVU`a‹NoZ

m

eih;ei\^NOe'Z€a

K

_Xe

K _ Ž N g \K

NaNoZ

m

YVh_ c\iUVOZ]Yc

m

_dcHN

m

\^UVNe!\

K

Z€\\

K N 

Z]xHN

m

cH_dYVh;a_dOU[Z]Y¥e

cH_deie^Neie*–=Z]a

g

x;h\

K N Ž _ m

£X_€’

J m

N’?N\

K

Na

t t G_da\

K

NS;j K Ž Z

m^¦§

š

Km

UVei\^_]¨=NYGOZ]cc‹U`a‹!’?Waj-\iUV_daG–ZcH_

Ž N m

’?W‹Y=\i_;_dY

’?_

m

j_dae^\

m

Wj\iUVaX\

K N 

Z]xHN

m

cH_dYVh;a_dOU¤Z€YVeo•

$d~

+

)X " Q"b|(“ )Q.

‹m

_dO Ze^\fZ

m

\iUVa”dN-j\i_

m Ž

UV\

K !#"

yd–7\

K

N,p

m

a_ Y g U c m

_;jNe^eXdNaN

m

Z]\^NeZ]a _ m \ K _ a_

m

O,Z]Y&xEZ€eiUVe%$'&

"

() ))*(+

&, _€’“\

K N

¢!m

h;YV_]”‘e^cEZ]j-N.-/&102-/&435

(*6/"87*9

3:

6;#<=9?>@

&=A

B Ž K N m N @

&CA

UVe\

K

Ne^N\b_€’

cH_dYVh;a_dOU¤Z€YVe _€’

g

N

m

NNY`N-eie!_

m N ;WEZ€Y#\i_#DFE y9Z€a

g

ZG'N-eieiN-a5xHN

m

—O,Z]\

m

UVkIH/&

"

$KJ

&

L$M& _€’ _ m g N m D eiW‹j

K

\ K Z]\

N$ & " $ & H &PORQS&UT +V

&

&UT W

J& )

3iy

6

 _ m

O_

m N g

N-\fZ]UVYVe_da\

K

UVe'Z]YVd_

m

UV\

K

Ž N m

N’?N

m

\^_

tdu i•

X

N\W‹e'Z]eie^WONQ\

K

Z€\ \ K N m UV

K \ §

O_dei\NUVdNa;”dZ]YVWNYG–

Ž

N“Z

m

N!UVa;\iN

m

Nei\^N

g

UVaG–;UVe e^NOU

§

eiUVOc‹Y`NZ]a

g

YVN\UZ

g

Na‹_d\iN

\ K N'j_

mim

N-eicH_da

g

UVaXe^cEN-j\

m

Z]Y=c

m

_onNj\^_

m

•\[2’]Z L^"

u– \

K

N-aI_

" Z +`a

Z

!

UVe Z]aNUVdNa;” Nj\i_

m

_€’b Z€eie^_5j-U¤Z€\iN

g

Ž

UV\

K

Y7•

X

N\We

g

N˜Ea‹N“\

K

N” Nj\i_

mdcL"

3feKEgZ 6`S

3heCEiZ 6; !

U’U3feEPZ 6L^"

uXZ]a

g

cN"

u_ \ K N m Ž

UVeiNd–‹\

K

Na

Ž N K

Z*”dN!\

K

NQ’?_ YVY`_

Ž

UVa~c

m

_ cE_ eiUV\iUV_da

j!…] lk# =ƒnm2„ m2 logprq:sutwvax/y{z|~}xd€

3h_

( - &

6L*x‚fƒ\x*xz

_

yza„#‚…vSxL†h‡ay{ˆ*x

- &

†ny{‚h‰†:Š\xn†

eiUVa

€

3f_

~(

-/&

6Œ‹

OUVa

9.>@

&=A

9

3hY 6\"

y

9

3:

6;c~ !

\fZ]a

€

3h_

~(*+6

3 6

ƒbvSx x=€

3f_

(+n6L‰Ž†K‚vaxNy{z{|}…xG*x‚fƒ\x*xz

_

‚vaxN†n‚fyn‚f‰5z{|Gx+ˆ‚h‘{= ’

j!…] = S“ˆ JK

N“” Nj\^_

md

joZ€a xHN

Ž m

UV\i\iN-a Z€e "•”

3f_ O—–

c6

– Ž UV\

KI”?^"

uXZ€a g – "

\fZ€a

€

3f_

(+ 6 •

X

N\˜

>•@

&=A Ž

UV\

K

˜M3fY 6"

yd–\

K

Na \K N”dNj\^_

m#™"

š

˜M3:

6; > - & Z€a

g

joZ€a xEN

Ž m

UV\^\iNa Z€e

—"

˜M3:

6

3h_ O›–

c6œ"

_ ˜M35

6c

•\QNajN

eiUVa

€

3f_

(*{6\"

UVa;’

'žMŸ

_/E  a!‹

–

˜M3:

6;c~ !

3?t

6

¡

(7)

_ cE_ eiUV\iUV_ a •`y~Neie^Na;\iU[Z]YVYVh ONoZ]aeQ\ Z€\!\ NXe^cEZ]jN-/& U`YVY7j_da;\fZ€UVa‘Na‹_dW UVa;’?_ OZ]\iUV_ a_da\ NXNUVdNa

”dN-j\i_

m

_P3U•Nd•e^UVa

€

3f_

(

-/&

6 u 6 U’ Ž

NjoZ]aj

K

_;_de^NZXcH_dYVh;a_dOU¤Z€Y

9

e^Wj K \ K

Z€\

9i>@

&=A (U9

3hY 6C"

yXZ€a

g

;9

3:

6;c

Z€e e^O,Z]YVYEZ€e cH_de^eiUVxYVNd•[2’bCUVe

g

U[Z€d_daHZ]YVU

¦

Z]xYVNd–€\

K

UVe

Ž

U`YVYxHN!cE_ eie^U`x‹Y`NQU’G\

K

N!cH_dYVh;a_dOU¤Z€Y 9Xm

NoZ€YVU

¦

Ne\

K N

OUVaUVOZ€kc

m

_ xYVNO

 

&CA

3 6U™"

OU`a

9%>%@

&CA

9

3fY 6\"

y

OZ]k

ž 9

3 6

3[P

6

Ž K N m

N

Ÿ

UVe'Zj_ OcEZ€j\ e^N\

Ž K

UVj

K

j_da;\fZ€UVae\

K

N~eicHNj\

m

WO_]’ Nk;jNc\YG•

‰ N Ž

UVYVY a‹_

Ž

’?_;jWe!_da \ K

Nei_dYVW\^UV_da _€’ \ K

NOUVaUVOZ€kbc

m

_dxYVNO›3[P 6 • X

N-\“W‹e|Z€eie^WON\

K

Z]\

™"

Ÿ C\

K N

j_ OcYVNONa;\ _€’

Ž

UV\

K m

Ne^cEN-j\\i_\

K

NNk;\iN-a g N g

j_dOcYVNkc‹Y¤Z€aN

Ÿ "

Ÿ 7?B

UVe'e^U`OcYVhXj_da‹aNj\^N g •

JK

Na

\ K N q

UVNO,Z€aaO,Z€ccUVa\

K

N_

m

NO {

– cG•w Na‹eiW

m

Neb\

K

NNk;U`e^\iNa‹jN_€’ Z’?Waj\^UV_da

"

3 6 Ž K

U`j

K

O,Z€ce

j_ a;’?_

m

O,Z€Y`YVh_ a;\i_

™"

Ÿ r\

K

N“N-k5\^N

m

UV_

m

_]’#\

K

N!WaUV\

g

UVei£

™" 7 > Ÿ (

"!

y B Z]a

K

UVj

K

efZ]\^UVe˜ENeQ\

K N

j_ a

g

UV\iUV_da‹e

3 6\"#

Z€a

g u !

YVUVO

%$'&

3

6

!

•)

3 (

6

p'e'Zj-_dae^N ;WNajN –=\

K N X

Z]W

m

Na;\'Nk;cEZ€aeiUV_da_]’

3

6

–Z€\U`a ˜EaUV\h5–

K

Z€e\

K

N!’2_

m O

3 6\"?”

O

”*)

O

”\

O

”!

!GO,+-+-+

Ž

UV\

K”/.

u )

3?{

6

JK

N!cH_dYVh;a_dOU¤Z€YcEZ

m \ 0 1

3 6\"•”2

13

1 1 O

”2 13

1 A 1 A O ) ))

O

”2 13

)

3?v

6

_]’&\

K N X

Z]W

m

Na;\Nk;cEZ€aeiUV_da_€’

3 3

6;6 1 " ” 2

13

1 1 O ” 2

13

1 A 1 A O )) )

O ” 2 13

) O ” 2

13

O

)) )

3f

6

UVejoZ]YVYVN g \K N 

Z]xHN

m

cH_dYVh;a_dOU[Z]Y;_]’

g

N

m

NN4 NaN

m

Z]\^N

g

x;h5!•

[a\

K

N'e^N ;WNY–;\

K N 

Z]xHN

m

cE_ Y`h;a_ OU[Z]YeG_€’

g

N

m

NN'4X NaN

m

Z]\iN

g

x;h6

Ž

UVYVYxEN

g

Na_ \iN

g

x;h 0 1 V

3 6 _ m

e^UVOcYVh 0 1

3

6

U¥’&\

K N m

NXZ

m

N!a_Z€OXxUVdWUV\^U¥Ne*•

_ Ž

YVN\87/3

6

xHN“\

K

N!UVa5” N

m

e^N!_]’ \ K

NQ’?Waj-\iUV_da

3

6

dUV”dN-aUVa 3(

6

Z]a

g

eiW‹ccE_ eiN' U`ej_ a5\Z]UVaN

g

UVa\

K N g

UVei£

9!;:

•

JK

Na {

–;cG•7you {

7<3

6

7/3

6

E=

" &

>

1?

) 0 1 3

6

1 T

3?z

6

Ž K N m

N\

K

Nj_ a5” N

m

 NajNUVe&WaU’?_

m O ’?_

m

Z€Y`Y

9@A:

• [abcHZ

m

\^UVjWY[Z m \ K

N’?Waj-\iUV_da67/3 67K

Z€e#Z]\GUVa;˜HaUV\h|Z

X

Z€W

m

Na;\

Nk;cEZ€aeiUV_da_€’ \K N!’?_

m O

7/3 6\"

– O—–

) O –

O – !

!LO )) ) Ž

UV\

K – " y

” )

3^you

6

š_dae^U g N m

UVa \ K N X Z]W

m

Na;\Nk;cEZ]a‹eiUV_daeG32z

6

Z€a

g

3^you

6

–=\

K N 

Z]xHN

m

cE_ YVh5a‹_dOU[Z€Y`ej*Z]a g Um

Nj\^YVh‘xHNXj_dOcW\^N

g

m

N-jW

m

e^UV”dNYVh’

m

_ O

0 )

3

6 " y 3^ydy

6

0

3 6 "

3 CE – ) 6;`

– 3^y

6

0 1

3 6 "

3 0 1 A

3

6 E 3– ) 0 1 A

3 6 O

)) )

Oi–

1 A 0 ) 3

66

E3 4NE y 6 – 1 A +6a`

– ( 4 @

3^yot

6

BDCFE G/H8I#JLK*MNI#OPHQISRTI#UWVXMKZYQVXM\[^]R_K*`aISbFbXVFc8M\I

X

N-\'WeQj_dae^U g N m \ K

NcEZ

m

\^U`j-WY[Z

m

joZ€eiN

Ž K N m

N6 UVe'\

K

NjYV_ eiN g g

U`e^£6dI3e

(Ff'6Œ" 7 > Ÿ ™

CEPe

‹AfSB

_]’#jN-a5\^N m e

Z]a g m Z g

UVWe f •

JK

Na\

K

NO,Z€c

7/3 6\"f

O e ( . y 3^yP

6

(8)

\ Z]ae ’?_ OeS&_da;\^_\ N#N-k5\^N UV_ _€’\dI3 e •Gš_dOcEZ UVad3^yP U`\ 3iyou Z€a 3iydy 3iy*t –*\ N Z]xHN cH_dYVh;a_dOU[Z]Y¥e

joZ€a,xHN

Ž m

U`\^\iNaUVa\

K

UVejoZ]e^N

0 1

3 6\"

CE=e

f

1 ( 4 @ u )

3^y ( 6

_ Ž

–dZ€eie^WON#\

K

Z]\&\

K N g

_dOZ]UVa UVe \ K NjYV_deiN

g

U`a;\^N

m

UV_

m

_]’E\

K

NNYVYVUVceiNœ3e ( W ('6

_€’Mj-Na;\iN

m

e€–*’?_;jUTe O W ( eME

W –

Z]a

g

O,Zn _ m eiN-OU

§

Z€k;U`e

–=\

K

Na\

K

NO,Z€c7 UVe'\

K

N d_dW‹£d_

Ž

ei£;U7\

m

Z€ae’2_

m

OZ]\^U`_ a

y]–=cG•&yoz v 7’

m

_ O _da;\i_\

K N

\ K N“N-k5\^N

m

UV_

m

_]’œ3e ( W ('6

dUV” Nax5h

7/3 6\"

O e O W !

y

( . y )

3^yo{

6

š_dOcEZ

m

UVa3iy*{

6 Ž

UV\

K

3iy*u

6

Z]a

g

3iy y 6§

3^yot

6

–;\

K N 

Z]xHN

m

cH_dYVh;a_ OU[Z]Ye j*Z]a xHN

Ž m

UV\i\^NaU`a\

K

UVejoZ]e^N

0 )

3

6 " y 3^yov

6

0

3 6 "

3 CEPe 6;`

3^y

6

0 1

3

6 "

CE e

0 1 A

3 6 E W !

! 0 1 A 3

6n(

4 @ )

3^yoz

6

JK

WeU’ Ž N g N˜EaN~\

K

NcH_dYVh;a_ OU[Z]Ye

1

x;h

) 3

6 " y 3 u 6

1 3

6 " y

W

1 0 1 3W O e

6(

4 @ y ( 3 y 6

Ž

N“N*Z]eiUVYVhXeiN-Nb\

K

Z]\\

K

N!cE_ Y`h;a_ OU[Z]YVe

1

eZ]\iUVe ’?h\

K

N\

K‹m

NN\^N m O m

N-jW mim

NajNX_€’ š K N-x5h;e

K

N”cE_ YVh5a‹_dOU[Z€Y`e

1 3

6\"

1 A 3

6

E

1 A ! 3

6n(

4 @ 3

{

6

Ž

UV\

K )

3 6\"

yZ€a

g

3 6 "

‹•

_ \

K

’?Waj\^UV_dae3iy-(

6

Z€a g 3

6 Z m N Ž

NYVYH£5a‹_

Ž

a’?_

m \ K

NU

m

_dc\^U`OZ]Y;c

m

_dcHN

m

\^U`N-e

Ž

U`\

KXm

Ne^cEN-j\\i_\

K

NQOUVaUVOZ]k

c m

_dxYVNO83[P

6

_ a

m

NeicHNj\^UV”dNYVh\

K

N~jU

m

jYVN

P Z€a

g

Z]e^h5Oc\i_ \iUVjoZ€Y`YVh_ a\

K

NNYVYVUVceiN y ( i•

BDC B G/H8I#JLK*MNI#OPHQISRTI#UWVXMKZc ]bD]`

 _ m

_ W

m

c‹W

m

cH_de^Nd–\

K N K

_ cEN!UVe#\

K

Z]\



Z€xEN

m

cH_dYVh;a_dOU[Z]YVe

K

Z*” N'Z]aEZ€YV_dd_ We#c

m

_dcHN

m

\^UVNe’?_

m

O_

m

NdNa‹N

m

Z€Y

g

_dOZ]UVaG•

JK

NQ’?_ Y`YV_

Ž

UVa

m

NeiW‹Y`\*–

g

W‹N\i_S;\Z

m

£dNZ]a g#

Z m

;Z t

i–;e K _ Ž

e\

K

Z€\\

K

UVeUVeUVa

g

NN g \ K

N!joZ€eiNbZ€\YVNoZ]e^\

Ž K

Na UVe

j_ a5” Nk=•

&œ; E…]œ prq:s†*†! /x‚vay‚

‰†4ˆ*‘{zax" ƒ ‰:‚…v

Y

^>

’ twvaxz‚vaxzS‘{ /y~}‰$#x+„ % y{*xU‡a‘~}'&~za‘( /‰:y~}†) 0 1

3 6L™"

0 1

3 6;`

0 1

3hY

6N†ny{‚h‰†+*&-,

  1 3

6Œ‹

O,Z€k

ž )

0 1

3

6 !

3fY

6 1 E y ‹

yŒE

./

2'0 3.1   1

3

6

34

@ y 6 3 t 6

S;UVajN

3hY 6 .

y –&\

K

N-_

m

N-O

•VyXe K _ Ž

eb\

K

Z]\~\

K

Na_

m

O,Z]YVU

¦ N g 

Z]xHN

m

cH_dYVh;a_ OU[Z]YVe 0 1

YVUVNZ]eih;Oc\i_ \iUVjoZ]YVYVh

xHN\

Ž

N-NaP 

1

3

6

Z]a gP

  1 3

6 – Z€a g \ K

UVeN-k5c‹Y¤Z€UVae!\

K

NU

m

aNoZ

m

_dc\^UVO,Z]YVUV\hd•N[a—eiNj\^UV_da P Ž

NXdUV”dNZa;WON

m

UVjoZ€Y

UVYVY`W‹ei\

m

Z]\^U`_ a_€’ \ K N_

m

NO 3

•Vy 6 •

 _ m

Z€\ YVNoZ€ei\eiW32j-U`N-a5\^YVhXY¤Z

m

 N4=– \

K

N!j_ aei\

m

Z€U`a;\_da\

K

N!j_ a5” Nk;UV\hX_]’ joZ€axHN

m

NY[Z]k;N

g

x;hZ]eie^WOUVa~\

K

Z€\

U`e_ aYVhX_]’&xE_ Wa g N g

xH_dWa g Z m h m

_ \fZ€\iUV_da t

[a—\

K

Nj_da;\^Nk;\b_€’ YVUVaNoZ

m

eih;ei\^NOe*–&\

K

NXj-_da

g

UV\^U`_ aPY

^>

U`e

m

NcY[Z€jN

g

x;h\

K

Nj_da

g

UV\iUV_da—u

^>

Ž K N m N5

UVe“Z

g

_dOZ]UVa\

K

Z]\Qj_da;\fZ€UVae'\

K

Nbe^cEN-j\

m

WO _]’œ•=[\QU`eQ£;a_

Ž a \ K

Z]\'\

K

Nj_da;”dN-k5UV\h Z]e^eiWOc\iUV_ a,_ a Z]a g \ K N

j_ a

g

UV\iUV_da—u

^>

™Z mN g

U$2j-WYV\“\^_

m

N-j_daj-U`YVNd•

JK

NeiUV\iWEZ€\iUV_da—Z]c‹cENoZ

m

e\i_xHNX’Z*” _

m

Z€xYVNUVa—\

K

NXj-_da;\iNk;\_]’\

K N

NUVdN-a5”dZ€YVWNc

m

_ xYVNO \ K Z]\UVa5\^N

m

N-ei\ieQWexENj*Z]We^N\

K

N!e^N\8 UVe Z|cE_ Y`h; _daXj_ aei\

m

Wj\^N

g

Z]e\

K

NQj_da;”dN-k

K

WYVYE_€’G\

K N

eicHNj\

m

WO _]’ \ K NXO,Z€\

m

UVk#H/& Z€a

g

Nk;jYVW

g

UVa\

K N m

UV

K \ §

O_de^\”dZ]YVWNGYG–=e^_\

K

Z€\“\

K

NXj_ a

g

UV\iUV_daY

^>

joZ€aa_ \

c m N”dNa;\Q\

K

N!eiN-\ ’m

_ O xEN-U`a‹Xj_da;”dN-kM•

Références

Documents relatifs

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST

Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST