0DWHULDá\SRPRFQLF]HGRüZLF]H
]Ä1RZRF]HVQ\FKPHWRGVWHURZDQLD´
F] ü =DGDQLDRSW\PDOL]DFML
û: 'REyURSW\PDOQHMZDUWR FLMHGQHM]PLHQQHMSU]\]DNáyFHQLDFK
SRPLDUyZZVND(QLND6WHURZDQLHQDHNVWUHPXP
û:
3U]HJOGD
OJRU\WPyZSURJUDPRZDQLDQLHOLQLRZHJR
û: 2SW\PDOL]DFMDVWDW\F]QDZREHFQR FL]DNáyFH
-DQ/HV]F]\VNL
.DWHGUD$XWRPDW\NLL(OHNWURQLNL
3ROLWHFKQLND%LDáRVWRFND
%,$à<672.
û:'REyURSW\PDOQHMZDUWR FLMHGQHM]PLHQQHMSU]\]DNáyFHQLDFK
SRPLDUyZZVND(QLND6WHURZDQLHQDHNVWUHPXP
:WUDNFLHüZLF]HQLDEDGDQHV SURJUDPÄQPVBP´ LWHUDF\MQHDOJRU\WP\RSW\PDOL]D
FMLMHGQRZ\PLDURZHMZNWyU\FKSRND*G\PQRZ\PNRPSOHFLHSRPLDUyZQDVW SXMHREOLF]HQLH
L XVWDZLHQLH QRZHM ZDUWR FL RGG]LDá\ZDQLD ZHM FLRZHJR = ]DáR*HQLD GRVW SQH V MHG\QLH
EH]SR UHGQLRSRPLDU\RSW\PDOL]RZDQHJRZVND(QLNDMDNR FL]DNáyFDQHUHDOL]DFMDPLZLHONR
FLORVRZHMR UR]NáDG]LHQRUPDOQ\P
0RGHORZDQ\LEDGDQ\MHVWWDN*H SOLNÄVBPV´Z 6LPXOLQNX XNáDGUHJXODFMLHNVWUH
PDOQHMR FKDUDNWHU]HFLJá\P:RGUy*QLHQLXRGSUREOHPXVWDFMRQDUQHJRZDUWR üRSW\PDOQD
VWHURZDQLDZWDNLPXNáDG]LHUHJXODFMLPR*H]PLHQLDüVL QLHSU]HUZDQLHDSURFHVRSW\PDOL]D
FMLRGE\ZDVL ZF]DVLHU]HF]\ZLVW\P:PRGHOX XZ]JO GQLDQDMHVWWDN*HG\QDPLNDVWHURZD
QHJRSURFHVX
35=('û:,&=(1,(0 QDOH*\]DSR]QDüVL
QDSRGVWDZLH]HVWDZLHQLD$] SRGVWDZRZ\
PLW\SDPLDOJRU\WPyZUR]ZL]\ZDQLD]DGDQLDSURJUDPRZDQLDQLHOLQLRZHJRMHGQHM]PLHQQHM
D WDN*HQDSRGVWDZLH]HVWDZLHQLD%]HVSHF\ILN UR]ZL]\ZDQLD]DGDQLDRSW\PDOQHJRGRERUX
]PLHQQHM OXERSW\PDOL]RZDQHJRSDUDPHWUX Z REHFQR FL ]DNáyFH SRPLDURZ\FK :VND]D
Q\PMHVWWDN*HX]XSHáQLHQLHZLHG]\WHRUHW\F]QHMZWU\ELHVWXGLyZOLWHUDWXURZ\FK XZDJLEL
EOLRJUDILF]QH]Z\ND]HPOLWHUDWXU\]DPLHV]F]RQRQDNRFXLQVWUXNFML
3\WDQLDNRQWUROQH
&RQD]\ZDP\]ELRUHPZ\SXNá\PLIXQNFM Z\SXNá SRGDM FLVáHGHILQLFMH
:MDNLHMV\WXDFML]DGDQLH]QDOH]LHQLDHNVWUHPXPIXQNFMLPDMHGQR]QDF]QHUR]ZL]DQLH"
-DNIRUPXáXMHVL
ZDUXQHN]DNRF]HQLDREOLF]H SU]\SRVáXJLZDQLXVL
DOJRU\WPHPLWHUD
F\MQ\PRSW\PDOL]DFML"
2EMD QLMQDZ\EUDQ\PSU]\NáDG]LHG]LDáDQLHXNáDGXUHJXODFMLHNVWUHPDOQHM
352*5$02:$1,( %$'$ 6<08/$&<-1<&+ Z WUDNFLH üZLF]HQLD ZVSRPDJDQH
MHVWSU]H]SURJUDP\0DWODEDXPLHV]F]RQHZSOLNDFK ÄQPVBP´LÄQPVBBDP´=DOHFD
VL
SU]HDQDOL]RZDQLHÄQPVBP´3URJUDPWHQ]DPLHV]F]RQRQDQDVW
SQHMVWURQLH:\]QD
F]DQ\MHVWZQLPSU]HELHJPLQLPDOL]DFMLIXQNFMLNZDGUDWRZHMMHGQHM]PLHQQHM
\ F XXRSW
S] UDQGQ
SU]\SRPRF\GZXDOJRU\WPyZLWHUDF\MQ\FKSU]\WRF]RQ\FKZ]HVWDZLHQLX$ DOJRU\WPXSR
FKRGQHM LDOJRU\WPXÄNURF]FHJR´]XVWDORQ ZLHONR FL SRWHQFMDOQHMSRSUDZNL
2GWZDU]DQ\MHVWSU]HELHJPLQLPDOL]DFMLIXQNFMLNZDGUDWRZHM
MHGQHM]PLHQQHM \ F XXRSW AS] UDQGQ
SU]\SRPRF\DOJRU\WPXSRFKRGQHM DOJRU\WP]]HVWDZLHQLD$
:SURZDG]DQHVWDáH BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
XRSW X RSW\PDOQDZDUWR üDUJXPHQWXLSXQNWVWDUWRZ\
F S] GUXJDSRFKRGQDIXQNFMLNZZHMLSR]LRP]DNáyFH
Z]P Z]PRFQLHQLHSRFKRGQHMZDOJRU\WPLH
GHO SU]\URVWDUJXPHQWXSU]\RGWZDU]DQLXSRFKRGQHM
1 OLF]EDSU]HOLF]DQ\FKLWHUDFML
Q OLF]EDX UHGQLDQ\FKSRPLDUyZIXQNFML
NURN VWDáDDOJRU\WPX]HVWDá ZDUWR FL SRSUDZNL
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
2EOLF]HQLDDOJRU\WPHPZNWyU\PSRFKRGQD]DVW SRZDQDMHVW
LORUD]HPVNRF]RQ\FKSU]\URVWyZ]VHULLQSUyE´
IRU L 1
IRU M Q\SOXV M F X L GHOXRSW AS] UDQGQ HQG
IRU M Q\PLQXV M F X L GHOXRSW AS] UDQGQ HQG
\S VXP \SOXV Q \P VXP \PLQXV Q
X L X L Z]P \S\P GHO \ L \S\P
HQG
LZ\SURZDG]HQLHLFKZ\QLNyZ
T >VXP DEV XXRSW VXP \ @ILJXUH
SORW X RE JULGKROGRQSDXVHSORW \ U SORW ]HURV 1
[ODEHO QULWHUDFML \ODEHO X L \ L KROGRII
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
7HUD]RGWZRU]\P\MHVWSU]HELHJPLQLPDOL]DFMLIXQNFMLNZDGUDWRZHM
SU]\SRPRF\DOJRU\WPX]NURNLHPSUyEQ\P
\ F X XRSW AS] VXP UDQGQ Q Q ]QDN VLJQ UDQGQ
IRU L 1
LI ]QDN L XQH[W X L NURN
IRU M Q\SOXV M F X L XRSW AS] UDQGQ HQG
\ L VXP \SOXV Q
HOVH
XQH[W X L NURN
IRU M Q\PLQXV M F X L XRSW AS] UDQGQ HQG
\ L VXP \PLQXV Q
HQG
LI \ L \ L X L XQH[W ]QDN L ]QDN L
HOVH
X L X L ]QDN L ]QDN L
HQG HQG
3U]HND]DQLH Z\QLNyZ
T >VXP DEV XXRSW VXP \ @ILJXUH
SORW X RE JULGKROGRQSDXVHSORW \ U SORW ]HURV 1
[ODEHO QULWHUDFML \ODEHO X L \ L KROGRII
3U]\WRF]\P\NLONDNRPHQWDU]\XáDWZLDMF\FKDQDOL] SURJUDPX
1DSRF]WNXXVWDODQHV ZDUWR FLZV]\VWNLFKZ\VW SXMF\FKZSURJUDPLHVWDá\FK:\UD(
QHZ\G]LHOHQLHWHJRIUDJPHQWXSURJUDPXXáDWZLDLFKPRG\ILNDFM
ZWUDNFLHEDGD V\PX
ODF\MQ\FK :DUWR FL SRF]WNRZH SDUDPHWUyZ EDGDQ\FK DOJRU\WPyZ V WX XVWDORQH
SU]\SDGNRZR
)UDJPHQW\SURJUDPyZZNWyU\FKEDGDQHV VNXWNLVWRVRZDQLDREXDOJRU\WPyZV WDN*H
Z\UD(QLH Z\G]LHORQH :\GDMH VL
*H ] SXQNWX ZLG]HQLD HNVSHU\PHQWDWRUD ]PLHQLü WX
PR*QD SU]HGH ZV]\VWNLP ÄGH JXVWLEXV QRQ HVW GLVSXWDQGXP´ LQVWUXNFMH ]ZL]DQH
] SU]HND]DQLHPRWU]\PDQ\FKUH]XOWDWyZ8PLHV]F]RQHV RQHQDNRFXREOLF]H LWHUDF\M
Q\FK]ZL]DQ\FK]ND*G ]UHSUH]HQWRZDQ\FKWXPHWRG
.D*G ] Z\VW SXMF\FK Z SURJUDPLH VWDá\FK PR*HP\ WDN*H XF]\QLü IXQNFM F]DVX WM
QXPHUXLWHUDFML :W\PSU]\SDGNXSRZLQQL P\MHGQDNRF]\ZL FLHXPLH FLüRNUH ORQ
LQVWUXNFM
ZHZQWU]LQVWUXNFMLF\NOX1SPR*HP\ZLQVWUXNFMLÄIRU´ZSURZDG]LüVLQXVR
LGDOQHSU]HPLHV]F]DQLHVL
SXQNWXRSW\PDOQHJRFKDUDNWHU\VW\NL]JRGQLH]UyZQDQLHP
XRSW L XRSWDPSOLWXGD VLQ SL L
XPR*OLZLDMFZWHQVSRVyEV\PXODFM SURFHVX OHG]HQLD]DSU]HVXZDMF\PVL HNVWUHPXP
'RVW SQD MHVW UyZQLH* PRG\ILNDFMD WHJR SURJUDPX Z SRVWDFL PSOLNX ÄQPVBBD´ SR
]ZDODMFD MHGQRF]H QLH X]\VNDü L SRUyZQDü Z\QLNLV\PXODFML SU]\ RSW\PDOL]DFML DOJR
U\WPHP GODRNUH ORQHJR]ELRUXZDUWR FLDQDOL]RZDQHJRSDUDPHWUX
0RGHORZDQLHMHGQHM]ZHUVML8.à$'85(*8/$&-,(.675(0$/1(- SR]ZDOD]EDGDü
ZSá\Z G\QDPLNL VWHURZDQHJR RELHNWX QD SU]HELHJL SURFHVyZ SRV]XNLZDQLD HNVWUHPXP QD
MHJR FKDUDNWHU\VW\FH VWDW\F]QHM 6WHURZDQ\ RELHNW VWDQRZL V]HUHJRZH SRáF]HQLH GZXZHM
FLRZHJRVWDW\F]QHJRHOHPHQWXQLHOLQLRZHJRR FKDUDNWHU\VW\FHNZDGUDWRZHM
\ F XXQHVW
SRáF]RQHJRV]HUHJRZR] OLQLRZ\PHOHPHQWHPG\QDPLF]Q\PRWUDQVPLWDQFML
*
V N
H[S V7R V7
=]DáR*HQLDXQHVW MHVWZLHONR FL QLHVWHURZDOQD=PLDQ\MHMZ\ZRáXM ]PLDQ\SRáR*HQLD
HNVWUHPXPQDFKDUDNWHU\VW\FHVWDW\F]QHMRELHNWXVWHURZDQLD: XNáDG]LH SU]HGVWDZLRQ\PQD
U\V VWRVRZDQ\ MHVW FLJá\ SURFHV SRV]XNLZDQLD DNWXDOQHJR SRáR*HQLD HNVWUHPXP ]H VWDá
SU
GNR FL ]PLDQZLHONR FLVWHUXMFHMX.LHUXQHN]PLDQ]DOH*\RGDNWXDOQHJR]QDNXLORF]\
QXMHMSRFKRGQHMLSRFKRGQHMZLHONR FLZ\M FLRZHM
3R GRSURZDG]HQLX GR VWHURZDQHJR RELHNWX ]DNáyFH R FKDUDNWHU]H SUREDELOLVW\F]Q\P
V\WXDFMD VL
NRPSOLNXMH 1DOH*\ ZSURZDG]Lü GRGDWNRZR GR XNáDGX ILOWU GROQRSU]HSXVWRZ\
Z FHOXZ\JáDG]HQLDSU]HELHJXSRFKRGQHMZLHONR FLZ\M FLRZHM
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
Mux
Generator
5\V0RGHOXNáDGXUHJXODFMLQDHNVWU HPXP
f(u)
5\V0RGHOVWHURZDQHJRRELHNWX]XQLPRGDOQ FKDUDN
WHU\VW\NDVWDW\F]Q
35=(%,(*û:,&=(1,$
1DOH*\SU]HSURZDG]LüEDGDQLDV\PXODF\MQHPDMFHQDFHOXGREyUSDUDPHWUyZ 'XD
Z DOJRU\WPLHSRV]XNLZDQLDPLQLPXPQDNZDGUDWRZHMFKDUDNWHU\VW\FHSURFHVX
\ F XX N
JG]LHVWDáDF! X N FRQVW
=DáR*HQLD VWDáHFV SUHF\]XMHZ\NáDGRZFD SU]\MPRZDQHZWUDNFLHEDGD V\PXOD
F\MQ\FK
3RPLDU\ZLHONR FLZ\M FLRZHM\EDGDQHJRSURFHVX OXEMHMSRFKRGQHM REDUF]RQHV
ORVRZ\PL]DNáyFHQLDPLRUR]NáDG]LHQRUPDOQ\P ]
1
V
6WRVRZDQ\MHVWDOJRU\WPVNRF]RQ\FKSU]\URVWyZ
X N X N ± DN >'I X @_
X N
'XN
%DGDQLDPR*HP\SU]HSURZDG]LüSU]\SRPRF\Ä
nms_1_a1
´OXEDQDORJLF]QHJRSUR
JUDPXRSUDFRZDQHJRVDPRG]LHOQLH:WUDNFLHGR ZLDGF]H V\PXODF\MQ\FKQDOH*\GRELH
UDü UDFMRQDOQ\ SR]LRP ZVSyáF]\QQLND Z]PRFQLHQLD RNUH ODQ\ SU]\ SRPRF\
ZVSyáF]\QQLND D L ZDUWR ü 'X WHVWXMFHJR SU]\URVWX DUJXPHQWX 3DUDPHWU\ GRELHUDP\
Z SURFHVLHPLQLPDOL]DFML X UHGQLRQ\FK]ZLHOXSRZWyU]H ÄVWUDW´ZWUDNFLHSURFHVXSR
V]XNLZDQLD HNVWUHPXP SU]\ SRPRF\ DOJRU\WPyZ VWDUWXMF\FK ] MHGQDNRZ\FK ZDUXQNyZ
SRF]WNRZ\FK6WUDW\PR*QDZ\UD]LüQDMSUR FLHMSRSU]H]VXP
ZDUWR FLZ\M FLRZ\FKED
GDQHMIXQNFML ZUDPDFK]DGDQHM]JyU\RJUDQLF]RQHMGRQ OLF]E\LWHUDFML
1DOH*\SU]HSURZDG]LüEDGDQLDV\PXODF\MQHPDMFHQDFHOXGREyUNURNX' Z DOJRU\WPLH
SRV]XNLZDQLDPLQLPXPZSRVWDFL
u(k+1) = u(k) -
'
sign[
'
f(u)]|
u(k)
, k=0,1,2,...
%DGDQLDPR*HP\SU]HSURZDG]LüSU]\SRPRF\SURJUDPXÄ
nms_1
´OXESU]\SRPRF\VD
PRG]LHOQLHRSUDFRZDQ\FKSURJUDPyZDQDORJLF]Q\FKGRÄ
nms_1_a1
´
3RGRERU]HZLHONR FLNURNX'QDOH*\SRUyZQDüHIHNW\G]LDáDQLDDOJRU\WPXWHJRW\SX]
SXQNWXZLG]HQLDVWUDWQDSRV]XNLZDQLHHNVWUHPXP]DOJRU\WPHPVWRVRZDQ\PXSU]HGQLR
6SUDZG]LüMDNR FLRZR UHMHVWUDFMDW\SRZ\FKSU]HELHJyZ SURFHV OHG]HQLD]DZROQR]PLH
QLDMF\P VL
SRáR*HQLHP SXQNWX RSW\PDOQHJR ZSURZDG]DQHJR QS SU]\ SRPRF\
X
N
$
VLQ L 6WRVXMHP\ SU]\ W\P DOJRU\WP ] SDUDPHWUDPL Z\EUDQ\PL
Z WUDNFLHGRW\FKF]DVRZHM UHDOL]DFMLüZLF]HQLD
1DOH*\ ]DPRGHORZDü XNáDG UHJXODFML HNVWUHPDOQHM ] U\V 3R ]DPRGHORZDQLX ]DUHMH
VWURZDü W\SRZH SU]HELHJL ZLHONR FL ZHM FLRZHM L Z\M FLRZHM D WDN*H WUDMHNWRUL ID]RZ
Z SURFHVLHVWHURZDQLD-DNLH]PLDQ\SU]HELHJyZREVHUZXMHP\ SU]\ ]PLDQDFKVWDáHMF]D
VRZHM LZLHONR FLRSy(QLHQLD ZPRGHOXVWHURZDQHJRRELHNWX"
=$'$1,(8=83(à1,$-&(
Î -DNL SURJUDP EDGD V\PXODF\MQ\FK QDOH*DáRE\ SU]HSURZDG]Lü PDMF QD FHOX XVWDOHQLH
]DVDG\GRERUXSDUDPHWUyZD N L' Z]DOH*QR FLRGGUXJLHMSRFKRGQHM F LRGZD
ULDQFML]DNáyFH V
SRPLDUyZIXQNFMLNU\WHULDOQHMZGDQ\P]DGDQLXRSW\PDOL]DFML"
5H]XOWDWHPEDGD V\PXODF\MQ\FKSRZLQQRE\üZ\SUDFRZDQLHL]ZL
]áHSU]HND]DQLH]DOHFH
QDWHPDWGRERUXSDUDPHWUyZ'X LD N SU]\UR]ZL]\ZDQLX]DGD RSW\PDOL]DFMLDQDORJLF]
Q\FKGRV\PXORZDQ\FKZWUDNFLHEDGD ZüZLF]HQLX=DOHFHQLDPRJ SU]\MüSRVWDüMDNR
FLRZQSPRJ E\üSU]HND]DQHZIRUPLHRGSRZLHG]LQDS\WDQLDW\SX
&]\]HZ]URVWHPGUXJLHMSRFKRGQHM F SDUDPHWU\ D N ' SRZLQQ\UDF]HMURVQü
F]\PDOHü"
&]\]HZ]URVWSR]LRPX]DNáyFH SRPLDURZ\FKV SRZLQLHQE\üVSU]
*RQ\]HZ]URVWHP
F]\WH* VSDGNLHPSDUDPHWUyZD N ' "&]\QDOH*DáRE\]PLHQLüZyZF]DVWDNW\N
Z\
]QDF]DQLDUHDOL]DFMLSRFKRGQHM"
635$:2='$1,(32:,112=$:,(5$û
,OXVWUDW\ZQHSU]HND]DQLHZ\QLNyZ±PLQZSRVWDFLWDEHO]X UHGQLRQ\PLZVND(QLNDPL
MDNR FLSURFHVyZRSW\PDOL]DFMLZ]DOH*QR FLRGZ\ERUXSDUDPHWUyZDOJRU\WPXDWDN*H
Z SRVWDFLJUDILF]QHJRSRUyZQDQLDW\SRZ\FKUHDOL]DFMLSURFHVyZRSW\PDOL]DFML
:QLRVNLLNRPHQWDU]H QDWHPDWSU]HSURZDG]RQ\FKEDGDPLQZ\SRZLHG( QDQDVW
SXM
FHWHPDW
.WyUD]HEDGDQ\FKZSPHWRGRSW\PDOL]DFMLRND]DáDVL VNXWHF]QLHMV]"
2GSRZLHG( QDOH*\X]DVDGQLüSRVáXJXMFVL
DUJXPHQWDPLRSDUW\PLQD FLVá\FKZVND(QL
NDFK HIHNWNRFRZ\OLF]EDLWHUDFMLLWS
û:
3U]HJOGD
OJRU\WPyZSURJUDPRZDQLDQLHOLQLRZHJR
*áyZQHVNáDGRZ GDQHJRüZLF]HQLDVWDQRZL SURJUDPRZDQHEDGDQLDV\PXODF\MQH
Z\EUDQ\FK DOJRU\WPyZ SURJUDPRZDQLD QLHOLQLRZHJR : EDGDQLDFK W\FK SURZDG]R
Q\FKSU]\SRPRF\SU]\JRWRZDQHJRSURJUDPX SOLNÄQPVB´ SU]HVWU]H SRV]XNLZD
QLDHNVWUHPXPMHVWGZXZ\PLDURZDDSRPLDU\IXQNFMLNU\WHULDOQHMQLH]DNáyFRQH:FHOX
]LOXVWURZDQLD SUREOHPDW\NL XZ]JO GQLHQLD RJUDQLF]H Z SU]HVWU]HQL SRV]XNLZDQLD
Z SURJUDPLHZSURZDG]RQRPR*OLZR üPRG\ILNDFMLIXQNFMLNU\WHULDOQHM
=DGDQLHP GRGDWNRZ\P MHVW UR]ZL]\ZDQLH LQQ\FK Z\EUDQ\FK ]DGD RSW\PDOL]DFML
Z URGRZLVNXSURJUDPRZ\P0DWODED
35=('û:,&=(1,(0 QDOH*\]DSR]QDüVL
]SRGVWDZRZ\PLURG]DMDPL]DGD SURJUDPR
ZDQLD QLHOLQLRZHJR D WDN*H ] SU]HJOGHP GHWHUPLQLVW\F]Q\FK DOJRU\WPyZ SURJUDPRZDQLD
QLHOLQLRZHJR]]HVWDZLHQLD&
3\WDQLDNRQWUROQH
¾ 3U]HGVWDZVIRUPXáRZDQLH]DGDQLDSURJUDPRZDQLDQLHOLQLRZHJR]RJUDQLF]HQLDPL
¾ -DN FHFK
SRVLDGDM HOHPHQW\]ELRUX3DUHWRGHILQLRZDQHJRZUDPDFKWHRULLRSW\PDOL]D
FMLZLHORZVND(QLNRZHM"
¾
: MDNL VSRVyE PR*QD X]\VNDü HNVSHU\PHQWDOQLH SU]\EOL*RQ ZDUWR ü JUDGLHQWX EDGDQHM
IXQNFMLSU]\PR*OLZR FLEH]SR UHGQLHJRSRPLDUXMHG\QLHMHMZDUWR FL"
¾
:\PLH SRGVWDZRZHW\S\DOJRU\WPyZVWRVRZDQ\FKSU]\UR]ZL]\ZDQLX]DGDQLDSURJUD
PRZDQLDQLHOLQLRZHJR
352*5$02:$1,(2%/,&=( ZWUDNFLHüZLF]HQLDZ\PDJD]DSR]QDQLDVL
]WHFKQLN
RGQDMG\ZDQLD PLQLPXP IXQNFML ZLHOX ]PLHQQ\FK SU]\ SRPRF\ PDWODERZVNLFK PIXQNFML
ÄIPLQX´ L ÄIPLQV´ ] SDNLHWX 2SWLPL]DWLRQ D WDN*H ]DSR]QDQLD VL
] SU]\WRF]RQ\P
QL*HMSURJUDPHPÄ
nms2
´1DOH*\]DSR]QDüVL WDN*H]WHFKQLN SU]\JRWRZ\ZDQLDGDQ\FKGR
UR]ZL]\ZDQLD LQQ\FK SRGVWDZRZ\FK ]DGD RSW\PDOL]DFML VWDW\F]QHM UHSUH]HQWRZDQ\FK
Z W\P SDNLHFLH SURJUDPRZ\P ]DGDQLH SURJUDPRZDQLD OLQLRZHJR ]DGDQLH SURJUDPRZDQLD
NZDGUDWRZHJRL]DGDQLHSURJUDPRZDQLDQLHOLQLRZHJR]RJUDQLF]HQLDPL
3\WDQLDNRQWUROQH
x :MDNLVSRVyEZSURZDG]DP\SU]\VWRVRZDQLXSURJUDPyZÄIPLQX´LÄIPLQV´ LQIRUPD
FM RSRVWDFLDQDOLW\F]QHMRSW\PDOL]RZDQHMIXQNFML"
x .LHG\ZSURZDG]HQLHZ\UD*HQLDDQDOLW\F]QHJRJUDGLHQWXRSW\PDOL]RZDQHMIXQNFMLQLHMHVW
NRQLHF]QH"
x :MDNLVSRVyEXVWDODVL
PDNV\PDOQ OLF]E
LWHUDFMLZSURFHVLHRSW\PDOL]DFML"
x :MDNLVSRVyEPR*QDX]\VNDüSR UHGQLHZ\QLNLSURFHVXRSW\PDOL]DFML"
*áyZQ\SURJUDPEDGD REOLF]HQLRZ\FKDOJRU\WPyZRSW\PDOL]DFMLIXQNFMLZLHOX]PLHQ
Q\FKXPLHV]F]RQRZSOLNX Ä
nms_2.m
´3U]HND]XMHRQ ZSRVWDFLOLF]ERZHMLDQDOLW\F]QHM
SU]HELHJPLQLPDOL]DFMLEDGDQHMIXQNFMLSU]\ ]DVWRVRZDQLX SRGVWDZRZ\FK PHWRG SURJUDPR
ZDQLD QLHOLQLRZHJR 3HáQ\ WHNVW SURJUDPX Ä
nms_2.m
´ XPLHV]F]RQR QD NRFX LQVWUXNFML
: W\PPLHMVFX]DPLHV]F]DP\MHG\QLHMHJRSRF]WHN
%DGDQLHDOJRU\WPyZPLQLPDOL]DFMLIXQNFMLGZX]PLHQQ\FK
3U]HGUR]SRF] FLHPREOLF]H QDOH*\]GHILQLRZDüIXQNFM GZX
]PLHQQ\FK>[ [ @RQD]ZLHPLQIXQLZ\UD*HQLHJUDGLHQWX
ZHGáXJSU]\WRF]RQHJRQL*HMZ]RUXGODIXQNFMLWHVWXMFHMI [
I [ [ A [ A [ [ A A
*5$' > [ A [ [ [ [ A [ A
[ [ [ A [ A EDQSORW [2/'; @
*5$' > [ A [ [ [ [ A [ A
[ [ [ A [ A EDQSORW [2/'; @
HFKRRIIFOFKHOSQPV
IRU N IRU O
[ N N [ O O
ZIXQNFML NO IHYDO PLQIXQ >[ N [ O @
HQG HQG
ILJXUH PHVK ZIXQNFML
VWDUW LQSXW RNUH OSXQNWVWDUWRZ\PLQLPDOL]DFML QS>@
GLVS 1DFL QLMNODZLV]!RWU]\PDV]ZDUVWZLFHEDGDQHMIXQNFML
3U]\DQDOL]LHSURJUDPXZLGRF]Q\PMHVW*HPR*QDSRG]LHOLüJRQDNLONDF] FL
¾ 1DSRF]WNX]DZDUWRLQVWUXNFMHVSRU]G]DQLDZ\NUHV\ IXQNFMLGZX]PLHQQ\FKQDNWyUHM
WHVWRZDQHV PHWRG\SURJUDPRZDQLDQLHOLQLRZHJR3U]HGVWDZLDQ\MHVW]DUyZQRMHMZ\NUHV
Z SU]HVWU]HQL WUyMZ\PLDURZHM ÄPHVK´ MDNL ZDUVWZLFH ÄFRQWRXU´ QD SáDV]F]\(QLH
DUJXPHQWyZ
¾ :GDOV]HMF] FLSURJUDPXÄ
nms_2.m
´ SDWU]'RGDWHN ZLGRF]QHMHVWPHQXL]ZL]DQH
] QLPLQVWUXNFMHVWHUXMFHGDOV]\PSU]HELHJLHPSURJUDPX:]DOH*QR FLRGZ\EUDQHMPH
WRG\XVWDZLDQHV SDUDPHWU\URERF]HSURFHGXU]ZHNWRUDÄ237,216´DQDVW
SQLHZ\ZR
á\ZDQD MHVW PIXQNFMD ÄIPLQX´ OXE ÄIPLQV´ 5yZQROHJOH GR REOLF]H LWHUDF\MQ\FK
U\VRZDQ\ MHVW SU]\ SRPRF\ PIXQNFML ÄEDQSORW´ SU]HELHJ Z\QLNDMFHM VWG WUDMHNWRULL
SRV]XNLZDQLDPLQLPXP
¾ 1DNRFXSURJUDPXSRGDMHVL JáyZQHZ\QLNLRSW\PDOL]DFMLLSU]\JRWRZXMHVL ZDUVWZLFH
SRGQDVW SQ\ SRWHQFMDOQ\ U\VXQHNWUDMHNWRULL
: WUDNFLH EDGD SU]HSURZDG]DQ\FK Z üZLF]HQLX Z\VW SXMH NRQLHF]QR ü VDPRG]LHOQHJR
ZSURZDG]DQLDQLH]QDF]Q\FKPRG\ILNDFMLÄ
nms_2.m
´
3U]\JRWRZDü QDOH*\ PIXQNFM Z\]QDF]DQLD ZDUWR FL PLQLPDOL]RZDQHM IXQNFMD ÄPLQ
IXQ´RNWyUHMPyZLVL
ZÄKHOSLH´SURJUDPX2NUH ODVL
M QDED]LHZ]RUXDQDOLW\F]QHJR
IXQNFML VNRQVWUXRZDQHM QD SRGVWDZLH RWU]\PDQHJR GR UR]ZL]DQLD XNáDGX GZX UyZQD
QLHOLQLRZ\FK
-HGQ\P]DUJXPHQWyZPIXQNFMLÄIPLQX´MHVWDQDOLW\F]QHZ\UD*HQLHJUDGLHQWXEDGDQHM
IXQNFML*5$':\UD*HQLHWRQDOH*\Z\]QDF]\üDQDOLW\F]QLHLZSLVDüQDVDP\PSRF]WNX
SURJUDPXZZ\PDJDQHM SDWU]SU]\NáDGSU]\WRF]RQ\ZÄKHOSLH´ NRQZHQFML
3R]RVWDáHPLHMVFD PRG\ILNDFML SURJUDPX R]QDF]RQH >
@ ZL*VL ] HZHQ
WXDOQ\PL ]PLDQDPLSXQNWXVWDUWRZHJRLSXQNWXVWDQRZLFHJRUR]ZL]DQLH]DGDQLD
35=(%,(*û:,&=(1,$
5R]ZL*SRVáXJXMFVL PIXQNFMDPLÄ
fmins
´L´
fminu
´ UHDOL]XMF\PLDOJRU\WP\SUR
JUDPRZDQLD QLHOLQLRZHJR XNáDG GZX QLHOLQLRZ\FK UyZQD DOJHEUDLF]Q\FK SU]\G]LHORQ\
VSR UyGZ\PLHQLRQ\FKSRQL*HM
[
[
[
[
[
[
± [
[
[
±
[
[
[
[
[
[
[
± [
[
[
[
±
[
[
[
[
±[
[
[
[
±[
:MDNLVSRVyEPR*QDRFHQLüGRNáDGQR üX]\VNDQHJRUR]ZL]DQLD"
:VND]yZND 1DOH*\]GHILQLRZDüPLQLPDOL]RZDQ IXQNFM MDNRVXP NZDGUDWyZOHZ\FKVWURQ
REXUyZQD:DUWR üDUJXPHQWXIXQNFMLZNWyU\PRVLJDRQD UyZQH]HUX PLQLPXPMHVW
MHGQRF]H QLHUR]ZL]DQLHPXNáDGXUyZQD
1DOH*\ SRVáXJXMF SDNLHWHP 2SWLPL]DWLRQ 0DWODED UR]ZL]Dü SU]\G]LHORQH
SU]H]SURZDG]FHJR]DGDQLH] ]DNUHVX SURJUDPRZDQLD OLQLRZHJRSURJUDPRZDQLD NZD
GUDWRZHJROXE SUREOHPX RSW\PDOL]DFML QLHOLQLRZHM ] RJUDQLF]HQLDPL QS ] MZ ] GRGDW
NRZ\PZDUXQNLHP [
[
!
=EDGDM SU]\ SRPRF\ SURJUDPX Ä
nms_2.m
´ JáyZQH ZáDVQR FL QS OLF]ED Z\OLF]H
ZDUWR FLEDGDQHMIXQNFMLQLH]E
GQ\FKGRZ\]QDF]HQLDUR]ZL]DQLDXNáDGX]]DGDQ GR
NáDGQR FL PHWRG SURJUDPRZDQLD QLHOLQLRZHJR UHSUH]HQWRZDQ\FK Z RWU]\PDQ\P SUR
JUDPLH 3RUyZQDM MDNR FLRZH SU]HELHJL WUDMHNWRULL SRV]XNLZDQLD HNVWUHPXP IXQNFML
Z UR]ZL]\ZDQ\PSUREOHPLH
:VND]yZND 1DOH*\ XWZRU]\üP SOLNL ] EDGDQ IXQNFM L MHM JUDGLHQWHP OXE RGSRZLHGQLR
]PRG\ILNRZDü ± Z\MD QLHQLD QD SRSU]HGQLFK VWURQDFK LQVWUXNFML RWU]\PDQ\ SURJUDP
REOLF]H3RGVWDZRZ\P URGNLHPVWRVRZDQ\PZWUDNFLHEDGD REOLF]HQLRZ\FKMHVWRG
QRWRZ\ZDQLH VNXWNyZ Z\UD*DQ\FK ]DUyZQR OLF]ERZR MDN L QD SU]HELHJX X]\VNLZDQHM
WUDMHNWRULL VWRSQLRZHJR]ZL
NV]DQLDOLF]E\LWHUDFMLSU]HSURZDG]DQ\FKEDGDQ PHWRG
635$:2='$1,(32:,112=$:,(5$û
¾ 5R]ZL]DQLDRWU]\PDQ\FK]DGD RSW\PDOL]DFML
¾ :\QLNLSU]HSURZDG]RQ\FKEDGD V\PXODF\MQ\FKDOJRU\WPyZ SURJUDPRZDQLDQLHOLQLR
ZHJR SU]HGVWDZLRQH Z LOXVWUDW\ZQHMIRUPLH QS SRUyZQDZF]R Z XNáDG]LH WDEHODU\F]
Q\PLJUDILF]QLH
¾ :QLRVNLLNRPHQWDU]HQDWHPDWSU]HSURZDG]RQ\FKEDGD
û: 2SW\PDOL]DFMDIXQNFMLZLHOX]PLHQQ\FK
ZREHFQR FL]DNáyFH SRPLDURZ\FK
:SURJUDPLHEDGD SU]HELHJyZSURFHVyZRSW\PDOL]DFMLJáyZQ\PMHVWSUREOHPZ\
]QDF]HQLDVWHURZDQLD]DSHZQLDMFHJRRVLJQL FLH ZVHQVLHVWDW\VW\F]Q\P HNVWUHPXP
ZVND(QLNDMDNR FLVWHURZDQHJRSURFHVX:HUVMDZLHORZ\PLDURZDWHJR]DGDQLDMHVWMHG
Q\P] JáyZQ\FKVNáDGQLNyZSUREOHPDW\NLVWHURZDQLDSURFHVHPWHFKQRORJLF]Q\PXNLH
UXQNRZDQHJR QD RVLJQL FLH PR*OLZLH QDMOHSV]\FK UHGQLR U]HF] ELRUF HIHNWyZ
VWHURZDQLD
:SURJUDPLHEDGD REOLF]HQLRZ\FKÄQPVBP´JáyZQ XZDJ ]ZUyFRQRQDG]LD
áDQLH DOJRU\WPyZ RSW\PDOL]DFML VWDW\F]QHM Z V\WXDFMDFK QLHGHWHUPLQLVW\F]Q\FK =QDM
GRZDQHMHVWWXPLQLPXPSHZQHMIRUP\NZDGUDWRZHMSRPLDU\ WXWDM REOLF]HQLD NWyUHM
V DGG\W\ZQLH ]DNáyFDQH SU]\ ]DVWRVRZDQLX JHQHUDWRUD OLF]E ORVRZ\FK ûZLF]F\ PD
PR*OLZR üÄMHGQRF]HVQHJR´SU]HSURZDG]HQLDLDQDOL]\Z\QLNyZRNUH ORQHMVHULLSU]H
ELHJyZRSW\PDOL]DFML
35=('û:,&=(1,(0 QDOH*\]DSR]QDüVL
QDSRGVWDZLH]HVWDZLHQLD'LVWXGLyZOLWHUD
WXURZ\FK ] SUREOHPDW\N UR]ZL]\ZDQLD ]DGDQLD RSW\PDOL]DFML VWDW\F]QHM Z REHFQR FL ]D
NáyFH SRPLDURZ\FK
3\WDQLDNRQWUROQH
¾ -DNLFLJF]DVRZ\QD]\ZDP\VWDFMRQDUQ\P"
¾ : MDNL VSRVyE V]HURNR ü SU]HG]LDáX XIQR FL SU]\ SRPLDU]H GDQHM ZLHONR FL IL]\F]QHM
Z REHFQR FLDGG\W\ZQ\FK]DNáyFH RUR]NáDG]LHQRUPDOQ\P ]DOH*\RGLFKZDULDQFMLLRG
OLF]E\Z\NRQDQ\FKSRPLDUyZ"
352*5$02:$1,( %$'$ 6<08/$&<-1<&+ Z WUDNFLH üZLF]HQLD ]QDF]QLH XáD
WZLD SU]\WRF]RQ\ QD NRFRZ\FK VWURQDFK LQVWUXNFML SURJUDP Ä
nms_3
´ SRF]WHN NWyUHJR
SU]HGVWDZLRQRSRQL*HM
6\PXODFMDSURFHVXPLQLPDOL]DFMLIXQNFMLGZX]PLHQQ\FK]SRPLDUDPL
]DNáyFRQ\PLDGG\W\ZQ ]PLHQQ ORVRZ S] UDQGQ
0LQLPDOL]RZDQDMHVWIRUPDNZDGUDWRZD I [ X $X S] UDQGQ
]PLQLPXPZSXQNFLHXRSW >@I XRSW
6WDáHZ\M FLRZH
$ $ $ $
HFKRRIIFOFKHOSQPV
IRU N IRU O
X N N X O O
INZ NO >X N X O @ $ >X N X O @
HQG HQG
GLVS 1DFL QLMNODZLV]ZFHOXRWU]\PDQLDZ\NUHVXEDGDQHMIXQNFML
SDXVHILJXUH PHVK INZ QU
S] LQSXW :SURZDG( SR]LRP]DNáyFH
GLVS 2EHMU]\MQDZ\NUHVLHVNXWNLZSURZDG]HQLD]DNáyFH
SDXVHPHVK INZS] UDQGQ
GLVS 1DFL QLMGRZROQ\NODZLV]
$QDOL]D]DZDUWR FLSURJUDPXÄ
nms_3
´ SDWU]'RGDWHN QLHVSUDZLDSU]\]QDMRPR FLSUR
JUDPXEDGD ]SRSU]HGQLHJRüZLF]HQLDZL
NV]\FKWUXGQR FL2ELHNRQVWUXNFMHSURJUDPRZH
V EDUG]R]EOL*RQH.RPHQWXMFQDMRJyOQLHMZ SURJUDPLH W\P PR*HP\ Z\UD(QLH Z\Uy*QLü
WU]\F]
FL3LHUZV]D]QLFKSU]\JRWRZDZF]DSR]ZDODQDREHMU]HQLHWUyMZ\PLDURZ\FKZ\
NUHVyZRSW\PDOL]RZDQHMIXQNFML ]DUyZQRZSRVWDFLGHWHUPLQLVW\F]QHMMDNL]QLHNV]WDáFRQHM
]DNáyFHQLDPL D WDN*H QD XVWDOHQLH SDUDPHWUyZ SURFHVX V\PXODFML SRV]XNLZDQLD PLQLPXP
:\ELHUDP\ WX PHWRG
REOLF]H OLF]E
LWHUDFML LOR ü SRZWyU]H ÄSRPLDUyZ ZLHONR FL Z\M
FLRZHMRELHNFLH´SDUDPHWURNUH ODMF\GáXJR üSRMHG\QF]HJRNURNXLWS:SR]RVWDá\FKF]
FLDFK SURJUDPX RGG]LHORQ\FK OLQLDPL NUHVNRZDQ\PL UHDOL]RZDQH V REOLF]HQLD
RSW\PDOL]DF\MQH PHWRG SURJUDPX NURF]FHJR VWDQRZLFHJR GZXZ\PLDURZ\ DQDORJ SUR
JUDPX RSW\PL]DWRUD ] üZLF]HQLD PHWRG VLPSOHNVyZ SURVW\FK D QDVW
SQLH PHWRG JUD
GLHQWXVWRFKDVW\F]QHJR
:\QLNL REOLF]H SU]HND]\ZDQH V Z SRVWDFL JUDILF]QHM MDNR WUDMHNWRULD SRV]XNLZDQLD
RSWLPXPL QLHNWyUHQDMEDUG]LHMLVWRWQH ZSRVWDFLOLF]ERZHM:DUWRSDPL
WDüUyZQLH* RW\P
*H Z\GUXN SHáQHJR SURWRNRáX REOLF]H PR*HP\ X]\VNDü QDMSUR FLHM SRSU]H] ]DNRF]HQLH
LQVWUXNFMLZ\]QDF]DMF\FKQRZHZDUWR FLDUJXPHQWyZEDGDQHMIXQNFMLZWUDNFLHRSW\PDOL]D
FMLSU]HFLQNDPL]DPLDVW UHGQLNyZ
35=(%,(*û:,&=(1,$
3U]HSURZDG( GR ZLDGF]HQLD V\PXODF\MQH LOXVWUXMFH VHNZHQF\MQ\ SURFHV GRERUX GZX
RGG]LDá\ZD ZHM FLRZ\FK X >X
X
@
7
ZSU]\SDGNXNZDGUDWRZHJRPRGHOXVWDW\F]QH
JRVWHURZDQHJRSURFHVX
\ I X X
7
$
X
SRPLDU\Z\M FLDNWyUHJRV ]DNáyFDQHDGG\W\ZQ\PLRGG]LDá\ZDQLDPLORVRZ\PLRUR]NáD
G]LHQRUPDOQ\P ] 1 V
&HOHPGR ZLDGF]H MHVWPLQLPDOL]DFMDZDUWR FL UHGQLHM
ZLHONR FL Z\M FLRZHM SURFHVX 3RVáXJXMHP\ VL SU]\ W\P SURJUDPHP Ä
nms_3
´
] PRG\ILNDFM SROHJDMF QD ZSURZDG]HQLX QD SRF]WNX SURJUDPX ]DGDQHM SRVWDFL GR
GDWQLRRNUH ORQHMPDFLHU]\
A
Z IRUPLHNZDGUDWRZHM
: WUDNFLH EDGD QDOH*\ SRUyZQDü HIHNW\ZQR ü SURFHVX RSW\PDOL]DFML PLHU]RQ VXPD
U\F]Q\PLVWUDWDPLQDSRV]XNLZDQLHHNVWUHPXPSU]\]DVWRVRZDQLXZV]\VWNLFKUHSUH]HQ
WRZDQ\FKZSURJUDPLHÄ
nms_3
´PHWRGSU]\]DVWRVRZDQLXUy*Q\FKZDUWR FLXVWDODQ\FK
Z\M FLRZR SDUDPHWUyZ DOJRU\WPyZ 3XQNW\ VWDUWRZH Z SáDV]F]\(QLH SRV]XNLZDQLD
XVWDODP\GHWHUPLQLVW\F]QLHOXEORVRZR
&]\SU]HELHJSURFHVXRSW\PDOL]DFMLLVWRWQLH]DOH*\RGSXQNWXVWDUWRZHJR":FHOXRGSR
ZLHG]LQDWRS\WDQLHSU]HSURZDG( V\PXODFM
SU]\SRPRF\W\FKVDP\FKDOJRU\WPyZVWDU
WXMF]SXQNWyZSRáR*RQ\FKQDUy*Q\FKRVLDFKHOLSVRLGDOQ\FKZDUVWZLFEDGDQHMIXQNFML
Z WHM VDPHM RGOHJáR FL RG SRF]WNX XNáDGX ZVSyáU]
GQ\FK 1DOH*\ WDN*H VFKDUDNWHU\]R
ZDü VWDFMRQDUQH SURFHV\ ORVRZH REVHUZRZDQH SU]\ EáG]HQLX SURFHVX SRV]XNLZDQLD
Z SREOL*XRSWLPXP
&]\SURFHVRSW\PDOL]DFMLLVWRWQLH]DOH*\RGSR]LRPX]DNáyFH"&]\ZSá\ZDRQQDUDFMR
QDOQ\ GREyU SDUDPHWUyZ Z VWRVRZDQ\FK DOJRU\WPDFK" 1DOH*\ SU]HSURZDG]Lü SURJUDP
EDGD V\PXODF\MQ\FKSR]ZDODMF\QDXVWDOHQLH]DOHFH RGQR QLH]PLDQ\ZDUWR FLNURNX
ZDOJRU\WPLHSU]\Z]UR FLHVSDGNXSR]LRPX]DNáyFH ZDQDOL]RZDQHMV\WXDFML
:5$0$&+û:,&=(1,$8=83(à1,$-&(*2
1DOH*\]DPRGHORZDüXNáDGUHJXODFMLHNVWUHPDOQHM]SU]HPLHQQ\PGRVWUDMDQLHPSRV]F]H
JyOQ\FKVWHURZD U\V 3R ]DPRGHORZDQLX ]DUHMHVWURZDü W\SRZHSU]HELHJL VWHURZDQLD
L Z\M FLDVWHURZDQHJRSURFHVX
du/dt
5\V2SW\PDOL]DWRU]SU]HPLHQQ\PGRVWUDMDQLHPVWHURZD
635$:2='$1,(32:,112=$:,(5$û
*áyZQHZ\QLNLSU]HSURZDG]RQ\FKEDGD V\PXODF\MQ\FK±PLQSU]HGVWDZLRQHSRUyZ
QDZF]RZXNáDG]LHWDEHODU\F]Q\PLJUDILF]QLH
:QLRVNLLNRPHQWDU]HQDWHPDWSU]HSURZDG]RQ\FKEDGD LRGSRZLHG]LQDS\WDQLDSRVWD
ZLRQHZSURJUDPLHüZLF]HQLD
&]\]DVWRVRZDQHPHWRG\GDá\]EOL*RQHZ\QLNL"&]\WH* QDOH*\MHX]QDü]DLVWRWQLHUy*QH"
.WyUD]HVWRVRZDQ\FKPHWRGRSW\PDOL]DFMLRND]DáDVL
VNXWHF]QLHMV]"
2GSRZLHG( X]DVDGQLMQDWOH FLVá\FKZVND(QLNyZ HIHNWNRFRZ\SU]\]DGDQHMOLF]ELHSR
PLDUyZEDGDQHMIXQNFMLVXPDU\F]QHVWUDW\ZWUDNFLHSRV]XNLZDQLDHNVWUHPXPLWS
¦
f
f
N
N
D
=HVWDZLHQLH
$
$OJRU\WP\RSW\PDOL]DFMLIXQNFMLMHGQHM]PLHQQHM
0HWRG\LWHUDF\MQH
V SRGVWDZ NRQVWUXNFMLDOJRU\WPyZ]QDMGRZDQLDHNVWUHPXPIXQNFML
3RGVWDZRZDLGHDW\FKPHWRG]ZL]DQDMHVW] RNUH OHQLHPW]ZIXQNFML LWHUDF\MQHM Z\
]QDF]DMFHMQDVW SQ WMZQDVW SQHMLWHUDFML ZDUWR ü DUJXPHQWX EDGDQHM IXQNFML
u(k+1)
QDSRGVWDZLHMHJRP XSU]HGQLFKZDUWR FL
u(k),u(k-1),...,u(k-m+1)
LZDUWR FLED
GDQHMIXQNFML
f(u)
DWDN*HF]DVDPLMHMSRFKRGQ\FK ZW\FKSXQNWDFK3XQNWVWDUWRZ\
Z\ELHUDVL QDSRGVWDZLHLQIRUPDFMLGRVW SQHMSU]HGUR]SRF] FLHPSURFHVXSRV]XNLZDQLD
3U]HGVWDZLP\]ZL (OHPHWRG\LWHUDF\MQHRJUDQLF]DMFVL GRQDMSURVWV]\FKLQDMF] FLHM
VWRVRZDQ\FK PLQ EH]SR UHGQLH ]DVWRVRZDQLD Z XU]G]HQLDFK WHFKQLF]Q\FK 5R]ZD*DQLD
E
G]LHP\ SURZDG]Lü MH QD WOH SUREOHPX PLQLPDOL]DFML IXQNFML
f(u),
UyZQRZD*QHJR SUR
EOHPRZLPDNV\PDOL]DFMLWHM*HIXQNFML]XMHPQ\P]QDNLHP
[
-
f(u)].
$OJRU\WP\SRFKRGQHM
Z\QLNDM ]H ]QDQHJR IDNWX *H PLQLPXP FLJáHM L SRVLDGDMFHM
FLJá SRFKRGQ IXQNFMLZ\SXNáHMZGyá]QDMGXMHVL
ZSXQNFLH]HURZ\PWHMSRFKRGQHMWM
Z WDNLPSXQNFLHX
ZNWyU\P
df(u)/du|
u=u*
=0.
7\SRZDSRVWDüDOJRU\WPXSRFKRGQHM
u(k+1) = u(k) -
D
(k) [df(u)/du]|
u(k)
, k = 0,1,2,...
:DUXQNLHPNRQLHF]Q\P]ELH*QR FLGDQHJRDOJRU\WPXGRRSW\PDOQHJR
u = u*,
QLH
]DOH*QLHRGZ\ERUXSXQNWXSRF]WNRZHJR
u(0),
MHVWZ\EyUZVSyáF]\QQLNyZV]HUHJXD
(k)
ZSRVWDFL]DSHZQLDMFHM
&HFK W SRVLDGDM QSV]HUHJL D
(k)=
D
=const,
OXE D
(k) =
D
/k
w
JG]LH
Z\NáDGQLNSRW JL
w
[0,1].
:DUXQHNGRVWDWHF]Q\]ELH*QR FLZL*HVL ]NRQLHF]QR FL RJUDQLF]HQLDVWDáHMD GRSR
]LRPXZ\QLNDMFHJR]PDNV\PDOQHMZDUWR FLE GUXJLHMSRFKRGQHMRSW\PDOL]RZDQHMIXQNFML
Z REV]DU]HSRV]XNLZD
:
D
< 2
E
-1
.
,VWRWQ\PGRSXV]F]DOQ\PXSURV]F]HQLHPMHVW DOJRU\WP]QDNXSRFKRGQHM NWyU\G]LDáD
Z\áF]QLHZRSDUFLXRMHGQR]QDF]QLHRNUH ORQ\NLHUXQHNSRV]XNLZD
u(k+1) = u(k) -
D
(k)sign[df(u)/du]|
u(k)
, k=0,1,2,...
:DUXQHN SXQNWRZHM ]ELH*QR FL DOJRU\WPX Z WHM ZHUVML Z\PDJD MX* MHGQDN ]G*DQLD
ZUD]]HZ]URVWHPQXPHUXLWHUDFMLNGRfV]HUHJXD N GR]HUD,QDF]HMGRFHORZRRWU]\PXMH
VL
RVF\ODFMHZRNyáVWDFMRQDUQHJRSXQNWX
u*.
3RGNUH OP\*H]MDZLVNRWHJRW\SXSU]\RJUD
QLF]RQHMGRSXV]F]DOQHMDPSOLWXG]LHRVF\ODFMLQLHPXVLE\üFHFK XMHPQ'RGDWNRZRSU]\
Z\VW
SXMFHMQSZZLHOXV\VWHPDFKWHFKQLF]Q\FK]PLHQQR FLSRáR*HQLDSXQNWXRSW\PDOQHJR
X]\VNXMHVL
QDZHWSU]\UDFMRQDOQ\PZ\ERU]HVWDáHMD
,
FHQQ ZáDVQR ü OHG]HQLDSRWHQFMDO
Q\FK]PLDQ0yZLP\ZyZF]DVRW]ZVWHURZDQLXQDHNVWUHPXP
$OJRU\WP1HZWRQD MHVWXGRVNRQDORQ ZHUVM DOJRU\WPXSRFKRGQHMZ\PDJDMF MHGQDN
UyZQLH* ]QDMRPR FLGUXJLHMSRFKRGQHM
u(k+1)=u(k)-[d
2
f(u)/du
2
]
-1
|
u(k)
[df(u)/du]|
u(k)
, k=0,1,2,...
0R*QD Z\ND]Dü *H MH*HOL IXQNFMD NU\WHULDOQD MHVW NZDGUDWRZ W]Q W\SX
f(u)=au
2
+bu+c;
WRG]LDáDMF]JRGQLH]DOJRU\WPHP1HZWRQDPLQLPXPMHM]QDMGX
MHP\MX* SRSLHUZV]HMLWHUDFML=QDMRPR üRGZURWQR FLGUXJLHMSRFKRGQHMSR]ZDODZyZF]DV
XVWDOLüMHGQR]QDF]QLHZDUWR üZVSyáF]\QQLNDD
.
$OJRU\WPVNRF]RQ\FKSU]\URVWyZ VWRVRZDQ\ MHVWZV\WXDFMDFKZNWyU\FKQLHW\ONR
GUXJDOHF]LSLHUZV]DSRFKRGQDIXQNFMLNU\WHULDOQHMQLHMHVWGRVW
SQD=]DáR*HQLDPR*HP\
SRVáX*\ü VL
ZyZF]DV Z\áF]QLH SRPLDUDPL VDPHM IXQNFML 3URZDG]L WR GR NRQLHF]QR FL
SU]\EOL*HQLDZDUWR FLSRFKRGQHMLORUD]HPVNRF]RQ\FKSU]\URVWyZ:UH]XOWDFLH
u(k+1) = u(k)-
D
(k)[
'
f(u) ]|
u(k)
/ 2
'
u , k=0,1, 2,...;
D
N
X
N
N
'
f
¦
f
SU]\F]\P
[
'
f(u) ]|
u(k)
= f(u(k)+
'
u) - f(u(k)-
'
u);
JG]LH '
u
PDá\SU]\URVWZDUWR FLDUJXPHQWX=ELH*QR üGRSXQNWX
u*
JZDUDQWRZDQDMHVWWX
SRSU]H]VSHáQLHQLHQDVW SXMFHJRGRGDWNRZHJRRJUDQLF]HQLD
:L*HVL WR]NRQLHF]QR FL ]PQLHMV]DQLDZUD]]HZ]URVWHPQXPHUXLWHUDFML
k
WHVWXM
F\FKSU]\URVWyZDUJXPHQWX'
u.
:SUDNW\FHUR]VGQ\NRPSURPLVPR*HVWDQRZLüUyZQLH* DOJRU\WP]QDNXVNRF]RQ\FK
SU]\URVWyZ G]LDáDMF\ QD ED]LH RNUH OHQLD ]QDNX SU]\URVWX EDGDQHM IXQNFML SU]\ WHFKQROR
JLF]QLHX]DVDGQLRQ\PSU]\UR FLHDUJXPHQWX'X
u(k+1) = u(k) -
D
(k) sign[
'
f(u)]|
u(k)
, k=0,1,2,...
5DFMRQDOQ\PZ\UD*HQLHPPR*OLZ\FKWXXSURV]F]H MHVWSRáF]HQLHNURNXWHVWXMFHJR]
URERF]\P 2WU]\PXMHP\ ZyZF]DV SURVW\ L F]
VWR VWRVRZDQ\ Z XU]G]HQLDFK WHFKQLF]Q\FK
DOJRU\WPW]ZUHJXODWRUDNURF]FHJR
u(k+1) = u(k) -
'
u sign[
'
f(u)]|
u(k)
, k=0,1,2,...
$OJRU\WP\ ]ZL]DQH ] UHGXNFM SU]HG]LDáX QLHRNUH ORQR FL
Z NWyU\FK
]QDMGXMH VL
PLQLPXP EDGDQHM IXQNFML XQLPRGDOQHM MHGHQ SXQNW HNVWUHPDOQ\ Z EDGDQ\P
SU]HG]LDOH ]ZL]DQHV ]QLHFRLQQ PHWRG\N3RGVWDZ LFKG]LDáDQLDQLHMHVWUXFKGRSDVR
ZDQ\ GRSRFKRGQHMMDNSRSU]HGQLROHF]LGHDV]\ENLHJR]PQLHMV]DQLDSRND*G\PSRPLDU]H
OXEREOLF]HQLX IXQNFMLNU\WHULDOQHMSU]HG]LDáXZNWyU\PPR*H]QDMGRZDüVL
SRV]XNLZDQH
PLQLPXP6WDUWQDVW
SXMH]GRVWDWHF]QLHGX*HJR]]DáR*HQLD]QDQHJRSU]HG]LDáX
u
[a,b],
ZNWyU\P]QDMGXMHVL PLQLPXPEDGDQHMIXQNFML
3U]HGVWDZLP\GZDHIHNW\ZQHDOJRU\WP\WHJRURG]DMX
$OJRU\WP]áRWHJRSRG]LDáX QD]Z ]DZG]L F]D FKDUDNWHU\VW\F]QHMGOD WHMPHWRG\ OLF]ELH J
=(sqrt(5)-1)/2 = 0,618,
Z\W\F]DMFHM ]GDQLHP VWDUR*\WQ\FK *UHNyZ GRVNRQDáH
SURSRUFMHNV]WDáWXSURVWRNWD3RMHJRSU]HDQDOL]RZDQLXáDWZR]DXZD*\ü*HV]HURNR üEDGD
QHJRSU]HG]LDáXPDOHMHSURSRUFMRQDOQLHGR J
k
= 0,618
k
.
-H*HOL
x SU]H]
u
a
i u
b
R]QDF]\P\GZDSXQNW\]ZQ WU]DUR]SDWU\ZDQHJRSU]HG]LDáXLXVWDOLP\
SRF]WNRZHLFKZDUWR FLSRSU]H]
u
a
(0)= b -
J
(b-a); u
b
= a +
J
(b-a);
x Z SXQNWDFK W\FK ] ]DáR*HQLD PDP\ PR*OLZR ü Z\NRQDQLD SRPLDUyZ OXE Z\OLF]HQLD
ZDUWR FL
f(u
a
) i f(u
b
) ;
WR DOJRU\WP]áRWHJRSRG]LDáXVSURZDG]DVL GRZ\NRQ\ZDQLDQDND*G\PNROHMQ\PHWDSLH
W\FKVDP\FK G]LDáD ]ZL]DQ\FK ] QRZ\P MHGQ\P SRPLDUHP OXE Z\OLF]HQLHP ZDUWR FL
EDGDQHMIXQNFML3U]HGVWDZLP\JRVWRVXMFNRQZHQFMH]DSLVXSU]\M
WHZ0DWODELH
']LDáDQLDVWDUWRZH =DNáDGDVL
LVWQLHQLHPIXQNFMLÄEDGDQDI´
JDPPD
VTUW
XD E±JDPPD
ED XE DJDPPD
ED
ID EDGDQDI XD IE EDGDQDI XE
2EOLF]HQLD BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
ZKLOH DEV ED !.U\WHULXP67238
]PQLHMV]HQLHEDGDQHJRSU]HG]LDáX
LI
IE!IDE XE XE XD IE ID
XD E±JDPPD ED ID EDGDQDI XD
HOVH
D XD XD XE ID IE
XE DJDPPD ED IE EDGDQDI XE
HQG
HQG
XRSW DE IRSW EDGDQDI XRSW Z\QLNREOLF]H
$OJRU\WP.LHIHUD Z\ZRG]LVL
WDN*H]LGHLUHGXNFMLSU]HG]LDáX]DZLHUDMFHJRPLQLPXPED
GDQHMIXQNFML:\PLHQLP\JáyZQHUy*QLFHZVWRVXQNXGRPHWRG\]áRWHJRSRG]LDáX
x .RU]\VWDVL ]OLF]E)LERQDFFLHJR
F
o
,F
1
,F
2
,...
GHILQLRZDQ\FKSRSU]H]
F
i
= 1, dla i = 0,1 ; F
i
= F
i-1
+ F
i-2
, dla i
t
2.
x .U\WHULXP6723XRNUH ODVL ]ZDUXQNX
F
k
<|b – a|/
H d
F
k+1
SU]\SR UHGQLF
WZLH]DGDQHM]JyU\OLF]E\LWHUDFMLN )DNWWHQPR*HE\üWHFKQLF]QLHZ\JRGQ\ZSU]\SDGNX
NRQLHF]QR FLVSHáQLHQLDZDUXQNX]ZL]DQHJR]RJUDQLF]HQLHPFDáNRZLWHJRF]DVXUR]ZL
]DQLD]DGDQLDRSW\PDOL]DFMLZV\VWHPDFKF]DVXU]HF]\ZLVWHJR
x 1DVWDUFLHZ\]QDF]DVL ZDUWR FLDUJXPHQWX
:
ua = a + F
k-1
(b-a)/F
k+1
; ub = a + b - ua ,
DQDVW SQLHZDUWR FLIXQNFML
fa = f(ua) i fb = f(ub
ZW\FKSXQNWDFK
x 3RGVWDZRZHG]LDáDQLDND*GHMLWHUDFMLV DQDORJLF]QH GR SU]\WRF]RQ\FK Z\*HM G]LDáD Z
DOJRU\WPLH]áRWHJRSRG]LDáX
-H*HOL
f(ub)>f(ua)
WR
: b =ub ; ub = ua ; fb=fa;
ua = a + b - ua ; fa = f(ua).
-H*HOL
f(ub)
d
f (ua)
WR
a = ua ; ua = ub; fa=fb ;
ub = a + b - ub ; fb=f(ub).
$OJRU\WPLQWHUSRODFMLNZDGUDWRZHM PR*HSRVáX*\ü]DSU]\NáDG]DVWRVRZDQLDPHWRG]OR
NDOQ DSURNV\PDFM IXQNFML Z SURFHVLH RSW\PDOL]DFML 2SLHUD VL
RQ QD VSRVWU]H*HQLX *H
V]F]HJyOQLHZRWRF]HQLXPLQLPXP EDGDQ IXQNFM
PR*QDF]
VWR]GRVWDWHF]Q GRNáDGQR FL
SU]\EOL*DüWUyMPLDQHPNZDGUDWRZ\P
: f(u) = a u
2
+ b u + c
NWyUHJRHNVWUH
PXPMDNZLDGRPR]QDMGXMHVL ZSXQNFLH
u* = - b/2a
.
3R]ZDODWRQDSRGVWDZLH]QDMRPR FLEDGDQHMIXQNFML
f(u)
ZWU]HFKSXQNWDFK
f(u
1
),
f(u
2
), f(u
3
)
]QDOH(ü ]D SR UHGQLFWZHP ZLHORPLDQX LQWHUSRODF\MQHJR /DJUDQJH¶D QD
VW SQWHRUHW\F]QLHOHSLHMSRáR*RQ ZDUWR üDUJXPHQWX
u(k+1)=(
E
23
f(u
1
)+
E
31
f(u
2
)+
E
12
f(u
3
))/2(
J
23
f(u
1
)+
J
31
f(u
2
)+
J
12
f(u
3
));
gdzie:
E
ij
= u
2
i
- u
2
j
;
J
ij
= u
i
- u
j
; {i,j}={2,3},{3,1}{1,2}.
:FHOX]DFKRZDQLDZáDVQR FLLQWHUSRODF\MQ\FKZDOJRU\WPLHSRZLQQRVL GRGDWNRZR]D
GEDüRWDN NRQW\QXDFM E\
: f(u
2
) < f(u
1
) i f(u
2
) < f(u
3
)
SU]\
u
1
< u
2
< u
3
.
.RQLHF]QR üVWRVRZDQLD SRPRFQLF]R UR]EXGRZDQ\FK]DOH*QR FLORJLF]Q\FKZFHOX]DJZD
UDQWRZDQLD ]ELH*QR FL DOJRU\WPX MHVW ]UHV]W FHFK FKDUDNWHU\VW\F]Q DOJRU\WPyZ G]LDáDM
F\FK QD ED]LH HNVWUDSRODF\MQR ± LQWHUSRODF\MQHM >@ $OJRU\WP\ WH Uy*QL VL
VWRSQLHP
VWRVRZDQHJR ZLHORPLDQX LQWHUSRODF\MQHJR 6WRVXQNRZR SRSXODUQ\P MHVW UyZQLH* DOJRU\WP
LQWHUSRODFMLV]H FLHQQHM
.U\WHULXPVWRSXZL*HVL SRGREQLHMDNZSRSU]HGQLRSU]HGVWDZLRQ\FKPHWRGDFK]V\
WXDFM ZNWyUHMRVLJDQDSRSUDZDRFHQ\PLQIXQNFMLQDMHGQ\PHWDSLH]PDOHMHSRQL*HM]DGD
QHJRVOXEJG\VDPSU]HG]LDá]PDOHMHGR °
u
3
- u
1
°
<
H
.
¦
f
f
N
N
D
D
N
N
f
¦
f
=HVWDZLHQLH
%
6SHF\ILNDRSW\PDOL]DFMLZREHFQR FL]DNáyFH SRPLDURZ\FK
:\]QDF]DQLH RSW\PDOQ\FK ZDUXQNyZ
UHDOL]DFML SURFHVX WHFKQRORJLF]QHJR Z\PDJD
SU]\]DáR*HQLXQLHVWDFMRQDUQR FLOXEQLH]QDMRPR FLSRVWDFLIXQNFMLNU\WHULDOQHM SU]HSURZD
G]HQLDRNUH ORQHJRSURJUDPXEDGD HNVSHU\PHQWDOQ\FKEH]SR UHGQLRÄQDRELHNFLH´:SUR
JUDPLH WHJR URG]DMX F]
VWR QLH RGWZDU]D VL
SHáQHM SRVWDFL IXQNFML NU\WHULDOQHM D MHG\QLH
]QDMGXMHVL
MHMHNVWUHPXP FL OHMU]HF]XMPXMF]UHJXá\ZSUDNW\FH]D]DGDZDODMFHUR]
ZL]DQLH]DGDQLDXZD*DQHMHVW]QDOH]LHQLHVL
ZGRVWDWHF]QLHEOLVNLP ]SXQNWXZLG]HQLDFHOX
RSW\PDOL]DFML RWRF]HQLXDNWXDOQHJRSRáR*HQLDSXQNWXHNVWUHPDOQHJR:SUDNW\FHVWRVXMHVL
ÄRVWUR*QH´ ]ZLHORNURWQLHQLHSRPLDUXZLHONR üNURNX ZHUVMHQDMSURVWV]\FKPHWRGGHWHUPL
QLVW\F]Q\FK
5R]ZD*P\SU]\GDWQR üGRWHJRFHOXSU]\WRF]RQ\FKZGRGDWNXSRSU]HGQLPDOJRU\WPyZ
LWHUDF\MQ\FKRSW\PDOL]DFMLMHGQHM]PLHQQHM
$OJRU\WPSRFKRGQHM PR*H E\üVWRVRZDQ\RF]\ZL FLHMHG\QLHSU]\GRVW
SQR FLSRPLDUyZ
SRFKRGQHMIXQNFMLNU\WHULDOQHM:ZDUXQNDFKQLHVNRUHORZDQ\FKUHDOL]DFMLVWDFMRQDUQ\FK] D
NáyFH VWRFKDVW\F]Q\FKSRPLDUyZSRFKRGQHM]ELH*QR üWHJRDOJRU\WPXZVHQVLH UHGQLRNZD
GUDWRZ\PWM]JZDUDQFM
E
{(u(k)-u*)
2
}
Æ
0, przy k
Æf
;
]DSHZQLDRGSRZLHGQLGREyUFLJXZVSyáF]\QQLNyZD N =JRGQLH]WZLHUG]HQLHP5RE
ELQVD0RQURRDSURNV\PDFMLVWRFKDVW\F]QHM>@V]HUHJWHQSRZLQLHQVSHáQLDüZDUXQNL
QSD
(k) =
D
/k
w
, gdzie: w
[0 , 1].
3RGNUH OP\WDN*H*HVNXWNLHPWHJRWZLHUG]HQLDMHVWWDN*H]ELH*QR ü UHGQLRNZDGUDWRZD
SURFHGXU RSDUW\FK QD X UHGQLDQLX RNUH ORQHM OLF]E\ SRPLDUyZ SRFKRGQHM 3RZWDU]DQLH GR
ZLDGF]H ] X UHGQLHQLHP Z\QLNyZ SURZDG]L ERZLHP MHG\QLH GR ]PQLHMV]HQLD ZDULDQFML
PLHU]RQHMZLHONR FL
-H*HOL ;
1
P V
WR ;
1
1
P V
1
SU]\]DFKRZDQLXLQQ\FKZDUXQNyZ]ELH*QR FLDOJRU\WPX
7 VDP XZDJ PR*QDRGQLH üUyZQLH* ZVWRVXQNXGR]ELH*QR FLDOJRU\WPXRSDUWHJRQD
]QDNXSRFKRGQHM
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
$OJRU\WPSU]\URVWyZ ZL*HVL
MX* ]SRPLDUDPLZDUWR FLEDGDQHMIXQNFML%H]SR UHGQLH
SRPLDU\SRFKRGQHMERZLHPQLH]DZV]HV ZSUDNW\FHGRVW
SQH8VWDODMFZDUWR üSRFKRGQHM
SU]\SRPRF\LORUD]XUy*QLFRZHJRSRZLQQL P\Z\NRQDüGZDSRPLDU\ZLHONR FLNU\WHULDOQHM
3RGNUH OP\]ZL]HNSDUDPHWUyZUR]NáDGXSUDZGRSRGRELHVWZDVXP\ Uy*QLF\ ]PLHQ
Q\FKORVRZ\FK]SDUDPHWUDPLVDP\FK]PLHQQ\FK
MH*HOL
X
1
1
(
P
1
,
V
1
2
), X
2
1
(
P
2
,
V
2
2
WR
X
1
+ X
2
1
(
P
1
+
P
2
,
V
1
2
+
V
2
2
),
DWDN*H
X
1
- X
2
1
(
P
1
-
P
2
,
V
1
2
+
V
2
2
).
$QDORJLF]QH GR SU]HGVWDZLRQHJR SRZ\*HM WZLHUG]HQLD DSURNV\PDFML VWRFKDVW\F]QHM
JUDQLF]QHWZLHUG]HQLHR]ELH*QR FLDOJRU\WPyZSU]\URVWRZ\FKZZDUXQNDFKSRPLDUyZREDU
F]RQ\FKEá
GDPLORVRZ\PL]RVWDáRZ\ND]DQHSU]H].LHIHUD:ROIRZLW]D>@
: SUDNW\FH NOXF]RZ VSUDZ MHVW MHGQDN QDMF] FLHM V]\ENR ü RVLJDQLD Z Z\QLNX
G]LDáDQLD GDQHJR DOJRU\WPX EH]SR UHGQLHJR RWRF]HQLD HNVWUHPXP OXE UyZQRZD*QD FHFKD
]ZL]DQD]H]GROQR FL OHG]HQLD]D]PLHQLDMF\PVL
SRáR*HQLHPWHJRHNVWUHPXP6NXWHF]
Q\ GREyU SDUDPHWUyZ Z DOJRU\WPLH SU]\URVWRZ\P DSURNV\PDFML VWRFKDVW\F]QHM Z GDQHM
RNUH ORQHMV\WXDFMLZ\PDJDZLHG]\]GR ZLDGF]HQLD LQIRUPDFMLD¶SULRUL 3UREOHPWHQPR*H
E\üUyZQLH* UR]ZL]DQ\SU]\SRPRF\RNUH ORQHJRSURJUDPXEDGD
$OJRU\WP\UHGXNFML SU]HG]LDáyZ QLHRNUH ORQR FL ]ZL]DQH ] EH]SR UHGQLP SRUyZQ\
ZDQLHPSRPLDUyZEDGDQHMIXQNFMLLGRNRQ\ZDQ VHNZHQF\MQLHZLHORNURWQ ZHU\ILNDFM KL
SRWH]W\SX+
ID!IE SU]HFLZNRDOWHUQDW\ZQ\P+
ID IE =HZ]JO GXQD]QDF]QLH
ZL
NV] ]UHJXá\ZDJ
Eá
GyZ SLHUZV]HJROXEGUXJLHJRURG]DMX GHF\GXMF\FKRRGU]XFH
QLXV]F]HJyOQLHQDVWDUFLH]QDF]Q\FKSU]HG]LDáyZGDOV]\FKEDGD
=DPLDVWG]LDáDQLDGHWHUPLQLVW\F]QHJRZVW\OX
SRND*G\PSRPLDU]HUHGXNXMGRW\FKF]DVRZ\SU]HG]LDáEDGD
G]LDáDQLHSRZLQQRSU]HELHJDüQDSRGVWDZLH
MH*HOL KLSRWH]
H
0
(fa>fb)
OXEDOWHUQDW\ZQ
H
1
(fa < fb)
PR*QDSU]\MüQD
GRVWDWHF]QLHZ\VRNLP SR]LRPLHLVWRWQR FL
[1-
D
]
WR UHGXNXMSU]HG]LDáEDGD WDNMDNZRGSRZLHGQLPDOJRU\WPLHGHWHUPLQLVW\F]Q\P
ZLQQHMV\WXDFML Z\NRQDMGRGDWNRZHSRPLDU\
3RGNUH OP\MHV]F]HUD]*HZHU\ILNDFMDKLSRWH]\SRZLQQDE\üSU]HSURZDG]DQDQDEDUG]R
Z\VRNLP V]F]HJyOQLHZSRF]WNRZHMID]LHREOLF]H SR]LRPLHLVWRWQR FL3R]DW\PZSUR
FHGXU]H RSW\PDOL]DFML SRZLQQR ]QDMGRZDü VL
G]LDáDQLD NRQWUROQH VSUDZG]DMFH Z RNUH OR
Q\FKRGVW
SDFKUyZQLH* ZDUWR FL UHGQLH IXQNFMLQD NUDFDFK DNWXDOQHJRSU]HG]LDáX EDGD
:\VW
SXMH WXERZLHPNRQLHF]QR üVWRVRZDQLDSRPRFQLF]\FK]DOH*QR FL ORJLF]Q\FK Z FHOX
]DJZDUDQWRZDQLD]ELH*QR FLDOJRU\WPXPLPRZ\VW
SRZDQLD]RNUH ORQ\PSUDZGRSRGRELH
VWZHPQLHVSU]\MDMF\FKVHNZHQFML]DNáyFH
=HVWDZLHQLH&
$OJRU\WP\RSW\PDOL]DFMLIXQNFMLZLHOX]PLHQQ\FK
5HJXODUQH PHWRG\ PLQLPDOL]DFML IXQNFML ZLHOX ]PLHQQ\FK NRQVWUXRZDQH V SRSU]H] VH
NZHQF\MQHSRZWDU]DQLHGZXG]LDáD
2NUH OHQLH DNWXDOQHJR WM XPLHMVFRZLRQHJRVL Z ELH*F\PSXQNFLH
u(k)
SU]HVWU]HQL
SRV]XNLZD ZHNWRUD
d
NLHUXQNXSRV]XNLZDQLDSXQNWXPLQLPXP
0LQLPDOL]DFMDIXQNFMLZW\PNLHUXQNX WMUR]ZL]DQLH]DGDQLDPLQLPDOL]DFMLIXQNFMLMHG
QHM]PLHQQHM
f(
D
) = f ( u(k) +
D
d ):
PLQ
D
f( u(k) +
D
d ) = f( u(k) +
D
* d ).
3UREOHP PLQLPDOL]DFML IXQNFML MHGQHM ]PLHQQHM ]RVWDá SU]HGVWDZLRQ\ Z ]HVWDZLHQLX $
$NWXDOQLH EDUG]R ]ZL
(OH SU]HGVWDZLP\ JáyZQH PHWRG\ UR]ZL]\ZDQLD ]DGDQLD RNUH OHQLD
NLHUXQNX
0HWRGD*DXVVD6HLGOD :HNWRUNLHUXQNXSRV]XNLZD MHVWUyZQROHJá\NROHMQRGRND*GHM
]RVLZVSyáU] GQ\FKSU]HVWU]HQLDUJXPHQWyZEDGDQHMIXQNFML
d(k)=[0,...,1,...0],
WM
d(1)=[1,0,...,0]; d(2)=[0,1,...,0];...
D* GR
d(n)=[0,0,...,1].
3U]H]
n
R]QDF]RQRWXOLF]E DUJXPHQWyZRSW\PDOL]RZDQHMIXQNFML
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGDQDMV]\EV]HJRVSDGNX :HNWRUNLHUXQNXSRV]XNLZD
d(k) =
N
X
X
I
JUDG
.
SRNU\ZDVL WX]JUDGLHQWHPEDGDQHMIXQNFML0HWRGDWDSRGREQLHMDNSRSU]HGQLDVWRVRZDQD
MHVWZVWRVXQNRZRQLHVNRPSOLNRZDQ\FKV\WXDFMDFK
:SUDNW\FHHNVSHU\PHQWDOQHM SU]\QLVNLPSR]LRPLH]DNáyFHVNáDGRZHJUDGLHQWXPRJ
E\ü]DVWSLRQHSU]H]GR ZLDGF]DOQLHZ\]QDF]RQHSU]\URVW\QS
'
f
i
= ( f(u(k)+
G
u
i
) - f(u(k) ) / 2
G
u
i
.
3U]\]QDF]Q\FK]DNáyFHQLDFKSRPLDUyZIXQNFMLNU\WHULDOQHMI X QDMF] FLHMVWRVXMHVL
ZHUVM
%R[D:LOVRQD PHWRG\QDMV]\EV]HJRVSDGNX:ZHUVMLWHMQDSRGVWDZLHRNUH ORQHM
OLF]E\SRPLDUyZSU]HSURZDG]RQ\FK ]JRGQLH]SODQRZDQ\PSURJUDPHPEDGD]DVW
SXMHVL
EDGDQ IXQNFM
I X MHM SU]\EOL*HQLHPOLQLRZ\P T X ZRWRF]HQLXDNWXDOQHJRSXQNWXSRV]X
NLZDQLDHNVWUHPXPX :VSyáF]\QQLNL
a
i
(i=1,...,n)
Z\UD*HQLXSU]\EOL*RQ\P
q(u) = a
0
+ a
1
u
1
+ a
2
u
2
+ ... + a
n
u
n
;
RGJU\ZDM URO
RV]DFRZDQLD VNáDGRZ\FK JUDGLHQWX IXQNFML NU\WHULDOQHM Z GDQ\P SXQNFLH
SU]HVWU]HQLSRV]XNLZDQLD
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGD 1HZWRQD5DSKVRQD 8VWDOHQLH ZHNWRUD NLHUXQNX SRV]XNLZD Z SXQNFLH VWD
QRZLF\PDNWXDOQHSU]\EOL*HQLHUR]ZL]DQLDZUDPDFKWHMPHWRG\Z\PDJDZ\OLF]HQLDJUD
GLHQWXL RGZURWQR FL KHVMDQXEDGDQHMIXQNFML
d(k) =
[
2
f(u) ]
–1
~
u(k)
grad(f(u))
~
u(k)
.
3U]\SRPQLMP\*HKHVMDQIXQNFMLZGDQ\PSXQNFLHMHVWPDFLHU] NZDGUDWRZ UR]PLDUX
]JRGQHJR ] OLF]E DUJXPHQWyZ PLQLPDOL]RZDQHM IXQNFML QS Z SU]\SDGNX IXQNFML GZX
]PLHQQ\FK
[
2
f(u)] = [
w
2
f /
w
u
1
2
,
w
2
f /
w
u
1
w
u
2
;
w
2
f /
w
u
1
w
u
2
,
6
2
f /
w
u
2
2
]).
0HWRGD1HZWRQD5DSKVRQDMHVWV]F]HJyOQLHVNXWHF]Q ZEOLVNLPRWRF]HQLXSXQNWXVWD
QRZLFHJR UR]ZL]DQLH ]DGDQLD 3RWHQFMDOQH ]DVWRVRZDQLH MHM Z SUDNW\FH HNVSHU\PHQWDOQHM
ZL*HVL ]SU]HGVWDZLHQLHPEDGDQHMIXQNFMLZRWRF]HQLXDNWXDOQHJRSXQNWX
u(k)
SU]HVWU]H
QLSRV]XNLZDQLDSU]\EOL*HQLHPNZDGUDWRZ\P
q(u).
0R*HWRQDVWSLüQSSRZ\NRQDQLX
SURJUDPXEDGD GUXJLHJRU] GXZRWRF]HQLXDNWXDOQHJRSXQNWXSU]HVWU]HQL
u(k).
=Z\QL
NRZHMSRVWDFLIRUP\NZDGUDWRZHM
q(u)= a
0
+ a
1
u
1
+a
2
u
2
+...+ a
m
u
m
+ a
11
u
1
2
+ 2a
12
u
1
u
2
+...+a
m,m
u
m
2
=
= a
0
+ a u + u
T
Q u ;
SR UHGQLRZ\QLND]DUyZQRRV]DFRZDQLHKHVMDQXMDNLRV]DFRZDQLHJUDGLHQWXIXQNFMLNU\WH
ULDOQHMZGDQ\PSXQNFLH
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGD JUDGLHQWX VSU] *RQHJR QD]\ZDQD MHVW UyZQLH* PHWRG )OHWFKHU¶D5HHYV¶D
:HNWRU NLHUXQNX SRV]XNLZD G N MHVW WX OLQLRZ NRPELQDFM JUDGLHQWX PLQLPDOL]RZDQHM
IXQNFMLZDNWXDOQ\PSXQNFLHSU]HVWU]HQLSRV]XNLZD
grad(f(u))
u(k)
LZHNWRUDNLHUXQ
NX]SRSU]HGQLHMLWHUDFML
d(k-1):
d(k)=
E
(k)d(k-1)
grad(f(u))
u(k)
;
:VSyáF]\QQLNZDJL]ZL]DQ\MHVW]HVWRVXQNLHPGZXQRUPJUDGLHQWyZ
E
(k) = [grad(f(u))
u(k)
]
T
grad(f(u))
u(k)
/[grad(f(u))
u(k -1)
]
T
grad (f(u))
u(k -1)
;
DNLHUXQHNSRF]WNRZ\
d(0)
MHVWGHILQLRZDQ\SRSU]H]JUDGLHQWZSXQNFLHVWDUWRZ\P
1DOH*\ SRGNUH OLü *H NRQLHF]Q\ QDNáDG EDGD HNVSHU\PHQWDOQ\FK ± Z\]QDF]HQLH JUD
GLHQWXEDGDQHMIXQNFMLZGDQ\PPLHMVFXSU]HVWU]HQLF]\QQLNRZHMMHVWZGDQHMPHWRG]LHWDNL
VDPMDNZPHWRG]LHJUDGLHQWXSURVWHJR(IHNW\ZQR üMHMMHVWQDWRPLDVW]UHJXá\Z\*V]D
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGD3RZHOOD SU]\VWRVRZDQLXNWyUHMZSLHUZV]\PÄRELHJX´ WMW]ZGX*HMLWHUDFML
DOJRU\WPX NLHUXQNL SRV]XNLZDQLD Z\]QDF]DVL
Z WHQ VDP VSRVyE MDN Z PHWRG]LH *DXVVD
6HLGODQDED]LHZHUVRUyZ
{e(1),e(2),...,e(n)}.
=HVWDZ\NLHUXQNyZZQDVW SQ\FK
ÄGX*\FK´LWHUDFMDFKOLF]RQHV QDWRPLDVW]XZ]JO
GQLHQLHPSU]\URVWyZZHNWRUDDUJXPHQWyZ
ZRELHJDFKEH]SR UHGQLRMHSRSU]HG]DMF\FKQS
x SRSLHUZV]\PRELHJXIRUPXMHVL ]ELyU
{e(2),e(3),...,d(0)} ,
ZNWyU\P
d(0) = u(n+1) - u(1);
x SRGUXJLPQDWRPLDVW
{ e(3),e(4),...,d(0),d(1)},
JG]LH
d(1)= u(2n+1)-u(n+1),...,
LWG
0HWRG\3RZHOODL)OHWFKHUD5HHYVDQDOH*GRJUXS\PHWRGRSDUW\FKQDNLHUXQNDFK
VSU] *RQ\FK>@
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
:PHWRGDFK]PLHQQHMPHWU\NL DNWXDOQ\ZHNWRUNLHUXQNXOLF]RQ\MHVWZVSRVyEDQDOR
JLF]Q\GRVWRVRZDQHJRZPHWRG]LH1HZWRQD5DSKVRQD
G N
' JUDG I X
N
X N
,
]W\P*H
D
k
MHVWWXMHG\QLHDNWXDOQ\PSU]\EOL*HQLHPRGZURWQR FLKHVMDQXIXQNFMLZSXQNFLH
u(k).
=DSRF]WNRZ PDFLHU]
D
0
SU]\MPXMHVL ]UHJXá\PDFLHU]MHGQRVWNRZ
I
n
3U]\EOL*HQLD KHVMDQX
D
k
PR*QD NRQVWUXRZDü QD V]HUHJ Uy*Q\FK VSRVREyZ 1LHNWyUH ]
QLFKED]XMFHQDZHNWRURZ\FKSU]\URVWDFKDUJXPHQWX
c
k
= u(k+1) – u(k) ;
LSU]\URVWDFKJUDGLHQWX
g
k
= grad(f(u))|
u ( k + 1)
- grad(f(u))|
u ( k )
;
SR ND*GHM LWHUDFML SU]HGVWDZLRQR SRQL*HM ZV]\VWNLH ZHNWRU\ UHSUH]HQWRZDQH Z SU]\WRF]R
Q\FKZ]RUDFKV NROXPQRZH
PHWRGD)OHWFKHUD3RZHOOD'DYLGRQD
D
k+1
= D
k
+ c
k
c
k
T
/ c
k
T
g
k
- D
k
g
k
g
k
T
D
k
T
/ g
k
T
D
k
T
g
k
;
PHWRGD,3HDUVRQD
D
k+1
= D
k
- (D
k
g
k
) (D
k
g
k
)
T
/ g
k
T
D
k
g
k
;
PHWRGD0F&RUPLFND
D
k+1
= D
k
+ (c
k
- D
k
g
k
) c
k
T
/ c
k
T
g
k
;
PHWRGD:ROID%UR\GHQD'DYLGRQD
D
k+1
= D
k
+ (c
k
-D
k
g
k
) (c
k
-D
k
g
k
)
T
/ (c
k
-D
k
g
k
)
T
g
k
;
3RVW SRZDQLHZSU]\SDGNXQLHUHJXODUQ\FKPHWRGPLQLPDOL]DFMLIXQNFMLZLHOX]PLHQQ\FK
QLH ZL*H VL
] RNUH ORQ\PVFKHPDWHP RJyOQ\P 2SUDFRZ\ZDQR MH LQG\ZLGXDOQLH : UROL
SU]\NáDGXSU]HGVWDZLP\DNWXDOQLHQDMEDUG]LHMSRSXODUQ PHWRG
VLPSOHNVyZ
0HWRGDVLPSOHNVyZ MHVWSURJUDPHPEDGD W\SXEH]JUDGLHQWRZHJR6LPSOHNVHPQD
]\ZDQ\MHVWUHJXODUQ\ZLHOR FLDQSRVLDGDMF\
(n+1)
ZLHU]FKRáNyZ
n
MHVWWXOLF]E DUJX
PHQWyZ IXQNFML NU\WHULDOQHM QS Z SU]\SDGNX GZX DUJXPHQWyZ VLPSOHNVHP MHVW WUyMNW
UyZQRERF]Q\WU]\ RSW\PDOL]RZDQHF]\QQLNLSURZDG] GRF]ZRUR FLDQXZSU]HVWU]HQLWUyM
Z\PLDURZHMLWG
1DVWDUFLH SRV]XNLZDQLDHNVWUHPXP Z\NRQXMH VL SRPLDU\ IXQNFMLZ ZLHU]FKRáNDFK UD
FMRQDOQLHZ\EUDQHJRVLPSOHNVXSRF]WNRZHJR1DVW
SQLHZQDMSURVWV]HMZHUVMLPHWRG\QD
ND*G\PNROHMQ\PHWDSLHPLQLPDOL]DFMLZLHU]FKRáHN]QDMPQLHMNRU]\VWQ\PSRáR*HQLHPRG
SRZLDGDMF\ QDMZL
NV]HM ZDUWR FL IXQNFML ]RVWDMH ]DPLHQLRQ\ QRZ\P SRáR*RQ\P V\PH
WU\F]QLHGRQLHJRSRSU]HFLZVWDZQHM FLDQLHVLPSOHNVX:Z\QLNXSRZWDU]DQLDWHJRG]LDáDQLD
VLPSOHNVSU]HVXZDVL
VWRSQLRZRZSU]HVWU]HQLF]\QQLNyZZSá\ZDMF\FKQDMDNR üRSW\PD
OL]RZDQHJR SURFHVX : V\WXDFML JG\ Z\QLN SRPLDUX PLQLPDOL]RZDQHM IXQNFML Z QRZ\P
ZLHU]FKRáNXMHVWJRUV]\RGSRSU]HGQLHJRPR*HP\]DNRF]\üSRV]XNLZDQLHOXENRQW\QXRZDü
SURJUDPSR]PQLHMV]HQLXUR]PLDUyZVLPSOHNVX(IHNW\ZQ\P]DNRF]HQLHPPR*HE\üUyZ
QLH* Z\NRQDQLH Z ]QDOH]LRQ\P RWRF]HQLX SXQNWX VWDFMRQDUQHJR SURJUDPX EDGD GUXJLHJR
U]
GX
=QDF]QLHEDUG]LHMUR]ZLQL W ZHUVM PHWRG\VLPSOHNVyZMHVWSU]HGVWDZLRQDV]F]HJyáR
ZRZ>@VLHüG]LDáD 1HOGHUD±0HDGD1DWHMPHWRG]LHRSDUWRPLQSRGVWDZRZ SURFHGXU
PLQLPDOL]DFMLIXQNFMLEH]RJUDQLF]H ÄIPLQV´ Z URGRZLVNX0DWODED
=HVWDZLHQLH
'
2SW\PDOL]DFMDZREHFQR FL]DNáyFH SRPLDUyZ
IXQNFMLNU\WHULDOQHMZLHOX]PLHQQ\FK
:\]QDF]DQLH RSW\PDOQ\FK ZDUXQNyZ UHDOL]DFML SURFHVX WHFKQRORJLF]QHJR Z\PDJD
SU]\ ]DáR*HQLX QLHVWDFMRQDUQR FL OXE QLH]QDMRPR FL D¶SULRUL SRVWDFL IXQNFML NU\WHULDOQHM
SU]HSURZDG]HQLD RNUH ORQHJR SURJUDPX EDGD HNVSHU\PHQWDOQ\FK EH]SR UHGQLR ÄQD RELHN
FLH´:SURJUDPLHWHJRURG]DMXQLHRGWZDU]DVL
FDáHMIXQNFMLNU\WHULDOQHMDMHG\QLH]QDMGXMH
VL
MHMHNVWUHPXPOXE FL OHMU]HF]XMPXMF]UHJXá\FHOHPMHVW]QDOH]LHQLHVL
ZGRVWDWHF]QLH
EOLVNLP ]SXQNWXZLG]HQLDFHOXRSW\PDOL]DFML RWRF]HQLXDNWXDOQHJRSRáR*HQLDSXQNWXHNV
WUHPDOQHJR=]DáR*HQLDERZLHPSRPLDU\ EDGDQHMIXQNFMLREDUF]RQHV QLHSRPLMDOQ\PL]D
NáyFHQLDPLFRLVWRWQLHNRPSOLNXMHSUREOHP]QDOH]LHQLDHNVWUHPXPZSRUyZQDQLX]V\WXDFM
GHWHUPLQLVW\F]Q
7HRULDJZDUDQWXMHMHG\QLHDV\PSWRW\F]Q ]ELH*QR ü WM]ELH*QR üSU]\OLF]ELHSRPLDUyZ
URVQFHMGRQLHVNRF]RQR FL DOJRU\WPyZW\SXDSURNV\PDFMLVWRFKDVW\F]QHM>@:SUDNW\FH
VWRVXMHVL
ÄRVWUR*QH´ ]ZLHORNURWQLHQLHSRPLDUXIXQNFMLZGDQ\PSXQNFLHQLH]QDF]QDZLHO
NR üSRSUDZNLQDND*G\PHWDSLH ZHUVMHQDMSURVWV]\FKPHWRGGHWHUPLQLVW\F]Q\FK3U]HGVWD
ZLP\ WX NLOND SU]\NáDGyZ SURJUDPyZ EDGD WHJR URG]DMX FK
WQLH VWRVRZDQ\FK SU]H]
WHFKQRORJyZLG\VSR]\WRUyZV\VWHPyZVWHURZDQLDLQDG]RUX
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGD%R[D:LOVRQD MHVWSURJUDPHPW\SXJUDGLHQWRZHJR5HDOL]XMHVL
M ZGZXHWD
SDFK : WUDNFLH HWDSX SLHUZV]HJR QD SRGVWDZLH QLHZLHONLHM OLF]E\ GR ZLDGF]H QDVW
SXMH
RSLVDQLHSU]\EOL*HQLHPOLQLRZ\PIXQNFMLNU\WHULDOQHM
Q(x)
ZRWRF]HQLXDNWXDOQHJRSXQNWX
SU]HVWU]HQLF]\QQLNRZHM
x(k); x
MHVWWXZHNWRUHPSRV]XNLZDQ\FKZLHONR FLVWHUXMF\FK
OXESDUDPHWUyZ
=ZUyüP\XZDJ *HZZ\UD*HQLXSU]\EOL*RQ\P]QDMGRZDQ\PZZ\QLNXUHDOL]DFMLSUR
JUDPXEDGD SLHUZV]HJRU] GX
yest = a
0
+ a
1
x
1
+ a
2
x
2
+ ... + a
n
x
n
;
ZVSyáF]\QQLNL
a
i
RGJU\ZDM URO SU]\EOL*HQLD SRFKRGQ\FK IXQNFML NU\WHULDOQHM Z GDQ\P
SXQNFLH
: HWDSLH GUXJLP QD SRGVWDZLH X]\VNDQHJR PRGHOX QDVW SXMH Z\]QDF]HQLH NLHUXQNX
G\VSRQXMHP\ RV]DFRZDQLHP JUDGLHQWX L NURNX SU]HPLHV]F]HQLD .URN WHQ MHVW QDMF]
FLHM
VWDá\POXEGRELHUDQ\PSURSRUFMRQDOQLHGR JUDGLHQWX6NXWHF]Q\PUR]ZL]DQLHPPR*HE\ü
UyZQLH* ]UHDOL]RZDQLHSURJUDPXEDGD IXQNFMLNU\WHULDOQHMZNLHUXQNXJUDGLHQWX
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
0HWRGDVLPSOHNVyZ MHVWSURJUDPHPW\SXEH]JUDGLHQWRZHJR6LPSOHNVHPQD]\ZDQ\MHVW
UHJXODUQ\ZLHOR FLDQSRVLDGDMF\
(n+1)
ZLHU]FKRáNyZ
n
MHVWWXOLF]E DUJXPHQWyZIXQNFML
NU\WHULDOQHM QS Z SU]\SDGNX GZX DUJXPHQWyZ VLPSOHNVHP MHVW WUyMNW UyZQRERF]Q\ WU]\
RSW\PDOL]RZDQHF]\QQLNLSURZDG] GRF]ZRUR FLDQXZSU]HVWU]HQLWUyMZ\PLDURZHMLWG1D
VWDUFLHZ\NRQXMHVL
SRPLDU\IXQNFMLZZLHU]FKRáNDFKUDFMRQDOQLHZ\EUDQHJRVLPSOHNVXSR
F]WNRZHJR
:QDMSURVWV]HMZHUVMLPHWRG\QDND*G\PHWDSLHRSW\PDOL]DFMLZLHU]FKRáHN]QDMPQLHM
NRU]\VWQ\PSRáR*HQLHP RGSRZLDGDMF\ SU]\ PLQLPDOL]DFML QDMZL
NV]HM ZDUWR FL IXQNFML
]RVWDMHQDVW
SQLH]DPLHQLRQ\QRZ\PSRáR*RQ\PV\PHWU\F]QLHGRQLHJRSRSU]HFLZVWDZQHM
FLDQLHVLPSOHNVX:Z\QLNXSRZWDU]DQLDRSHUDFMLWHJRW\SXVLPSOHNVSU]HVXZDVL
ZSU]H
VWU]HQLF]\QQLNyZZSá\ZDMF\FKQDMDNR üRSW\PDOL]RZDQHJRSURFHVX
3RV]XNLZDQLH]RVWDMH]DNRF]RQHZV\WXDFMLJG\Z\QLNSRPLDUX
Q(x)
ZQRZ\PZLHU]
FKRáNXMHVWJRUV]\RGSRSU]HGQLHJR2F]\ZL FLHZV\WXDFMLWHJRW\SXPR*OLZDMHVWUyZQLH*
NRQW\QXDFMDEDGD SR]PQLHMV]HQLXUR]PLDUyZVLPSOHNVX±SDWU]DOJRU\WPGHWHUPLQLVW\F]Q\
]]HVWDZLHQLD&
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
3URJUDPRZDQLHHZROXF\MQH PR*HP\ UyZQLH* UHDOL]RZDü Z ZLHOX ZHUVMDFK >@ ,VWRWD
UR]ZL]DQLDSUREOHPXRSW\PDOL]DFMLZW\PSU]\SDGNXMHVWGZXID]RZD
:SLHUZV]HMID]LHND*GHJRHWDSXG]LDáD SU]HSURZDG]DVL ORNDOQHEDGDQLDIXQNFMLFHOX
QSZSRVWDFLSURJUDPXFDáNRZLWHJROXEF]
FLRZHJRZRWRF]HQLXSXQNWXSU]\MPRZDQHJRQD
SRGVWDZLHLQIRUPDFMLSRVLDGDQHM D¶SULRUL OXE Z Z\QLNX SRSU]HGQLFK SRPLDUyZ ]D GRVWD
WHF]QLHGREU\SXQNWFHQWUDOQ\
:ID]LHGUXJLHMRNUH ODVL QDSRGVWDZLHW\FKZ\QLNyZQRZ\SXQNWFHQWUDOQ\ZRNyáNWy
UHJR SU]HSURZDG]L VL
EDGDQLD Z NROHMQ\P HWDSLH &HORZ\P MHVW SU]\ W\P VNRU]\VWDQLH ]
PR*OLZLHQDMZL
NV]HMOLF]E\MX* SRVLDGDQ\FKSRPLDUyZ:SUDNW\FHSRVWXODWWHQUHDOL]XMHVL
QDMF]
FLHM SRSU]H] SU]HVXZDQLH QRZHJR FHQWUXP GR QDMNRU]\VWQLHMV]HJR ] ZLHU]FKRáNyZ
SODQXSRSU]HGQLHJRHWDSX
3RGNUH OP\*HG]LDáDMFZUDPDFKPHWRG\NLHZROXF\MQHMPR*HP\SRND*G\PHWDSLH]
PLHQLü QLH W\ONR SXQNW FHQWUDOQ\ GR ZLDGF]H OHF] WDN*H ]DNUHV\ ]PLDQ SRV]F]HJyOQ\FK
F]\QQLNyZDWDN*HREUyFLü QSR
SU]HVWU]H SRV]XNLZDQLD
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
=HVWDZLHQLH
(
3DNLHWRSURJUDPRZDQLD0DWODED 237<0$/,=$&-$
/LVWDQDMF] FLHMUR]ZL]\ZDQ\FK]DGD ZUDPDFK23<0$/,=$&-,
0LQLPDOL]DFMDIXQNFMLMHGQHM]PLHQQHM
[ IPLQ IXQNFMD
3U]HSURZDG]DQDMHVWPHWRGDPLLQWHUSRODFMLNZDGUDWRZHMLV]H FLHQQHM]HNVWUDSRODFM
0LQLPDOL]DFMD IXQNFML ZLHOX ]PLHQQ\FK PR*QD E\ü SU]HSURZDG]RQD SU]\ SRPRF\ P
IXQNFMLVWRVXMFHMPHWRG
V\PSOHNVX1DOGHUD0HDGD [ IPLQV
IXQNFMD
OXERSDUWHMQDMHGQHM]PHWRG]PLHQQHMPHWU\NL%)*6 [ IPLQX IXQNFMD
0LQLPDOL]DFMDIXQNFMLSU]\RJUDQLF]HQLDFK
[ FRQVWU IJ
ED]XMHQDMHGQ\P]ZDULDQWyZVHNZHQF\MQHJRSURJUDPRZDQLDNZDGUDWRZHJR 643 Z
ND*GHMLWHUDFMLDSURNV\PXMHVL
WXKHVMDQIXQNFML/DJUDQJH¶DLOLQHDU\]XMHRJUDQLF]HQLD
3URJUDPRZDQLHOLQLRZH
[ OS F$E
RSLHUD VL
QD MHGQHM ] ZHUVML PHWRG\ VLPSOHNVRZHM XPR*OLZLDMF PLQ ]QDMGRZDQLH
SLHUZV]HJRUR]ZL]DQLDED]RZHJR
3URJUDPRZDQLHNZDGUDWRZH
[ TS +F$E
UR]ZL]XMH]DGDQLH
min (x
T
H x /2 – c
T
x); przy A x
d
b;
VWRVXMF NRPELQRZDQ\ VWRVRZDQ\ L Z LQQ\FK IXQNFMDFK DOJRU\WP Z NWyU\P NLHUXQHN
RNUH ODQ\MHVWZZ\QLNXU]XWRZDQLDJUDGLHQWXQDRJUDQLF]HQLDDGáXJR üNURNXSRV]XNL
ZDQLDZW\PNLHUXQNXRNUH ODVL
PLQSU]\SRPRF\W]ZIXQNFMLNDU\
8NáDGUyZQD QLHOLQLRZ\FK I [
[ IVROYH IXQNFMH
MHVWUR]ZL]\ZDQ\PHWRGDPL*DXVVD1HZWRQDL/DYHQEHUJD0DUTXDUGWD
3RGVWDZRZH]DVDG\VWRVRZDQLDPIXQNFML237<0$/,=$&-,
:SURZDG]HQLHEDGDQHMIXQNFMLLRJUDQLF]H QDVW
SXMHSRSU]H]VSUHF\]RZDQLHRGSRZLHG
QLHJRPSOLNX1S
IXQFWLRQ >IJ@ IXQ [
I H[S [
[ A
[ A
J [
[ [ [
J [
[
*UDGLHQW\PRJ E\üRNUH ODQH QXPHU\F]QLHSRSU]H]ZHNWRULORUD]yZUy*QLFRZ\FK MH
*HOL
options(9) = 0
OXE SRXVWDOHQLX
: options(9) = 1
DQDOLW\F]QLH
1SJUDGLHQW\IXQNFML
f
L
g
E G REOLF]DQHDQDOLW\F]QLHSRX]XSHáQLHQLXSRGDQ\FKZ\*HM
GHNODUDFMLÄIXQ´SRSU]H]
IXQFWLRQ >GIGJ@ JUDG [
GI >H[S [
[ A
[ A
[
H[S [ [ @
GJ >[ [
[
[ @
:\ZRá\ZDQLH SURFHGXU RSW\PDOL]DF\MQ\FK QDVW
SXMH SR ]DGHNODURZDQLX EDGDQHM IXQN
FMLRJUDQLF]H LHZHQWXDOQLHJUDGLHQWyZ
/LF]ERZHXVWDOHQLHZDUWR FLZ\PDJDQ\FK SDWU]ÄKHOS\´ DUJXPHQWyZWDNLFKMDN ZDU
WR ü VWDUWRZD ]QDMGRZDQHJR ZHNWRUD QS
x = [1, -1]),
RJUDQLF]HQLD ]PLHQQ\FK
vlb, vub, options
LSDUDPHWUyZJOREDOQ\FK
p1, p2,...
QDMZ\*HM PR*H
QDVWSLüZVDP\PWHN FLHZ\ZRáDQLDOXESU]HGQLP
1S5R]ZL]DQLH]DGDQLD]RJUDQLF]HQLDPLPR*HP\XUXFKRPLüSRSU]H]
[ >@ RSWLRQV >@ YOE >@ YXE >@
[ FRQVWU CIXQ
[RSWLRQVYOEYXE OXE
[ FRQVWU CIXQ
>@>@>@>@
1S'HNODUDFMDVDPHMIXQNFMLPR*HE\üUyZQLH* ]DZDUWDZMHGQ\P]DUJXPHQWyZ
'RSXV]F]DOQ\PMHVWZ\ZRáDQLHSURFHGXU\PLQLPDOL]DFMLZSRVWDFL
[ IPLQX C[ A[ A >@>@C> [ @
:HNWRUXVWDODQ\FKD¶SULRULSDUDPHWUyZ ÄRSWLRQV´ ZUR]ZL]\ZDQ\P]DGDQLXSR
VLDGDQDVW SXMFHVNáDGRZH
GLVSOD\ RSFMRQDOQLH RSWLRQV NRQWUROXMH VSRVyE SU]HND]\ZDQHM QD PRQLWRU]H
LQIRUPDFMLSRND*GHMLWHUDFML SU]\RSWLRQV Z\QLNREOLF]H MHVWZ\ ZLHWODQ\
VWRS[ RSFMRQDOQLHRSWLRQV
±
GRSXV]F]DOQDSUHF\]MD]PLHQQHMQLH]DOH*QHM[
VWRSI RSFMRQDOQLHRSWLRQV
±
GRSXV]F]DOQDSUHF\]MDRSW\PDOL]RZDQHMIXQNFML
XFK\EJ RSFMRQDOQLHRSWLRQV
±
SRGDMHSUHF\]M XZ]JO GQLDQLDRJUDQLF]H
DOJRU\WP RSFMRQDOQLHRSWLRQV Z\EyUJáyZQHJRDOJRU\WPXZGDQ\P]DGDQLX
GDOJRU\WP RSFMRQDOQLH Z\EyUPHWRG\RNUH OHQLDNLHUXQNXSRV]XNLZD
DDDOJRU\WP RSFMRQDOQLH Z\EyUPHWRG\SRV]XNLZDQLDPLQLPXPZNLHUXQNX
VSHF\ILF]Q\VWRVXMHVL
MHG\QLHGODIXQNFMLPLQLPD[ L DWWJRDO
JUDGLHQW RSFMRQDOQLH VSRVyE]DGDZDQLDJUDGLHQWX Uy*QLFRZRDQDOLW\F]QLH
OLF]QLN REOLF]H IXQNFML
OLF]QLNJUDGLHQWX OLF]EDREOLF]H JUDGLHQWXIXQNFML
OLF]QLNRJUDQLF]H RJyOQDOLF]EDZ\OLF]DQ\FKIXQNFMLáF]QLH]RJUDQLF]HQLDPL
RJUUyZQR FLRZH RSFMRQDOQLH OLF]EDRJUDQLF]H W\SXUyZQR FLRZHJR Z\PLHQLD
Q\FKSU]\Z\ZRáDQLXPIXQNFMLQDSLHUZV]\FKPLHMVFDFK J J
PD[LWHUDFML RSFMRQDOQLH RSWLRQV Q RJUDQLF]D OLF]E LWHUDFML Q MHVWWX
OLF]E QLH]DOH*Q\FK]PLHQQ\FK
VSHF\ILF]Q\VWRVXMHVL
MHG\QLHGODIXQNFMLDWWJRDO
PLQSU]\URVWX RSFMRQDOQLHRSWLRQV
RNUH ODPLQLPDOQ\SU]\URVWSU]\Uy*
QLFRZ\PZ\OLF]HQLXJUDGLHQWXIXQNFML
PD[SU]\URVWX RSFMRQDOQLH RSWLRQV RNUH ODSU]\URVWDUJXPHQWXSU]\Uy*QL
FRZ\PZ\OLF]HQLXJUDGLHQWXIXQNFML
VWHSVL]H RSFMRQDOQLHOXEPQLHM GHILQLXMHZLHONR üNURNXZSLHUZV]HMLWHUDFML
3U]\NáDG\GHNODUDFMLSDUDPHWUX RSWLRQV RSWLRQV OH
RSWLRQV
'RGDWHN
7HNVWQLHNWyU\FKPSOLNyZVWRVRZDQ\FKZüZLF]HQLDFK
7HNVWPSOLNX ÄQPVBBD´
%DGDQ\MHVWSU]HELHJPLQLPDOL]DFMLIXQNFMLNZDGUDWRZHM
MHGQHM]PLHQQHM \ F XXRSW AS] UDQGQ
SU]\SRPRF\DOJRU\WPX ]]HVWDZLHQLD$
0RGHORELHNWXVWDW\F]QHJRL]DNáyFH
XRSW X RSW\PDOQDLSRF]WNRZDZDUWR üDUJXPHQWX RSFMD
F GUXJDSRFKRGQDQDFKDUDNWHU\VW\FH
S] RGFKVWDQG]DNáyFH RUR]NáQRUPDOQ\P]]HURZ UHGQL
:SURZDG]DQHVWDáH
Q OLF]EDX UHGQLDQ\FKSRPLDUyZIXQNFML RSFMD
X LQSXW :SURZDG( SXQNWVWDUWRZ\DOJRU\WPX
1 OLF]EDSU]HOLF]DQ\FKLWHUDFML
1 LQSXW :SURZDG( OLF]E DQDOL]RZDQ\FKLWHUDFML
GHO SU]\URVWDUJXPHQWXSU]\RGWZDU]DQLXSRFKRGQHM
Z]P Z]PRFQLHQLHSRFKRGQHMZDOJRU\WPLH
SZ]P LQSXW &]\EDGDQLHZNDZ]PSRFKRGQHMZDOJRU\WPLH"WDNQLH
LISZ]P SGHO
ZZ]P LQSXW :SURZDG( ZHNWRU >Z]PBPLQGHOBZ]PZ]PBPD[@
HOVH
SGHO LQSXW %DGDQLHSU]\URVWXDUJSU]\RNUH OSRFKRGQHM"WDNQLH
LISGHO
ZGHO LQSXW :SURZDG( ZHNWRU >GHOBPLQGHOBGHOGHOBPD[@
HQG
HQG
2EOLF]HQLDGODZ]PRFQLHQLD
LISZ]P T >@
11 IL[ ZZ]P ZZ]P ZZ]P
IRULL 11Z]P LL ZZ]P LL ZZ]P X X GHO GHO
2EOLF]HQLDLWHUDF\MQHSRFKRGQD]DVW SRZDQDMHVWLORUD]HPUy*QLFRZ\P
IRUL 1
IRUM Q\SOXV M F X L GHOXRSW AS] UDQGQ HQG
IRUM Q\PLQXV M F X L GHOXRSW AS] UDQGQ HQG
\S VXP \SOXV Q \P VXP \PLQXV Q
X L X L Z]P LL \S\P GHO \ L \S\P
HQG
T LL >VXP DEV XXRSW VXP F XXRSW A @
HQG
LZ\SURZDG]HQLHLFKZ\QLNyZ
ILJXUH SORW Z]PT JULGOHJHQG \ VWHURZDQLH P Z\MVFLH
[ODEHO ZVNZ]PRFQLHQLD \ODEHO VWUDW\QDSRV]XNLZDQLH KROGRII
WLWOH ZSO\ZZ]PRFQLHQLDQDSR]LRPZVND]QLNDMDNRVFL
HQG
2EOLF]HQLDGODSU]\URVWXDUJXPHQWX
LISGHO T >@
11 IL[ ZGHO ZGHO ZGHO
IRULL 11GHO LL ZGHO LL ZGHO X X Z]P Z]P
2EOLF]HQLDLWHUDF\MQHSRFKRGQD]DVW SRZDQDMHVWLORUD]HPUy*QLFRZ\P
IRUL 1
IRUM Q\SOXV M F X L GHO LL XRSW AS] UDQGQ HQG
IRUM Q\PLQXV M F X L GHO LL XRSW AS] UDQGQ HQG
\S VXP \SOXV Q \P VXP \PLQXV Q
X L X L Z]P \S\P GHO LL \ L \S\P
HQG
T LL >VXP DEV XXRSW VXP F XXRSW A @
HQG
LZ\SURZDG]HQLHLFKZ\QLNyZ
ILJXUH SORW GHOT JULGOHJHQG \ VWHURZDQLH P Z\MVFLH
[ODEHO NURNSURENL \ODEHO VWUDW\QDSRV]XNLZDQLH KROGRII
WLWOH ZSO\ZNURNXSURENRZDQLDQDSR]LRPZVND]QLNDMDNRVFL
HQG
7HNVWPSOLNX ÄQPVB´
%DGDQLHDOJRU\WPyZPLQLPDOL]DFMLIXQNFMLGZX]PLHQQ\FK
3U]HGUR]SRF] FLHPREOLF]H ]GHILQLXMIXQNFM GZX]PLHQQ\FK>[ [ @
RQD]ZLHPLQIXQ3U]\PHWRGDFK]ZL]DQ\FK]PIXQNFM IPLQXQDOH*\
ZSURZDG]LüZ\UD*HQLHJUDGLHQWXZDSRVWURIDFKX]XSHáQLRQHEDQSORWHP
QDPLHMVFHZ]RUXSU]\WRF]RQHJRSRQL*HMGODIXQNFMLWHVWXMFHM
I [ [ A [ A [ [ A A
*5$' > [ A [ [ [ [ A [ A [
[ [ A [ A EDQSORW [2/'; @
*5$' > [ A [ [ [ [ A [ A [ [
[ A [ A EDQSORW [2/'; @
HFKRRIIFOFKHOSQPV
IRU N IRU O
[ N N [ O O
ZIXQNFML NO IHYDO PLQIXQ >[ N [ O @
HQG HQG
ILJXUH PHVK ZIXQNFML QU
VWDUW LQSXW RNUH OSXQNWVWDUWRZ\PLQLPDOL]DFML QS>@ GLVS 1DFL QLMNODZLV]
RWU]\PDV]ZDUVWZLFHEDGDQHMIXQNFML
QU SDXVHILJXUH QU FRQWV H[S
FRQWRXU [[ZIXQNFMLFRQWV Z
[ODEHO [ \ODEHO [
WLWOH 0LQLPL]DFMDIXQNFMLGZX]PLHQQ\FK
GUDZQRZ KROGRQ
SORW VWDUW VWDUW UR
WH[W VWDUW VWDUW 3XQNWVWDUWRZ\
SORW UR
WH[W 3RV]XNLZDQHUR]ZL]DQLH
ZKLOH
FOF
GLVS :\ELHU]MHGQ ]QDVW SXMF\FKPHWRGPLQLPDOL]DFMLIXQNFML
GLVS EH]RJUDQLF]H
GLVS %UR\GHQ)OHWFKHU*ROIDUE6KDQQR
GLVS 'DYLGRQ)OHWFKHU3RZHOO
GLVS 0HWRGDQDMV]\EV]HJRVSDGNX
GLVS 0HWRGD6LPSOHNVyZ1HOGHUD0HDGD
GLVS :\M FLH
PHWRGD
ZKLOH PHWRGD PHWRGD!
PHWRGD >@
ZKLOH aOHQJWK PHWRGD
PHWRGD LQSXW :\ELHU]QXPHUPHWRG\
HQG
HQG
LI PHWRGD IRU L QUILJXUH L KROGRIIHQG
UHWXUQ
HQG
237,216 [ VWDUW
LI
PHWRGD 237,216
HOVH
LI PHWRGD 237,216
HOVH LIPHWRGD 237,216 HQG
HQG
HQG
O 237,216
237,216 LQSXW PDNV\PDOQDOLF]EDLWHUDFML
GLVS
LI PHWRGD
I PLQIXQ
GLVS >[RSWLRQV@ IPLQX I[237,216*5$'
>[RSWLRQV@ IPLQX I[237,216*5$'
HOVHLI PHWRGD
I > [ A [ A [ [ A A EDQSORW [ @
GLVS >[RSWLRQV@ IPLQV I[237,216
>[RSWLRQV@ IPLQV I[237,216
HQG
[RSW [
GLVS VSULQWI ?Q:DUWR üIXQNFMLZRVWDWQLHMLWHUDFML J RSWLRQV
GLVS VSULQWI /LF]EDREOLF]H ZDUWR FLIXQNFML J RSWLRQV
GLVS VSULQWI /LF]EDREOLF]H ZDUWR FLJUDGLHQWX J?Q?Q RSWLRQV
GLVS
GLVS ]REDF]RWU]\PDQ\SU]HELHJLQDFL QLMGRZROQ\NODZLV]
SDXVHILJXUH QU QU QU
FRQWRXU [[ZIXQNFMLFRQWV Z
[ODEHO [ \ODEHO [
WLWOH 0LQLPL]DFMDIXQNFMLGZX]PLHQQ\FK
GUDZQRZKROGRQ
SORW VWDUW VWDUW UR
WH[W VWDUW VWDUW 3XQNWVWDUWRZ\
SORW UR
WH[W 3RV]XNLZDQHUR]ZL]DQLH
HQG
7HNVWPSOLNX ÄQPVB´
6\PXODFMDSURFHVXPLQLPDOL]DFMLIXQNFMLGZX]PLHQQ\FKSU]\SRPLDUDFK
]DNáyFRQ\FKDGG\W\ZQ ]PLHQQ ORVRZ S] UDQGRUR]NáDG]LHQRUPDOQ\P
0LQLPDOL]RZDQDMHVWIRUPDNZDGUDWRZD I [ X $X S] UDQG
]PLQLPXPZSXQNFLHXRSW >@I XRSW
:\ELHUDQHSDUDPHWU\
$ $ $ $ $ LQY $
HFKRRIIFOFKHOSQPV
IRUN IRUO
X N N X O O
INZ NO >X N X O @ $ >X N X O @
HQG HQG
GLVS 1DFL QLMNODZLV]ZFHOXRWU]\PDQLDZ\NUHVXEDGDQHMIXQNFML
SDXVHILJXUH PHVK INZ QU
S] LQSXW :SURZDG( SR]LRP]DNáyFH
GLVS 2EHMU]\MVNXWNLZSURZDG]HQLD]DNáyFH QDZ\NUHVLHEDGDQHMIXQNFML
SDXVHPHVK INZS] UDQGQ
GLVS 1DFL QLMNODZLV]
ZKLOH
FOF
GLVS
GLVS :\ELHU]MHGQ ]QDVW SXMF\FKPHWRGPLQLPDOL]DFML
GLVS
GLVS 0LQLPDOL]DFMDHZROXF\MQD
GLVS 0HWRGDVLPSOHNVyZSURVW\FK
GLVS
0HWRGDJUDGLHQWXVWRFKDVW\F]QHJR
GLVS :\M FLH
PHWRGD
ZKLOH PHWRGD PHWRGD!
PHWRGD >@
ZKLOHaOHQJWK PHWRGD
PHWRGD LQSXW :\ELHU]QXPHUPHWRG\
HQG
HQG
LI PHWRGD IRUL QUILJXUH L KROGRIIHQG
UHWXUQ
HQG
GLVS
1 LQSXW :\ELHU]OLF]E LWHUDFML
VWDUW LQSXW 3RGDMSXQNWVWDUWRZ\PLQLPDOL]DFML QS>@
IROG VWDUW $ VWDUW S] UDQGQ
LIPHWRGD
GHOWD LQSXW 2NUH OZLHONR üSRMHG\QF]HJRNURNX
ILJXUH QU FRQWV A
FRQWRXU XXINZFRQWV Z
[ODEHO [ \ODEHO [
WLWOH 0LQLPL]DFMDIXQNFMLPHWRGDHZROXF\MQD
GUDZQRZ KROGRQ
SORW VWDUW VWDUW RU
WH[W VWDUW VWDUW 3XQNWVWDUWRZ\
SORW RU
WH[W 3RV]XNLZDQHPLQLPXP
[ VWDUW [ VWDUW OLI >@
IRUL 1
[ L [ L GHOWD
IQHZ >[ L [ L @ $ >[ L [ L @S] UDQGQ
OLI >OLIIQHZ@
LIIQHZ!IROG[ L [ L GHOWD
IQHZ >[ L [ L @ $ >[ L [ L @S] UDQGQ
OLI >OLIIQHZ@
LIIQHZ!IROG[ L [ L
HOVHIROG IQHZ
HQG
HOVHIROG IQHZ
HQG
[ L [ L GHOWD
IQHZ >[ L [ L @ $ >[ L [ L @S] UDQGQ
OLI >OLIIQHZ@
LIIQHZ!IROG[ L [ L GHOWD
IQHZ >[ L [ L @ $ >[ L [ L @S] UDQGQ
OLI >OLIIQHZ@
LIIQHZ!IROG[ L [ L
HOVHIROG IQHZ
HQG
HOVHIROG IQHZ
HQG
SORW >[ L [ L @>[ L [ L @ U [ L [ L R (UDVHPRGH QRQH
GUDZQRZ
HQGLQVWUIRU
QU QU ILJXUH QU SORW OLI JULG
GLVS =QDOH]LRQHSRáR*HQLHRSWLPXP [RSW >[ 1 [ 1 @
GLVS :DUWR üIXQNFMLZRVWDWQLHMLWHUDFML I IQHZ
GLVS 2ULHQWDF\MQDOLF]EDREOLF]H IXQNFML OHQJWK OLI
GLVS QDFL QLMNODZLV]E\ZUyFLüGRPHQX SDXVH
KROGRII
HQGREOLF]H PHWRG HZROXF\MQ
LIPHWRGD [ VWDUW [ VWDUW OLI >@
GHOWD LQSXW 2NUH ONUDZ G( VLPSOHNVX ERNWUyMNWD
QQ LQSXW 2NUH OZLHORNURWQR üSRPLDUXIXQNFML
ILJXUH QU FRQWV A
FRQWRXU XXINZFRQWV Z
[ODEHO [ \ODEHO [
WLWOH 0LQLPL]DFMDIXQNFMLPHWRGDVLPSOHNVX
GUDZQRZ KROGRQ
SORW VWDUW VWDUW RU
WH[W VWDUW VWDUW 3XQNWVWDUWRZ\
SORW RU
WH[W 3RV]XNLZDQHPLQLPXP
G VTUW GHOWD
XW >[GHOWD[G [[G [GHOWD[G@
IRUL QQISRP L XW $ XW S] UDQGQ
OLI >OLIISRP L @
HQG I PHDQ ISRP
IRUL QQISRP L XW $ XW S] UDQGQ
OLI >OLIISRP L @
HQG I PHDQ ISRP
IRUL QQISRP L XW $ XW S] UDQGQ
OLI >OLIISRP L @
HQG I PHDQ ISRP
>IXLG[@ VRUW I
XFROG XW LG[ XW LG[ XW LG[
IRUL 1
XSRP XW LG[ XW LG[
XQRZH XSRP XSRPXW LG[ SUyEDRGELFLD
IRUL QQISRP L XQRZH $ XQRZH S] UDQGQ
OLI >OLIISRP L @
HQG IQRZD PHDQ ISRP
LIIQRZD IX XW LG[ XQRZH
I LG[ IQRZD
HQG
>IXLG[@ VRUW I
XFQHZ XW LG[ XW LG[ XW LG[
SORW >XFROG XFQHZ @>XFROG XFQHZ @ U
XFQHZ XFQHZ R (UDVHPRGH QRQH
GUDZQRZ
XFROG XFQHZ
HQGLQVWUIRU
QU QU ILJXUH QU SORW OLI JULG
GLVS =QDOH]LRQHSRáR*HQLHRSWLPXP XRSW XFQHZ
GLVS :DUWR üIXQNFMLZRVWDWQLHMLWHUDFML IRSW PLQ I
GLVS /LF]EDREOLF]H IXQNFML OHQJWK OLI
GLVS QDFL QLMNODZLV]E\ZUyFLüGRPHQX SDXVH
KROGRII
HQG REOLF]H PHWRG VLPSOHNVX
LIPHWRGD
Z]P LQSXW 2NUH OZ]PRFQLHQLH
GHOWD LQSXW 3RGDMSU]\URVWSU]\Z\]QDF]DQLXJUDGLHQWX
ILJXUH QU FRQWV A
FRQWRXU XXINZFRQWV Z
[ODEHO [ \ODEHO [
WLWOH 0LQLPL]DFMDIXQNFMLPHWRGDJUDGLHQWXVWRFKDVW\F]QHJR
GUDZQRZ KROGRQ
SORW VWDUW VWDUW RU
WH[W VWDUW VWDUW 3XQNWVWDUWRZ\
SORW RU
WH[W 3RV]XNLZDQHPLQLPXP
[ VWDUW [ VWDUW OLI >@
IRUL 1
[GHO L [ L GHOWD
IQHZ >[GHO L [ L @ $ >[GHO L [ L @S] UDQGQ
[GHO L [ L GHOWD
IQHZ >[ L [GHO L @ $ >[ L [GHO L @S] UDQGQ
UJUDG >IQHZIROGIQHZIROG@GHOWD
[ L [ L Z]P UJUDG L [ L [ L Z]P UJUDG L
[ L [ L Z]P UJUDG [ L [ L Z]P UJUDG
IQHZ >[ L [ L @ $ >[ L [ L @S] UDQGQ
OLI >OLIIQHZ@
IROG IQHZ
SORW >[ L [ L @>[ L [ L @ U [ L [ L R (UDVHPRGH QRQH
GUDZQRZ
HQGLQVWUIRU
QU QU ILJXUH QU SORW OLI JULG
GLVS =QDOH]LRQHSRáR*HQLHRSWLPXP [RSW >[ 1 [ 1 @
GLVS :DUWR üIXQNFMLZRVWDWQLHMLWHUDFML I IQHZ
GLVS 2ULHQWDF\MQDOLF]EDREOLF]H IXQNFML OHQJWK OLI
GLVS QDFL QLMNODZLV]E\ZUyFLüGRPHQX SDXVH
KROGRII
HQGREOLF]H PHWRG JUDGLHQWXVWRFKDVW\F]QHJR
QU QU
HQG
8:$*,%,%/,2*5$),&=1(L/,7(5$785$
:LHG]DSRGVWDZRZDZ]DNUHVLHQLHOLQLRZHJRSURJUDPRZDQLDPDWHPDW\F]QHJR RSW\PDOL
]DFMLVWDW\F]QHM MHVWSU]HND]\ZDQD]ZL
(OHZQLHNWyU\FKSRGU
F]QLNDFK]PHWRGQXPHU\F]
Q\FKQS>@3UREOHPDW\NDMHGQRZ\PLDURZHMRSW\PDOL]DFMLáF]\VL
FL OH ]HZ]JO
GXQD
]QDQ UyZQR ü
df(u)/du|
u=u*
= 0)
]SUREOHPUR]ZL]\ZDQLDUyZQDQLDQLHOLQLRZHJRVWD
QRZLF\P SRGVWDZRZ\ SUREOHP DQDOL]\ QXPHU\F]QHM 5R]ZLQL
FLH WHM SUREOHPDW\NL
Z NLHUXQNX UR]ZL]\ZDQLD ]DGD RSW\PDOL]DFML Z REHFQR FL ]DNáyFH ]QDMGXMHP\ SU]HGH
ZV]\VWNLPZPRQRJUDILDFKSR ZL
FRQ\FKDSURNV\PDFMLVWRFKDVW\F]QHM QSMHGHQ]UR]G]LD
áyZ >@ = QLHNWyU\FK SRGU
F]QLNyZ SRUXV]DMF\FK WHPDW\N
RSW\PDOL]DFML Z V\WXDFMDFK
QLHGHWHUPLQLVW\F]Q\FKZDUWRSROHFLüV\VWHPDW\]XMF\ZLHG]
SRGVWDZRZ >@OXERGZRáXM
F\VL
GREDGD RSHUDFMLWHFKQRORJLF]Q\FKZSU]HP\ OHHOHNWURPDV]\QRZ\P QS>@
6SHF\ILF]Q\PW\SHPWHFKQLNL SU]\EOL*RQHJR UR]ZL]\ZDQLD]DGD RSW\PDOL]DFMLV DOJR
U\WP\HZROXF\MQH]ZDQHWH* JHQHW\F]Q\PL7HFKQLNDREOLF]H WHJRURG]DMXUR]ZLQ
áDVL
Z
ODWDFK G]LHZL
üG]LHVLW\FK ,QWHUHVXMFH VDPH SU]H] VL
Z\NáDG\ ] DOJRU\WPyZ HZROXF\M
Q\FKZ SRGU
F]QLNX >@ SRSU]HG]RQH V ]ZL
]á\P SU]HND]DQ\PZ GRVW
SQHM IRUPLH SU]H
JOGHPSRGVWDZRZ\FKSUREOHPyZRSW\PDOL]DFML
8NáDG\UHJXODFMLHNVWUHPDOQHMSU]HGVWDZLDQHV ZQLHNWyU\FKSRGU F]QLNDFKWHRULLUHJXOD
FML0RGHOXNáDGX]EOL*RQHJRGREDGDQHJRZ üZPR*HP\QS]QDOH(üZNVL*FHQDWHPDW
UHJXODWRUyZ>@]DSHZQH]QDQHM]SRSU]HGQLFKVHPHVWUyZVWXGLyZ3RGVWDZRZHPIXQNFMH
SDNLHWX2SW\PDOL]DFMLQDWOHVWRVRZDQ\FKZQLFKPHWRGSURJUDPRZDQLDSU]HGVWDZLRQHV Z
MHGQ\P]UR]G]LDáyZ>@
-6HLGOHU$%DGDFK:0RGOLV] 0HWRG\UR]ZL]\ZDQLD]DGD RSW\PDOL]DFML:17
:DUV]DZD
:)LQGHLVHQ-6]\PDQRZVNL$:LHU]ELFNL 7HRULDLPHWRG\REOLF]HQLRZHRSW\PDOL
]DFML3:1:DUV]DZD
$%MRUFN*'DKOTXLVW 0HWRG\QXPHU\F]QH3:1:DUV]DZD
-$UDEDV :\NáDG\]DOJRU\WPyZHZROXF\MQ\FK:17:DUV]DZD
/7U\EXV 5HJXODWRU\ZLHORIXQNF\MQH:17:DUV]DZD
(3DMN.:LHF]RUNRZVNL 3RGVWDZ\RSW\PDOL]DFMLRSHUDFMLWHFKQRORJLF]Q\FKZSU]\
NáDGDFK3:1:DUV]DZD3R]QD
$=DOHZVNL 5&HJLHáD 0DWODE ± REOLF]HQLD QXPHU\F]QH L LFK ]DVWRVRZDQLD:\G1D
NRP3R]QD
Document Outline
- Materia³y pomocnicze do æwiczeñ
- z „Nowoczesnych metod sterowania”
- BIA£YSTOK, 2002
- Algorytmy optymalizacji funkcji jednej zmiennej
- Specyfika optymalizacji w obecnoœci zak³óceñ pomiarowych
Wyszukiwarka
Podobne podstrony:
NowoczesneMetodySterowania ZadaniaOptymalizacji
socjologia zadanie Co mnie – człowieka nowoczesnego
Zadania-Abs Fl-cw.zaoczni 09-10, Nowoczesne medoty instrumentalne
Brzozowski Drogi i zadania nowoczesnej filozofii
Brzozowski Stanisław DROGI ZADANIA NOWOCZESNEJ FILOZOFII
Brzozowski, Stanisław Drog i i zadania nowoczesnej filozofii w Kultura i życie , Warszawa 1973
Zadania z treścia
Prezentacja 2 analiza akcji zadania dla studentow
Technologia informacji i komunikacji w nowoczesnej szkole
Przedmiot i zadania dydaktyki 4
zadanie 1 v 002
Przedmiot dzialy i zadania kryminologii oraz metody badan kr
KOLOKWIUM 2 zadanie wg Adamczewskiego na porownawczą 97
CELE I ZADANIA EDUKACJI MEDIALNEJ(1)
ochrona atmosfery zadania
zadania
więcej podobnych podstron