HAL Id: inria-00073536
https://hal.inria.fr/inria-00073536
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.
A Simplex-Like Algorithm for Interval Linear Systems
Olivier Beaumont
To cite this version:
Olivier Beaumont. A Simplex-Like Algorithm for Interval Linear Systems. [Research Report] RR- 3153, INRIA. 1997. �inria-00073536�
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
A Simplex-Like Algorithm for Interval Linear Systems.
O. Beaumont
N ˚ 3153
Avril 1997
TH `EME 4
6879;:=<>?A@CBEDGF
HI%J
KML;KON;PRQ=STLUWVYXGZ[S]\_^.KZ(\a`WZbSTL;SYc[XGZbST\a^
d
K,cbe=cfZ
J
KL&KMcg\_L&`WV]KhiKMc
jlk
\amnKZpo(VYX
d
ST^
q
X!`r`\
k Z d K k
KMg I K k g I
Ks^taurvxw!uyPzo({
k
STV|vM}_}~PvM`rXaaKMc
2 'MGWaay
^yZ
I
SYc+`rX!`K kx
Kc I \
sI
\
SZ,SYc+`\_c[cfSTWVTK&Zb\ UrcbKg\_^{_Kh`\aVTe I K d k
X\
k
cb\aVT{=S]^r
VTST^WKMX
k
ST^ZbK
k
{GX!VcfeicfZbKML&c
S]Z
I
\_UiZOUcfST^W2`
k
Kxg\a^
d
SZ[S]\_^W^WST^WrOK
k
cfZOc I \
I
\ Zb\
d K k
ST{aK
k
\aL
Xa^AK^gVT\_cbU
k
K2\!pOf p¡£¢M¤[¡¦¥
X§`\aVTe I K d k
\_^
pI
STg
I
g\_^_ZX!ST^rc+Z
I
K.g\a^={_Kh
I
UrV]V¨\!(Z
I
Kcb\aVTUiZ[S]\_^
cbKZx
HI
KM^
X©cfSTL&`WVTKh=ª«VTST¬aK;L;KZ I \ d
K^rXaWVTKMcOUrcOZb\r^
d
X^WK
\aUiZ[K
k
ST^rgVTUrcbS]\_^% \
k
KM\G{aK kM
ZI K
g\a^rcfZ
k
X!ST^Z[c¨\aWZ[X!ST^WK
d
LX®eK+UrcbK
d
Zb\g\_L&`WUiZbK,Xa^ST^W^WK
k
ST^rgVTUrcbS]\_^\a¨pOf p¡£¢M¤[¡¦¥
¯ °±C²[³´ Gµ'a
ST^ZbK
k
{GX!VV]ST^WKMX
k
cfeicfZbKLc
cbS]L&`WVTKhXaV]_\
k
SZ I L
¶Y·;¸
¹bº«»x¼
¸
¹¾½'¿Tº£À«ÁÂ
ÃĦŮÆÈÇÈÉÇ«ÊMÇ'Ë®ÌlÍ%ÌnΣϮÌnЫΣÏxÌlÌnžĦÅxÑTÒÓÐÈÔ(Õ[Ç«É×ÖÊ®ÌÌØÇÙÚMÆÈÇfÛÌfÔpÌnÆ%Ü'Ý£ÞÌfÕ[ÇÈÒÓÉ×ЫÌnÆ%ßÌnÔ(ÕÉ×ÝYàÒÓá!ÌfÕÊ®ÔpÒÓÅ!â'É×ЫÉ×Æ«ÕxãÑTÐ
Unit´e de recherche INRIA Rennes
IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) T´el´ephone : (33) 02 99 84 71 00 – T´el´ecopie : (33) 02 99 84 71 71
1 021 3 (1#M$(*,|(1#*() M$;'()+*,M(15
° °
X!^rcgKZ2X
k
Z[STgV]K
^W\_UrcL&\a^Z
k
\a^c;g\aL&L&K^ZS]V3KMcfZ&`\_c[cbS]WVTK d
UiZbSTVTSTcbK k d
Kxc&`\_V]eª
I%J
K d k
KMcg\_^{_KhiKMc`\_U k;k
Kxcf\_U d k
KV]Kxc&cfeicfZ
J
KML;Kxc,VTS]^
KMXaS
k
KMc d
ST^ZbK
k
{GX!VTVTKMc;cbXa^rcUWZbSTV]SYcfK k d
K`
k
Kxg\a^Wª
d
SZ[S]\_^W^WKL&KM^_Zx3\aUrcL&\a^Z
k
\a^rcOZ[\aUiZ d
Xa\
k d
g\aL&L&K^Z,Z
k
\aUr{aK kx¨J
X.`X
k
ZbS
k d
UW^AcfU
k
ª£K^rcbKLWVTK
d
K Of p¡£¢M¤[¡¥
UW^8`\_V]e I%J
K d k
KUWS g\a^Z[S]KM^_Z+V
KM^rcbKLrV]Kcb\aVTUiZbST\a^(\aUc+UiZbSTVTSTcb\a^rc+XaV]\
k
cOUW^AXaVª
_\
k
SZ
I
L&K
d
K2Zne=`K.cfSTL&`WVTKhiK`\aU k d
KZbK
k
L&ST^WK
k
UW^-^W\aUW{_KMXaUAcfU
k
ª£K^rcbKLWV]K
d
K2VYXcf\_V]UiZ[S]\_^%
K
`rV]Urc
V]Kxcg\_^Z
k
X!ST^_Z[KMc\_iZbKM^UrKMc,`KUr{aK^Z KZ
k
K&UiZbSTVTSTc
KKxc+`\aU k d
KZ[K
k
L;ST^WK
k
UW^Acb\aUrcfª«KM^rcbKLrV]K d K
pOf p¡£¢M¤[¡¥Ó
!s´ x ² #"°
cbeicnZ
J
KL&Kxc VTST^
KMXaS
k
KMc d
ST^_Z[K
k
{GX!VTV]Kxc
XaV]_\
k
SZ
I
L&K
d
U cbS]L&`WVTKhiK
u
1
"%$;'| ¨;$
o VT\!Z\!
\ k ¬ I
XacOKK^
d
\a^rKST^y\
k d K k
Zb\.cf\_V]{_KST^_Z[K
k
{GX!V'V]ST^WKMX
k
cfeicfZbKLc; (¡3254 ¤[¡(6®¡£
q \ I ^
Xa^
d87 k
KST^W\G{=STg
I
]v:9®¡
Na¡
I
X_c`
k
\G{aK d Z I
X!ZlZ
I
KOgMX!VYgUWVYXGZ[S]\_^\!pOf p¡£¢M¤[¡¦¥
Z I
KOcfLX!VTVTKMcfZ \®h;Z
I
XGZ
g\a^Z[XaS]^c3X!VTV'Z
I
K&cb\aVTUiZbST\a^rc¾\!3 p¡;2<4 ¤¡
SYcOX j ªI X k d ` k
\_WV]KML.>=¾^§Z
I
K;\!Z I K
kOI
X!^
d
cfKM{aK
k
XaV
XaV]_\
k
SZ
I
Lc¨\aiZX!ST^_\\
d k
KxcfUrVZc
Kxcf`KMgSTXaV]VTe
pI
KM^Z I K d
SYX!L&KZ[K
k
\a p¡SYcpcbL&XaV]V ®¡
6!¡«
^Z
I
SYcl`X!`K kx_
K3`
k
\a`\_cbK(X,^rK
X!VTa\
k
S]Z I L
pI
SYg
I
SYcrX_cfK
d
\a^VTST^WKMX k ` k
\a
k
XaL&L;ST^W
Zlg\a^rcbSTcfZ[c
\alX!^ S]ZbK
k
X!ZbST{aKc[g
I
KL&K
pI
STg
I
g\a^cfS d K k
c(X_c3ST^W`WUiZX!^§KM^rgVT\_cbU
k
K,\!|Z
I
Kcb\aVTUiZ[S]\_^ \!(Ob (¡«¢x¤¡¦¥
Xa^
d k
KZbU
k
^rcpX.UrcbUrXaV]VTe2KZbZbK
k
¥¨K^gVT\_cbU
k
K_
HI
STc;X!VTa\
k
S]Z
I L g\_^={aK
k
aKxcZb\
X k d
XcbUW`K
k
cfKZ\apZ
I
K.g\_^{_Kh
I
UWV]V\apZ
I
K©UW^WS]ZbK
d
cb\aVTUiZbST\a^cbKZ
Ob (¡«¢x¤¡¦¥
^©Z
I
STc`X!`K kxW
KUrcbKOZ
I
K\_V]VT\
ST^W&^W\!ZXGZbST\a^c%?
Ob (¡«¢x¤¡¦¥@4BA:2 ¢DCi BE8 (¡«¢CW¤E8¤¡«¢( 2F4 ¤HG
I
f p¡£¢M¤[¡¥J4LKMi£Of p¡£¢M¤[¡¥f¥Ó¢
pI
K k K>KM
d
KM^W\!Z[KMcZ
I
K,g\_^={aKh
I
UWVTV
K,X!VYcb\
d
KM^W\!Z[K+=e§Ob (¡«¢x¤[¡¥ Z I K+S]^ZbK
k
{GXaV
I
UrV]V%\a Z I
K,cb\aVTUiZbST\a^%
HI
K3ST^_Z[K
k
{GX!VV]ST^WKMX
k
cbeicnZ[KLc
K¾g\a^rcbS d K k
S]^Z
I
SYc `X!`K k X k K d
K^WK
d
=e;Z
I
K3\_V]VT\
S]^r,^W\!ZXGZ[S]\_^rc%?
p¡N4 PO5QE ,¢[ SRTQ% p¡
¤[¡)4¤ ¢U ¤b¡V4¤WOXQ ¤®¢[¤YRSQl¤[¡£¢Ql¤Z4 U ¤@O ¤
[
\ ]
^ `_$ * ;Ó0. |;$ |.*O p$;*,$.1 |. p;$&)
|. $ 1' |;$ 1 p
ZSYcl¬=^W\
^Z
I
XGZ¨Z
I
KOcb\aVTUiZbST\a^2\!EZ
I
K¾`
k
\arV]KML (¡32F4s¤¡SYc¨S]^aK^rK
k
X!Vr^r\!Zpg\a^={_Kh%6x¡£|o3c X k Xac
KX
k
K3\a^rV]eST^ZbK
k
KMcfZbK
d
ST^pOf p¡£¢M¤[¡¥
W
K¾LX®eUrcbK
I
f p¡£¢M¤[¡¥¨Xac¨S]^Z[K
k
L;K
d
SYXGZbKcbKZM@=¾KZfZbVTS%X!^
d
jlk
XaaK
k
×~M¡`
k
\G{aK d Z I
K\aVTV]\
ST^W;Z
I
K\
k
KLa?
bdc °=´
° fe
gCW hE8 p¡£¢iCr¤E8¤[¡«¢¾ D2j4 ¤ ¥kml D2nO ¤lpoPQE /l2qlHRTQ ¤
HI
STclKhi`
k
KMc[cfST\a^
d
\=KMcl^W\!Z¨V]KxX dd
Sk
KMgÓZ[V]eZb\X,`\aVTe I K d k
\a^
KxgXaUrcfK3\!CZ
I
KOX!rcb\aVTUiZ[K({GX!VTUWK
pI
SYg
I
Ur^
d K k
VTS]^WKxc¨Z
I
KX_gÓZpZ
I
XGZpZ
I
K,cf\_V]UWZbST\a^.cfKZSYcpUrcbUrXaV]VTe^W\aZ3X;g\a^={aKh2cfKZ6G¡«
r
KZUc%Xac[cfUrL;KZ
I
XGZ
K¬=^W\
X!^KM^rgVT\_cbU
k
K|\!EOb (¡«¢x¤[¡¥
®pI
SYg
I
LX®e3Kl\aWZ[X!ST^WK d
\
k
ST^rcnZX!^rgK
=eZ
I
K,X!VTa\
k
S]Z I L ` k
\a`\_cbK
d
e
q
UWL&` !¡«|K,aST{aKX;
k
cnZ
k
KMcbUWV]Z
pI
STg I ` k
\G{=S
d
KxcpX
X®eZ[\&aKZ
k Sd
\a Z I
K,X!rcb\aVTUiZ[KO{GXaV]UWKxc
s ° AXe "t
Ob (¡«¢x¤[¡¥vu 2 ¢ U 2r¡
N
x y
xi
__ xi
__
S]_U
k
K;v?Q=SZ[UrXGZ[S]\_^
pI
K^.\!Z I 2 X!^
d U
2 X k
K¾`\_cbSZ[S]{_K
'# J
4 lU
2 lO l2 l
U
2DO 2
'+#
4 U
2 l2 l+O 2 lU
2 l
U
2ZO 2
!
2
%#+.!->
O
*+ #%# {
2
i '+&+
2
E2 ¢ U 2
¡£¢Hl2 lo
2 R
0
´%´
Sk
cnZ3gMXacbK ? U
2 o 9 ZI
K^ 4 O+va¢
D4L9
Xa^
d
%2 R
4 OZ2F4 l2ql
Q=KMg\a^
d
gMXacbK ?2
9 Z I
KM^
4v_¢
D4L9
X!^
d
%2 R
4 2F4 l2ql
HI
Sk d gX_cfK?
U
2%2
o 9 Z I
K^ 4
!#"%$!
"
! "'& ! " ¢ 4
&(
!")!
"
! "*& ! "
+
S]Y9 o-, o U 2 Z I
K^ ,R
O l,l4
&( ! "
!"& !
"/.
U 2 O0,¥
9W
Z I Urc
l,)lpo ,R
+
S]2
o-, o 9 Z I
K^ ,R
DO l,l4 (
! "
!
"*&
! " .
1, O<2
¥ 9W
Z I Urc
l,)lpo ,R
32T
^©X_gÓZ
cbK{_K
k
X!VEcbSZ[UrXGZ[S]\_^rcLX®e\iggU k ?
s ° A465 0*+#&( 7 t -%%98
4 At2 ¢):W¥<E<;
(
¢n2 E 2 ¢ U
2¡£¢=: o l2qlG
;- t
+ ,: *+ #0?>@#*A@%,B
4hA=t2¢%:W¥Ó¢2 E82 ¢ U
2 ¡£¢ 9 oC:Fo 2 R GD2
w
xi
__ xi
__
x y
! "#$&%! "
' ( *)+
xi __
x y
xi __
,*+ -./0 0! "#1324$&%!
"5
2
6
´%´
2
9\
k U
2>oS9
Z I
KM^
8 4 B
X!^
d Z I
K+`
k
\_`K
k
ZneSYc¨Z
k
ST{SYX!V«
K,Xac[cbUWL&KO^r\
ZI
XGZZ2
o 9/o
U
2a
HI
K3cfKZ
B
apI
SYg
I
SYc
d
Kr^WK
d
=eV]ST^WKMX
k
ST^WKUrXaV]S]ZbSTKMc
SYclg\a^={aKh
X!^
d Z I
K3`
k
K{=ST\aUrcVTKL&LX+STL;`rV]STKMc
8 u B
HI
K k K\
k K
KMi
8
¥Zu B
q
KMgST`
k
\igX!VTVTe
V]KZ(Urc(g\a^rcbS d K k
¢[¤¥ E A=t2 ¢):W¥Ó¢Z2 E82 ¢ U
2 ¡«¢ :Fo 2 R G
8
r
KZUrcg\_^rcfS d K k
^r\
Z I
KVTS]^WK : 4 2 R «¤O i¥
ipI
SYg
I
ST^ZbK
k
cbKMgÓZc: 4P2 Xa^
d : 4 OZ2.XGZZ
I K
`\aST^Z[c+!¢[¤Ó¥@4
&"
&"
nv_¢vx¥¨Xa^
d
(
¢¤
(
¥@4
&"
$ "
O+v_¢v®¥
k
KMcb`KxgÓZ[S]{_KVTea
QiS]^rgK&¤DO o
%
K\_iZ[XaS]^ 2
o o
( o U
2 X!^
d Z I K k
K\
k K
¢¤ ¥>E 8
(
¢¤
( ¥ E 8
Xa^
d
¢[¤¥ E KMi
8
¥Ó
K
I
X®{_K¾`
k
\G{aK d Z I
XGZKMi
8
¥W4
B
2]
HI
K k K\
k K
C
K;gMX!^Khi`KxgÓZOZ
I
X!ZOZ
I
K\aVTV]\
ST^WcbSTL;`rV]Khf p¡£¢M¤[¡«¢x2 ¢ U 2¡¥
k
K`
k
KMcbK^Zc¾Xa\=\
d
Xa`W`
k
\®hiS]LXGZ[S]\_^\!
I
f p¡£¢M¤[¡¥Ó
K d
K^W\aZbKOe
Z I K d
STXaa\_^rX!VL&X!Z
k
Sh
pI
\_cbK
d
SYX!_\a^rXaVKM^_Z
k
STKMc
X k K¾Z
I K
c3X!^
d ZI
K+{_KMgÓZ[\
k
\!'Z I K
cZ
I
K^
K d
Kr^WKb (¡«¢x¤¡«¢x2 ¢ U
2r¡¦¥X_c¨\aVTV]\
c:?
f p¡£¢M¤[¡£¢M2 ¢ U 2¡¥
D28O5QE
2 o U¤YRSQE
D2 RSQE
2 ¤ O5QE
K,gXa^^r\
K^=UW^gSYXGZbKOZ
I
KZ
I
KM\
k
KL \a Z I SYcp`rX!`K
k
bdc °=´ ° 4 !
Ob (¡«¢x¤¡¦¥ u 2 ¢ U 2r¡
0 5 %#
'+&+
Ob (¡«¢x¤¡¦¥ ugA:23 D28O5QE
2 o U¤YRTQ%
¢O D2 RSQE
2 ¤ O5QE
G !
4! " & ! "
! "'& ! "
'+#
4
! " ! " & ! "
! "
! "'& ! " 0
´%´
?
HI
SYc¨Z
I
KM\
k
KL SYc(X d Sk
KxgÓZ(g\_^rcfK UWK^rgK\!
HI
K\
k
KML#v,X!^
d r
KML;LX©vap
K
I
X®{aK;cfKMK^8X!\G{aK I \
Xa^yKM^rgVT\_cbU
k
K\!OOb (¡«¢x¤[¡¥OVTKMX
d
c3Z[\ XcbS]L&`WVTKhZ
I
X!Z
d
KMc[g
k
S]KMc
X8cbUW`K
k
cfKZ&\a
I
f p¡£¢M¤[¡¦¥
HI
K§cbKZ\!`
k
\aWVTKLc LXGh
!! #"%$&('*)+$
'*)+$! )
!,'.- 2
X!^
d
L&S]^
!!! "%$&('*)+$
'*)+$! )
!/'+- 2
gXa^
K.cf\_V]{_K d
\
k
S]^rcfZ[Xa^rgK
=eAX!`W`rV]e=ST^W
[0
ZbSTL&KMcZ
I
K.cbSTL;`rV]KhAL;KZ I \ d [
¡«
HI
K k K\
k K
X^WK
KM^rgVT\_cbU
k
K¾\a¨pOf p¡£¢M¤[¡¥¨STcp\_iZ[XaS]^rK d
Xa^
d
Xa^.SZ[K
k
XGZbST{aKc[g
I
KL&K+gXa^©K
d
KM{aKVT\a`K d
HI
K VTS]L&S]Z|\!rZ
I
SYc S]ZbK
k
X!ZbST{aKX!VTa\
k
S]Z
I L SYc'UrcfUX!VTV]e,X¾a\=\
d
KM^rgVT\_cbU
k
Kl\! Of p¡£¢M¤[¡¥¨^W\aZ'Z[\=\¾VYX
k
aK®¥
X_c¾c I \
^ ST^§Z
I
KVYXacfZOcbKMgZbST\a^%21(^W\
k
ZbUW^rX!ZbKMV]e
S]ZSTc
d
S43gUWV]Z¾Z[\r^
d
X2g I X k
XagÓZ[K
k
S45xXGZ[S]\_^©\aZ
I
STc
VTSTL;S]Z(ST^©\
k d K k
Zb\cfZbU
d
eSZc(XagMgU
k
Xagea
HI
K+VTS]L&S]Z(STcpUcfUrXaV]VTe^W\aZ(KUrXaVCZb\.Ob (¡«¢x¤[¡¥Ó
\ k
ST^rcfZ[Xa^rgK
rpI
KM^
K+¬=^W\
Xa^©KM^rgVT\_cbU
k
K+\!pOb (¡«¢x¤¡¦¥(S]^
pI
SYg
I
KMX_g I 2
¬aKKM`rc(Xg\a^rcfZ[Xa^Z
cbSTa^
S]Z2STcKMXacbeAZ[\A`
k
\G{aK©Z
I
XGZZ
I
KX!VTa\
k
S]Z I L d
KMc[g
k
STK
d
X!\G{aK.`
k
\G{S
d
Kxc;Z
I
K§KhWXagÓZ2cb\aVTUiZ[S]\_^
pOf p¡£¢M¤[¡¥X!¦ZbK
k
\_^WVTe \a^WK.cnZ[K`%
^ d
KK d
ST^-Z
I
SYc&gXacbK
Z I
K cbKZ[cFA=t2¢%:W¥Ó¢ 2 E 2 ¢ U
2¡£¢ : o
~
%2jR
G X!^
d
At2 ¢):W¥Ó¢ 2 E 2 ¢ U
2¡£¢ :Po l2qlG©X
k
KKUrXaV
Xa^
d
b (¡«¢x¤[¡£¢M2 ¢ U
2r¡¥STc;X!^AKhWXagÓZ
k
K`
k
KxcfKM^_ZXGZ[S]\_^2\! Z I
K,cb\aVTUiZ[S]\_^©cbKZ(\a¨ p¡;2j4 ¤[¡
ipI
SYg
I
SYcST^Z
I
SYc(gX_cfK+g\_^={aKhE
^Z I K \_V]VT\
ST^W¾cbKMgZbST\a^
!
K¨`
k
KxcfKM^ZX¾L&\
d
SgMXGZ[S]\_^\aWZ
I
KpXaV]_\
k
SZ
I
L`
k
KMcbK^ZbK
d
X!\G{aK
S]^;\
k d K k
Z[\&`K
k
\
k L S]Z(ST^.X
k
KMX_cf\_^rX!WVTKX!L&\aUW^Zp\!'ZbSTL&Ka
Ó ! |. ($(.1'$ 5
^.Z
I
K`
k
KM{ST\aUc(cfKxgÓZ[S]\_^
r
K I X®{_K+`
k
KxcfKM^ZbK
d
Xa^ SZ[K
k
XGZ[S]{_K+L;KZ I \ d
pI
STg
I
cb\aVT{aKxcpS]^Z[K
k
{®XaVVTS]^rKMX
k
cbeicnZ[KLcM
HI
STc3L&KZ
I \ d kK _UrS
k
KMc(Z
I
K,KhiKMgUWZbST\a^§\a
[0
cfSTL&`WV]Kh.XaV]_\
k
SZ
I
Lc d U k
ST^WKxXag
I
cnZ[K`§\a
Z I KXaV]_\
k
SZ
I
L.Q=ST^rgKX!^KhiKMgUiZbST\a^A\apZ
I
K©cfSTL&`WVTKhAX!VTa\
k
S]Z I L k
KUWS
k
Kxc
k
\aUW
I
V]e&
0
¥\a`rc
[
¡v [ ¡ ZI
KZ[\!ZX!VXaL;\_UW^Z(\!
\ k
¬`K
k
S]ZbK
k
XGZ[S]\_^SYcpZ I K k
K\
k
K\a'\
k d K k
&
0
¥Ó
HI
K,X!VTa\
k
S]Z
I
L&c
` k
\_`\cfK
d
=e
q
UrL;` G¡ Xa^
d
(KMUWLX!STK k 6G¡
k
KUWS
k
KX!^ XaL;\_UW^Zp\!
\ k
¬\a'\
k d K k w 0
lyK,c I \
ST^©Z
I
SYc(cbKMgZbST\a^
I \
Z[\;`K
k
\
k L X&cfZbK`.\a Z I
K,X!VTa\
k
S]Z
I L ST^©ZbSTL&K &
0
¥Ó
K.`
k
\a`\_cbKXyZ IWk
KKª£cfZbKM`-XaV]_\
k
SZ
I
L/S]^ \ k d K k
Z[\\_iZ[XaS]^ X!^-\aUiZ[K
k
S]^gVTUrcfST\a^
HI
K
k
cnZcfZbKM`
g\a^rcbSYcnZcpS]^ycf\_V]{=ST^W2X!^§Xa`W`
k
\®hiS]LXGZ[K+`
k
\aWVTKL.
HI
Kg\a^
d
SZ[S]\_^rcpZ
I
XGZ
d
Kr^rKKh=Z
k
KML&XaV`\aST^Z[c(\a
Z I K+X!`W`
k
\®hiSTL&X!ZbKO`
k
\aWVTKL X k
KcbUW`W`\_cbK
d
Zb\&K+gV]\cfKOZb\;Z
I
\_cbK
pI
STg I d
Kr^WKKh=Z
k
KLXaVE`\_S]^Z[c\a
Z I
KKhWX_gÓZ`
k
\_WVTKL.
HI
KcfKxg\_^
d
cfZbKM`g\a^rcbSYcnZcS]^UrcbS]^W©Z I K k
KxcfUrVZc\!lZ
I
K&
k
cnZ,cfZbKM`yZ[\ cf\_V]{_K
Z I
KOKhWXagZ¨`
k
\arV]KML©
HI
KOVYXacfZcfZbKM`2SYc¨X;g\
kbk
KMgZbST\a^cfZbKM`
X!^
d
V]KxX
d
clZb\&X`
k
\G{_K
d
\_UiZbK
k
ST^rgV]UrcbST\a^%
^\
k d K k
Zb\&X!`W`rV]e&aKM^WK
k
X!VZ
I
K\
k
KLc \!%VTS]^rKMX k ` k
\a
k
XaL;L&ST^W
K3`K
k
\
k L Z I
K¾\aVTV]\
ST^Wg
I
X!^W_K
\a {GX
k
SYX!WVTKMc
pI
SYg
I
S]^
d
UrgK+UrcbK+\!'`\cfS]ZbST{aK{GX
k
SYX!WVTKMcMyK,cbKZ : 4 28O<2 ?
f p¡£¢M¤[¡£¢M2 ¢ U 2¡¥
/:/OXQE
:8o U¤YRTQ%
O È POXQ%
¥2 4 U¤
/: RTQE
: ¤ OXQ%
O È SRTQ%
¥2 4 ¤
: 9
HI
K¾
k
cfZ3cnZ[K` g\_^rcbSTcfZ[c¨ST^.cb\aVT{=S]^rZ
I
K+`
k
\aWVTKL /: 4 ¤ ¢ ¤U
¡£
HI
K;`\aVTe I K d k
\_^
pI
SYg
I
g\
k[k
Kxcf`\a^
d
c(Z[\©Z
I
K&`
k
\aWVTKL 9: 4 ¤¢ ¤U
¡lgMX!^Kg\_^rcbS d K k K d
X_cOX
L&\
d
S]gX!ZbST\a^©\!'Z
I
K,cbSTL;`rV]Kh
pI
SYg I d
Kr^WKxc f p¡£¢M¤[¡£¢M2 ¢ U 2W¡¥?
/:Fo U¤!
/:
¤
: 9
¢(X!^
d
b (¡«¢x¤¡«¢x2 ¢ U 2¡¦¥
9:dO5QE
:8o U ¤!
9:RSQE : ¤
: 9
o¾cVT\a^WXacWQ% 4SYclcbL&XaV]Vg\_L;`X k K d
Zb\;
K(LX®e;Khi`KMgÓZ ZI K3\aVTV]\
ST^W,cbSZ[UrXGZ[S]\_^
\aiZX!ST^WK
d
\
k ?
L4#"
v O+v
v v $
¢QE 4 " v v
v v%$
¢[¤Z4'&
" v
v($
X!^
d
Q&¤Z4 " v
v%$
x2
1
-1
1 2
x1
! "$#%&'
()*+%,-., /102 3-45%-6 7+ 3-. ,8#9 .-. ;:< .=0> 4 .*+ ?,8 .@A>-6B,8CBC,DA2E<
"F+G 6H I+ J 6KL 4M?F#<B'NC,.O
()&APB-QR0> JSD,8B'TB .N 4'UV+ #: .MS, MSWK
XZY[\]^
_M3
XWY[\Q]^
:DU` . 4M? + ;, 6
Aa' 4bN,c+<WD I,W V0R+ 4,TN7 4-C+#: .MdMSWK
XZY[\ ] ^
eM?'
XZY[\ ] ^ Ogf + ,8
4,', 0R'a:< J, 4&? 6KLG,8 .#<,.O
h ##9B, I+WD02 JiLW0kjlPj
^$mn
Q/&##ZKLMSW'B`Ao+ JLT 4, IAqprO
s5 .WT 44/L 6ptl
uvvvw
x2y
xcz
OOO
xc{
|.}
}}
~
5pa&l
uvvvw
3x y
3x>z
OOO
3x>{
|.}
}}
~
/<+ ;W0t#8'B&Aq+ ;MS-. 4,Gp
FpE O
( AVj0> 4 + .KB-Q*LT 4, AVp;/70> 0>BB:'a/;0R+ .##UL'ke+ bB<=U
p ]
lkQ"F0R+ 4 ;Q"FQ"
"
]
lj3Q"F0R+ 4 ;Q"FQ"
"
f +L,4/N'VB .grM?''M?.
] ^ /LAaj
^Pmn
02 G+ZTB 7-.,% .>6"
n l
"n
S+ @-6 4,#<B
-.,8CN>Dx
nQ] n O( A<j
^$mn;
/02 >+<ZT @-.,% .Q"
n
lkQ"
n
;+ R-6B,8CBqx
n.] n O
:LTL,'UB/L##9B, @-C+%-6 4, ##UV0R+ .-6B, 47+ J,U,= .MB,,-6%W S7+ M?'M?'W
'BA
]B^
O
h '-. Ij,2'U?`##ZK'MSW'B?AF+ ILT .C, DAapr/B+ G 4,',>:C 4` D ,8 B/B:4/
B,c02 I,+'F, . G+ @ 6KLR,8 -Q'B,./L02 GB'US<,8 G+ 4,C,cB:' 4?A$BR " B,c%-.,2