!" # ')(+*-,&.
:
; (<.
$
%
# &
/0,01 2436587972
=?>
@ ; (A1B,C>
D =EF,01HG"=JI GK> EL=9MN,0O P1Q(<.
R ; =JSCEF,&1 V
T U (&P EF,W?E
X Y\[^]&_`Ya]&[b?cedgfihkj<]&l"m8Y\noh8p q hcJb?lsr ntp q u
Z
m [ntpbCvxwybz[{d
[bNr?Y\[
nm&l|m&p9c }~[bNr€‚ƒmY„b?[‚X…kr?Y„b?† r 3-@ ; ‡1 GˆSk=-‰ : Š IŒ‹
 =9S Š 1
MŽ=e1‚=9O-‰  =9IR =?>g ,0I‘Iˆ3“’”‰–•—3‚˜ I Š  ,&š
1 ™›TVUJUœ“€•
ž
]8Y\ltn‘pbeŸ
T8•
“(A1¡
=`1¢‰
5~•ƒ¥‡OC*-,&1¦>§,
2~•
’~W+=9.
£ 1B(/9I¤,&.
>
: ((<I¨>
S9I¤,>
© • RZ(O-ª+I Š >ŽG«(<Oz>
¬
­”4‡®
A„
¯
, =`1Q, °‚Gs*u,0O = . (‰J,&I =9Ou‰
G¨E±> Sk=`1²=9. ,EF,01¦>C3
P³1Q(.
 ; G´ª ;  , ªu=JIµª Š I´=EF, E ; , >§(<I Š E¶G«(O3· ; G´ª ;
GˆO E ; Gµ> ‰Gµ>§ª Š >F>ŽG«(O GK> = S0(<S Š I´=E¶G´(<O P Š O-ªŽE¶G«(O (&P
>§(<. , >§(A1—EC•
Gµ>
: ¸ S0G´ªu=JI‘I ¸ 3 = PH(A1¡ =`1¢‰ Se1Q(</9I¤,&.
 ,0I‘IŒ‹NS0(0>§,?‰43 G¹•o,9•¡3 ; =z> = Š O0G´º Š , >§(I Š E¶G«(<O  ; G´ª ;
*k=e1HG´,> ªC(OCE¶GˆO Š ( Š >|I ¸  G¨E ; E ; , Sk=`1²=J. ,EF,01»> (&P
E ; , S`1Q(</9I¤,0. • : ; 1Q,¼, >gEL=9Ou‰A=`1¢‰ ,€W+=9. S9I¤,>?½
¾6¿ À #Á ­”4
! Ã< À Ä # €
šÅ # €A
€# ÆVÇ £–(S€‹
Š IÈ=E¶G«(O GK> °1Q( Š S0,z‰ =uª?ªC(1¢‰‚GˆO-° EF( = >gE1 Š ªŽE Š 1²=JI
S`1Q(S&,01¡E ¸ > Š ª ; =?> =u°J, (1  ,kG´° ; E GˆOCEF(
‰GK>‹
m
ª+1Q,EF, >gEL=-°,C>?•
£–(S Š I´=CE¶G«(O
GK> ‰J,EF,&1
. GˆOV,z‰
=E
‰Gµ>§ª&1Q,EF, E¶Gˆ. ,>
=9Ou‰ S&(S Š I´=CE¶G«(O
tj , j = 0, 1, . . .
(+P E ; ,
‹\E ; >gEL=-°
, =CE E¶Gˆ. ,
GK>
• ¥ÉE Gµ>
j
t
n (t )
=?>F> Š . ,?‰ E ; =E GˆO-‰‚Gs*0G´‰ Š =JI¨>  G¨kE ; GˆO = j >gk
EL=-°, =`1Q,
G´‰J,&OCE¶G´ªV=9Iʕ Ë ,?ªEF(1 (&P =JItI!>gEL=-°, S0(<S Š IÈ=E¶G«(Oz> =E
h
im •
E¶Gˆ. ,
GK> E ; ,
‹^*u,?ªEF(1
tk
m
n(tk ) = nj (tk )
j=1
¯ , =`1Q, °‚Gs*u,0O = ̧S`1Q(  ,?ªE¶G«(O . =CE1‡G¨W¼Í
E ; =CE
A(tk )
. = ¸ *k=e1 ¸  G¨E ; E¶Gˆ. ,8• ¯ , =`1Q, °‚Gs*-,&O GˆO0G¨E¶G=9IzªC(O€‹
‰GsE¶G«(<O
=9Ou‰ S0(<S Š IÈ=E¶G«(O ‰ ¸ Ok=9. G´ª>¼½
n(t0) = n0
n(tk+1) = A(tk )n(tk )
Î
¿
Ï Á .
=CE1‡G¨W¼Í
Õ
G¨EF,&1²=CE¶G«(O
>gEL=JOCE
™§Ðx> ; ,01(A1ȳ>gEL=9Ou‰A=`1¢‰
 •
R =?>g ,0I‘I´
Se1B(ª¼,>F>
(&P E ; ,
GK>
βj
Pj
½ E ; ,
/0Gˆ1¡E ; 1{=CEF,
½ E ; ,

½
Ø
>gEL=u°J,
AgÙ<Ç
P Š O-ªŽE¶G«(O?>
/0,
GˆO
=
β3
···
P
• • 3•
βm
• • •
G m Pm
>gEL=-°,
GsE ; GˆO|‹a>gEL=u°J,
/0,E
 ,?,&O|‹a>ÓEL=-°J,
Gj
EF(
j
,€W+=9.
P‡(1
.
P
β2
 1
 G 2 P2

G3
A=



; ,&1Q,
E ; GK>
>gEL=CE¶G«(<Ok=`1 ¸ 3)G֕o,8•—3


¥HO
>ÑGµÒ,Ž‹Óª+I´=z>F>ÑG¤Ô,z‰
A
S9I¤,
E ; ,
Gµ>
ªC(O€‹








j
> Š 1×*0G¤*k=9I~1{=CEF,
E1²=9O?>ŽG¨E¶G«(O
1{=CEF,
(+P >gEL=u°J,
P³1Q(.
j
>gEL=u°J,
j+1
Sk=`1²=9.
(&P
E¶Gˆ.
‰J,?ª+1Q,-=?>ŽGˆO-°
,EF,01¦>
3
ªC( Š IK‰
/0,
Pj G j
™\,8•‘°–•—3
. G´° ; E
n
βj
P Š O-ªŽE¶G´(<O (+P
•
nj (tj )
,
(A1
βj
,€*-,&O
3
Ú
ۃ¿
$
#"Ü €# "k%eC$
šÅ + €# A“
! Ã<
!
~Å· # |
=uª?ªC(A1Q‰‚GˆO-°
=u°J,
EF(
=
(1  ,eG´° ; E
I´=CE¶G«(<O
GK>
# ÆuÇ
GˆOEF(
GÊOV,z‰
ÆVÇ À
#Á
£–(<S Š I´=E¶G´(<O
=E
S0(S Š I´=CE¶G«(O
Gµ>
> Š ª ;
>gEL=-°,C>¼•
‰‚GK>›ª+1Q,EF,
E¶GÊ.
(+P
‹\E ;
E ; ,
‚u
°1Q( Š S0,z‰
S`1Q(<S0,01—E ¸
‰‚GK>›ª+1Q,€EF,
m
­”4
< Ä # €
>gE1 Š ªŽE Š 1²=JI
‰,€EF,&1
.
=9Ou‰
€# À
‚ !
Ä # €
Ï Ý6 # =z>
£4(S Š ‹
,>
tj , j =
>ÓEL=-°J,
=CE
0, 1, . . .
j
E¶Gˆ. ,
GK>
• Þ¥ E GK> ?
= >F> Š . ,?‰ E ; =E GˆO-‰G¤*0G´‰ Š ‹
tk
nj (tk )
=JIs>  G¨E ; GˆO = g
> EL=-°, e
= 1Q, ´G ‰J,&OCE¶G´ªV=9Iˆ3,€W`ªC,0SE P‡(1 =
‰,|*u,0I¤(<S9.
>gEL=u°J,
n(t)

,0OEL=9IGˆOu‰J,W3
S0(<S Š I´=E¶G´(<Oz>
h
im •
= nj (t)
j=1
=E
IGˆMN,
=u°J,9•
E¶Gˆ.
,
t
£–(S Š I´=CE¶G«(O
Ë ,?ªŽEF(A1
Gµ>
E ; ,
‰ ¸ Ok=J.
m
(&P
=9I‘I
‹^*u,zªŽEF(1
G´ª|>?½
dnj (t)
= Rj (t)−Mj (t)−µj (t)nj (t), j = 1, . . . m, t > 0
dt
; ,&1Q,
µj
Rj
½ E ; ,
½
‰,-=E ;
½
1Q,zª+1 Š GsE.
Mj
j+1
.
1{=CEF,
,&OCE
=CE Š 1²=CE¶G«(<O
GˆO
1²=CEF,
1²=EF,
>ÓEL=-°J,
EF(
j
>gEL=u°J,
P³1Q(.
j
>gEL=-°,
j
EF(
>gEL=u°J,
ß
Ø
AgÙ<Ç
E ¸ S&GȪV=9I‘I ¸ 3 ,
,W8S0,zªŽE
E ; =CE
Rj (t) = Mj−1(t)
=JO-‰

Rt
− t−τ µj (s)ds
j
Mj (t) = Rj (t − τj )e
GK> E ; , ‰ Š 1{=CE¶G«(O (&P >gEL=-°,
•
τj
j
; ,&1Q,
¿
Ï Á 
’‰4•¡3
mY\[
náà
 =JI R =z>g ,0I‘Iˆ3
>gEL=u°J,
.
™
€•
nA(t)
; ,&1Q,
™ q
fih+j]0lmY\noh9p
@GˆOk= Š ,01
(‰,0IP‡(1  Š *u,&O0GˆI¤,C>
q
hcJb?lsr 3 5O-‰
™
™\57J7!Tu€•Ê
nJ (t)

â
=9Ou‰
E
 (e‹
=u‰ Š IãE±>
dnJ
= RJ (t) − MJ (t) − µJ nJ (t)
dt
dnA
= RA(t) − µAnA(t)
dt
RA(t) = MJ (t)
MJ (t) = RJ (t − τ )e−µJ τ .
ä
僿
$
Ï
­”–
# “A
šÅ + €# !AA
æ @GÊ.
*-,&1¦>ŽG«(<Oz>Vç
S9I¤,
XZY\[^]&_`Ya]&[b?c
@–,?,
fihkj<]0lm8Y\n‘h9p
GˆO-°!38R ì D
@‚‹Ñ'ƒ@ƒ ªC(.
1Q,&O-‰‚G¨E¶G«(O?>?•´î
S9I¤,W
=uª?ªC(A1Q‰‚GˆO-°
,EF,01
x
EF(
> Š ª ;
S0(<S Š IÈ=E¶G«(O
=
*+=JI Š ,
>§,&1×*k=CE¶G«(O
I´=ñ
ê …‚p&m0†
@–,01‡G«,>¼3`@¥»â
x
ÆVÇ
nt_Lr 3-듕JD
D
Y„h
•eR Š > ; ‹
™›TVU9U0퓝P‡(16.
£–(S Š I´=CE¶G«(O
GK>
ï,E
Sk=`1²=9.
n(x, t)
/0,
=JOk=9I ¸ >ŽGK>
E¶Gˆ.
,
¸ G«,0IK‰&>
t
•
â
>ŽGˆ.
E ; ,
£ ò
‹
E ; ,
™FS0(<S Š IÈ=E¶G«(Okðe>gE1 Š ªE Š 1Q,-
=9O-‰
(1Q,
ª+I´=z>F>ÑG¤Ô,z‰
>gE1 Š ªE Š 1Q,
=-°,93 >ŽGKÒC,9•
‰,0O?>ŽG¨E ¸
€# éHpY\[hcA]0_-Y\noh8p
ª¼(<OCE¶GˆO Š ( Š >
=z>
>gE1 Š ªŽE Š 1Q,
p
è
Ä +
À
‚ !
!Â
SJIK,
=E
ªC(O€‹
’

∂n ∂
+
(γ(x, t)n(x, t)) = −µ(x, t)n(x, t), x ≥ 0, t > 0
∂t ∂x
GsE ;
E ; ,
OV(O€‹NI¤(ªu=JI~/0( Š O-‰=e1 ¸
n(0, t) =
=JO-‰

γ
β
Z ∞
β(x, t)n(x, t) dt, t > 0
0
GÊO&G¨E¶G"=JI“ªC(O-‰G¨E¶G´(<O
; ,&1Q,
n(x, 0) = n0(x), x ≥ 0,
½ >gE1 Š ªŽE Š 1{=9I“°1Q(z
½
ªC(Ou‰‚G¨E¶G«(O
E ;
1{=CEF,
/0Gˆ1¡E ; 1²=CEF,
ó
Ø
AgÙ<Ç
@E1Q,eGsP‡,&1
.
ªV=z>§,
.
(‰J,&I
GK>
ªu=JI‘IK,?‰
E ; ,
@GˆO9MN(e‹
>gE Š ‰G´,?‰
>|S0,?ª`G"=JI
(‰,0I4™›TVU9ô0œ“€•
¿
Ï Á ; GK>
:
(&P4E ; GK>
; ,
:
.
.
(>ÓE

(‰,0IzGK> E ; ,
,&1¦>gEF,&1 =u°J,ыa>gE1 Š ªE Š 1Q,?‰
.
G´‰J,&I ¸
D
ªJ˜ ,0Ou‰<1‡G´ª+Mñ‹^*u(O
(`‹
(‰,0I
∂n
∂n(x, t)
+
= −µ(a, t)n(a, t), a ≥ 0, t > 0
∂t
∂a

GsE ;
E ; ,
OV(O€‹NI¤(ªu=JI~/0( Š O-‰=e1 ¸
n(0, t) =
=JO-‰
Z ∞
ªC(Ou‰‚G¨E¶G«(O
β(x, t)n(x, t) dt, t > 0
0
GÊO&G¨E¶G"=JI“ªC(O-‰G¨E¶G´(<O
n(a, 0) = n0(a), a ≥ 0,

a
β
; ,&1Q,
½
½
=-°,
™
>§(
E ; =CE
°1Q(?
E ;
1²=CEF,
GK>
ª+I¤,-=`1
I ¸
1

/0Gˆ1¡E ; 1²=CEF,
õ
8
Ž
; ,
:
=
S`1Q(/JI¤,0.
 ,
SJIGÊOu°
(&P E ; ,
>ñ=J.
*k=e1HG´( Š >
>ŽGˆ/JI ¸
I¤,&.
.
=
S0(GˆOCE±>
> Š /?>§,E
3EF(
E ; ,
; ,01B,
I Š E¶G«(O
5~•Z¯
,

.
= ¸
GsE ;
; =E
#"ú
GK>
¥HO
(&P
>¼½
Gs*u,0O
=JO-‰
(&P E ; ,
=E¶G«(O
EF(
=E
S0(>³‹
S`1Q(/€‹
E ; ,
 (A1Q‰0>?½
Se1Q(</9I¤,&.
÷
> ¸ >gEF,&.
Ou(‚GK>§,-
(+E ; ,01
E ; ,
=
,€EF,&1¦>
=JS9S`1Q(zWGˆ.
,EF,01¦>¼•
/0,

Sk=`1²=9.
=9O
! $
­Z“!
T8• :
(&P E ; ,
>ø(I Š E¶G«(O3!
=z> P‡(<I‘IK(?
S0(<S Š I´=E¶G´(<O
™FS0(>F>ÑGÊ/JI ¸
Sk=`1²=9.
”
ª¼(<Oz>ŽG´‰J,&1)GK>
1Q,?ª¼(+*-,&1
=0GˆO0GˆOu°
ö
1Q,Ž‹
°‚Gs*u,0O
ù
.
! €# 9&Ç
Š IûE¶GˆS9I¤,
™H¥‡I‘IŒ‹NS0(0>§,?‰<Ou,>F>
>ø(I Š E¶G«(O
(1 Ou(
GˆOC*-,&1¦>§,
S`1Q(/JI¤,0.
(&P
GˆI‘I~=9I‘.
(>gE
ª¼,&1¡EL=&GÊOJI ¸
/&,
GˆI‘IZ=9I‘.
(>gE
ªC,01¡EL=&GˆO9I ¸
/0,
°Gs*-,&O
>§(`‹
€•
Se=e1¡E¶G=JI
‰A=EL=“•
2~•Z¯
,

°‚Gs*u,0O
P„= Š IûE ¸
‰A=EL=“•
ü
â
ItIxGK>
Ou(+E
( Š 1
(?
I¤,u=-‰
EF(
:  (
/e=z>ŽG´ª
T9•
ò
O
IK(0>gEC•
/0G=z>ø,C>
=JO
,
; (S0,
=9/0( Š E
=uª?ªC,&SCE¶Gˆ/9I¤,
E ; =E
E ; ,
E ; ,
‰=CEL=
Oe=CE Š 1Q,
=9SJSe1B(zWGˆ.
(&P
=EF,
S9I Š >
E ; ,

=9O?>g
GˆI‘I
,&1ý•
=9SJSe1Q(A=-ª ; ,>?½
,EF,01
.
I´=`1‡GKÒV=CE¶G«(O
¯
GˆO0GK>gE¶G´ª
Se=e1²=J.
þ
P Š ‰9°,
P
=-ªŽEF(A1¦>
,EF,&1¦>ñ”E ; =CE .
™\,8•‘°–•—3y1Q,?° Š ‹
=JMñ, E ; ,
S`1Q(/JIK,&.
 ,0I‘IŒ‹NS0(0>§,?‰–•
5–•
@EL=CE¶GK>gE¶G´ªu=JI‰=CEL=
‚+ ­Z„
ÿ
:
ԂEE¶GˆO-°4•
; ,
P‡(1¡
=e1¢‰
S`1Q(/JIK,&.
®
–gÙ
I¤((M>
IGˆMñ,

AP (N ) = 0
; ,&1Q,
Se=e1²=J.
AP
,EF,&1¦>
GˆO¼*u,&1¡E
E ; Gµ>
GK>
>§(.
P
•
,
:
(<S0,01{=CEF(1‰,0S0,&O-‰GˆO-°
; ,
(S0,&1²=EF(A1ZEF(
P‡(1¡
=e1Q‰
>ø(Iã*-,
(O
Se1B(/9I¤,&.
PH(A1
N
E ; ,
•
Gµ>
EF(
¯
,
ªV=9O
,€W8S`1Q,C> >
E ; GK>
> ¸ .
/0(<I"G´ªV=9I‘I ¸
=z>
N = F (P )

; ,&1Q,
I¤,&.
E ; ,
(<O
‰J,&S0,&O-‰,0Ouª¼,
°,€E±> S`1Q(.
P
‰J(,>|OÈE
E ; Gµ>
ԂE
1B,0S`1Q,>§,0OE±>
H
EF(
N
G¨E±>
E¶G«(O
=E
,WS&,01‡Gˆ.
E ; ,
ªC,&1¡EL=0GˆO
E ; ,
P‡(1¡
EF(
=9O
=`1¢° Š .
,0OEC•
,0OE±>?•
¥ P E ; ,
(S&,01²=EF(A1
‰‚GK>›ª+1Q,E¶GKÒV=CE¶G«(O
,WS0,&1‡Gˆ.
(<S0,01{=CEF(1 PH(A1
(+EF,?‰
(&P
,0OEL=9IyPH(A1
.
S0(‚GˆOCE±>ñ
GˆOC*-,&1¦>ŽG«(O
(+P
E ; ,
S`1Q(/€‹
ì Š E
E ; ,
™\,9•‘°–•—3
n
E ; ,&O
=e1Q‰
‰=CEL=
,€*+=JI Š =|‹
=9SJSe1B(S`1‡G=CEF,
GK>
F (D) = n = HN.
Ouª¼,
 ,
n

,
; =¼*u,
; =C*-,
=9O
S0,&1×PH(A1
.
,€W8S0,&1‡Gˆ.
,?‰
( Š 1
,WS0,&1‡Gˆ.
,&OCEL=JI~=9S9S`1Q(?WJGˆ.
,0OE±>¼3
=CE¶G«(O
y≈
•
@EF,&Sz>
¾6¿
EF(
EL=JMñ,9½
# €
€# $
’ •‘°–•—3
IGˆMñ,
GˆO
µ(s)
IGˆMñ,
=
/&GȪ
>|SJIGÊOV,
3
,C>ÓE¶GÊ.
=CE¶GˆOu°
=z> > Š .
S0(I ¸ Ou(<.
(&P
“A„!
=
,
=
=
yŽ Å #  A8< ¿
ªC(OE¶GÊO Š ( Š >
P‡(1
.
P‡(1
G=9Ie(+P =
ª¼,&1¡EL=0GˆO
ªC,01—EL=0GˆO
‰J,?°<1Q,¼,8•
E ; ,
Sk=`1²=9.
,€EF,&1
Sk=`1²=9.
,EF,01
‰J,?°<1Q,¼,
(1 =
¬
ª Š ‹
Ø
AgÙ<Ç
; ,
:
S0(0>F>ŽGˆ/0GˆIG¨E ¸
; GK>
:
‰GK>›ª&1B,€E¶GKÒV=E¶G´(<O
(+P
>gEF,&S
ªC(O¼*u,&1¢°‚GˆO-°
pJh‚Y
GK>
>§,E GK> =9I‘1Q,u=-‰ ¸
EF(
Ou,?ª¼,>F>ñ=`1 ¸
E ; ,
ª¼(1
1Q,?ªE
E ; ,
Sk=`1²=9.
GsP
‰GK>›ª&1B,€EF,J3+,8•‘°–•—39=
.
=E1‡G¨W
‰J,&I´= ¸ ,?‰`‹Ó‰GZ,&1Q,&OCE¶G=9IZ,zº Š =CE¶G«(<O
=
E ; ,
¥HO
Sk=`1²=J.
,EF,01»>
=`1Q,
MeOu(?
 ,
,&O-‰
Š S
=9I‘IZªV=z>ø,C>
>ŽG«(<Ok=9Iu*u,zªŽEF(A1
ۃ¿
; GK>
:
n
EF(
Se1B(&*0Gȉ,C>
=
N
•
“(A1
E ; ,
‰A=EL=
=9Ou‰

( Š 1 =JS9S`1Q(?WJGˆ.
2–•
.
.
SJI¤,93
Ï Á A
8Ž ; ,&1Q,
ª¼(<Oz>gEL=JOCE±>?•
ÔO0G¨EF,
‰Gˆ.
,&O|‹
1Q,&Se1Q,>§,&OCE±>
P
•
PH(A1
GsP
Se=z>F>ÑGÊOu°
N = n(a, t)
P1Q(<.
GˆO
=JO
E ; ,
‰A=EL= n
>|S9IGˆOu,>?•
EF(
(‰J,&I¶(1
A8< ¿
(‰J,&Iˆ3 =JO-‰
=CE¶G«(O
=
,?ª ; =9O&Gµ>€.
n = 1,. . . , s
GsE ;
GsE ;
.
•
µ
,€EF,&1
(‰J,&I
/0,
=-‰,zº Š =EF,0I ¸
Se1Q(+*G´‰,C> ‰Gµ>§ª&1Q,EF,
1, . . . , m

.
EF(
! ·Å # Â
,W&=J.
=u°J,ыa>gE1 Š ªE Š 1Q,z‰
‰=CEL=
E ; =CE
p
O
(ƒ,01šE ; ,
> ; ( Š IK‰
3-
,
:
; Gµ>
n(s, t)
A
#
,W8S0,01HGÊ. ,&OCEL=9I
o P‡(1
n(a , tn)
j=
ªC( Š jIK‰ GˆOEF,01
S&(I´=CEF,
GˆOEF,01
S&(I´=CE¶G«(O
Gµ>
•
# €
# Â\–
f (p) = n.
, W8S0,&1‡Gˆ. ,&OCEL=JIA‰A=EL=
© • £ I Š ° GÊO €
=JO-‰ §
> (Iã*u, E ; , > ¸ >ÓEF,0.
ªV=9OA•
y
f (p) = y
GˆO
S9I´=-ªC,
=?>
(+P
/0,>gE
n
¸ ( Š
¬ý¬
È
# “A
D
.
,E ; (‰
GˆO0Gˆ.
Ä ¶% (+P ª ; (‚G´ª¼,
GKÒC,
E ; ,
GK>
ö
8Ҧ%
&“ŽÆuÇ
IK,u=z>ÓE
Wp = y
>›º Š =`1Q,>¼3‚E ; =CE
Gµ>C3EF(
1Q,>ŽG´‰ Š =9I
ky − W pk2 = pT W T W p − 2pT W T y + yT y

GsE ;
1Q,>|S0,?ªE
EF(
p
,?º Š =CE¶G«(O?>
3
; GȪ ;
I¤,u=-‰&>
EF(
E ; ,
OV(A1.
=JI
W T W p = W T y.
Ø
T9• ¯
AgÙk+Ç
,
; =¼*u,
.
> Š
>§(<I Š E¶G«(O
= ¸
ª`G«,0OEI ¸
GK>
5–• â > > Š .
,WS&,zªŽE
WTW
GK>
>gE¶GˆI‘I ; =¼*u,
.
=
=JO ¸
Se1Q(</9I¤,&.
‰A=EL=
° Š =`1²=9OEF,?,?‰43~G¨E
GˆO-°
> Š
EF(
/0,
ª`G«,0OEI ¸
.
.
S0(GˆOE±>?•
= ¸
Ou(kE
=JO ¸
‰=CEL=
(+P P Š I‘Iª¼(<I Š . O
GÊOC*-,&1¡E¶Gˆ/9I¤, =JO-‰
W
GsP  ,
‰J(
¯
/0,
; GˆI¤,
=
Š O0G´º Š ,8•
S0(‚GÊOE±>Z ,
1²=9OJM036>ø(
−1
p = WTW
Ou(kE
E ; =CE
WTy
¬HÎ
•
2–• ¥‡O
E ; ,
E¶G«(O
ªu=?>§,
GK>ƒE ; ,
*k=9I Š ,
(&P
/0,>gE Š O9/&G=z>§,?‰
p
3
Se1B(&*0Gȉ,z‰
‰Gµ>ÓE1‡Gˆ/ Š EF,?‰
0
=9Ou‰
(&P P Š ItIªC(I Š .
1²=JO-‰(.
ÔO&G¨EF,
O
1²=JO9M03<E ; Gµ>
,>gE¶Gˆ.
>§(I Š ‹
=CEF(1A(&PE ; , E1 Š ,
E ; ,
,01
1B(A1
*-,?ªEF(1
GsE ;
GK>
G´‰J,&OCE¶G´ªV=9I‘I ¸
,€W8S0,?ªEF,?‰
*k=9I Š ,
ªC(&*k=`1‡G=9O-ªC,9•
Ä ”! # “A ¶% A„
&AÆuÇ
f (p) =
y
â °A=0GˆO3
E ; ,
Š >§,
E ; ,
I¤,u=z>gE
>›º Š =`1Q,>
Gȉ,-=A½
.
GˆO0Gˆ.
GµÒ,
1B,C>ŽG´‰ Š =9I
F (p) = ky − f (p)k2 .
D
(‰J,&IŽE ; , P Š O-ªŽE¶G«(O
I"GˆOV,-=`1
I ¸ =z> g
> EL=`1¡E¶GˆO-° P1Q(<.
f
=9O-‰ S`1Q(ªC,?,?‰‚GˆOu° EF( . GˆO&GÊ. Š .
=CE
p0
p = p0 + α
 GsE ;

; ,&1Q,
Gˆ.
J
GµÒ,
GK>
f (p0 + α) = f (p0) + Jα
E ; ,
ëk=uª¼(</0G=9O
(+P
f (p)
•
@–(
 ,
.
GˆO|‹
ky − f (p0) − Jαk2 = kδ − Jαk2
¸ G«,0IK‰‚GˆOu°
>gEF,&S
E ; Gµ> GK> =JO
(+P
.
G¨EF,01²=E¶Gs*-,
G¨EF,01²=EF,C>
Š .
p
•
α =
.
−1
•
T
J J
JT δ
,€E ; (‰ ¸ G«,0IK‰GÊOu°
p0 , p 1 , . . .
ª¼(<O¼*u,01¢°GˆO-°
=
P
EF(
ªC( Š 1»>§,93
>§,?º Š ,0Ouª¼,
E ; ,
(<SCE¶Gã‹
¬HÚ
; ,&1Q,
:
=`1Q,
S`1Q(/JI¤,0.

G¨E ;
(&P  =uª¼(</0G=9O?>
>§,?º Š ,0Ouª¼,
ªC(Ou‰‚G¨E¶G«(Ou,?‰
Z,&.
>
,?‰ ¸ ½
PH(A1ZE ; ,
.
.
PH(A1
.
Š I´=
/0,
Š O0GsP‡(1
.
,€E ; (‰
EF(
> Š ª¼ª¼,¼,z‰4•
=e1Qº Š =`1¢‰eEL
((‰
T
,W8S9I´=0GˆOz> GsEC½ G¤PE ; ,
Oe=9OCE3- ,
=e1Q,
‰G´,&OCE
>gEF,&S
EF,&1
.
GK>
=JS9S`1Q(zWGˆ.
,>F>§,&OCE¶G=9I‘I ¸
/0,?ªV= Š >ø,
‰J(<.
GˆOe=9OCE3)
=E¶G«(O
GK>
P Š O-ªŽE¶G«(OA• Ì
EF(
E ; ,
I¸

1Q,&S9I´=uª¼,
,&I‘I
E ; ,
/ ¸
α = J J + λI
â > ¯
=JS9S`1Q(“=uª ; ½
Š >gE
™§ï“,|*u,&O9/0,&1¢°`‹¶D
G¨EF,&1²=CE¶G«(O
E ; Gµ>
−1
J T δ.
Ô1¦>gEZEF,&1
.
EL=9MuGÊOu°
=
‰J(?
−2J T δ = ∇F
,
E ; ,
=e1B,
‰(GˆO-°
I¤,u=z>gE
=u‰  Š >gEF,?‰
•
E ; ,
Gã‹
O ; GˆI‘I°1²=|‹
¥ P E ; ,
=
Ô1¦>gE
º Š =-‰1²=CE¶G´ª
(/  ,?ªE¶Gs*u,
>›º Š =`1Q,C>
GˆO
GK> ‰J(<.
P‡(I‘I¤(?
GˆO-°

= ¸ ½
λ
GsP
I¤,E
=9O-‰
F (p + α) < F (p),
λ = λ/10
p =
3!,&I¨>§,
=JO-‰ I¤,u=¼u
* ,
Š Ouª ; =9Ou°J,?‰–•´Í
p+α
λ = 10λ
p
¥HO¼*u,01¦>ø,
Gˆ.
E ; ,?(1‡GK>gE">
S0(GˆOCE
(+P *G«,
½ 
,
=`1Q,
.
GˆO|‹
GµÒ-GÊOu°
kδ − Jαk2 + λ kpk2 ,
>§(
E ; ,
Š IÈ=`1‡GKÒC,
r]kmV[bNr
ª ¸ ª&I¤,
=9Ou‰
.
GK>?½
PH(A1¡
Š S‰A=EF,8•
:
=`1¢‰
; GK>
>ø(Iã*-,J3 I"GˆOV,-=`1‡GKÒC,J3 1Q,?°`‹
GK>
E ; ,
h“]8YHj<]Y
,€E ; (‰–•
lobm¼r?Y
¬
ß
Ï Á 8
¬ ä
Aˆ“ # A
¬‡ó