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Calc Num Prato 02

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Calc Num Prato 02

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  4. 4. (VHPSLR (UURUH GL DSSURVVLPD]LRQH 1HOO·HVHPSLR SUHFHGHQWH VL DYHYDQR D E H GRSR 6H YRJOLDPR RWWHQHUH XQ·DSSURVVLPD]LRQH GHOOD UDGLFH [ LWHUD]LRQL DYHYDPR RWWHQXWR FRQ XQ HUURUH LQIHULRUH DG XQD FHUWD VRJOLD GL WROOHUDQ]D SUHILVVDWD SRQLDPR x * ≈ c 8 = 8 . 882813 1 b−a §b−a· (b − a ) ≤ Ÿ 2k ≥ Ÿ k ≥ log 2 ¨ ¸ /·HUURUH FRPPHVVR q PDJJLRUDWR GD PP P JJ 2k © ¹ 1 1 ( b − a ) = 8 (12 − 6 ) ≈ 0 . 0234 ,Q SDUWLFRODUH LO QXPHUR GL SDVVL FKH GREELDPR HVHJXLUH 2 k 2 FRQ LO PHWRGR GL ELVH]LRQH SHU RWWHQHUH O·DFFXUDWH]]D ULFKLHVWD FRLQFLGH FRQ O·DSSURVVLPD]LRQH SHU HFFHVVR GL §b−a· log 2 ¨ ¸ © ¹ DOO·LQWHUR SL YLFLQR (VHPSLR RPSOHVVLWj FRPSXWD]LRQDOH (VHPSLR GHWHUPLQDUH LO QXPHUR GL LWHUD]LRQL GHO PHWRGR GL ,Q TXHVWR FDVR OD FRPSOHVVLWj FRPSXWD]LRQDOH VL PLVXUD LQ ELVH]LRQH QHFHVVDULH SHU ULVROYHUH O·HTXD]LRQH QXPHUR GL YDOXWD]LRQL GL IXQ]LRQH x 3 + 4 x 2 − 10 = 0 3HU LO PHWRGR GL ELVH]LRQH ILVVDWD XQD VRJOLD GL WROOHUDQ]D OD FRPSOHVVLWj FRPSXWD]LRQDOH q GL QHOO·LQWHUYDOOR @ FRQ XQD WROOHUDQ]D ª § b − a ·º 6L KD « log 2 ¨ ¸» « © ¹» 1 2k ( ) ( b − a ) ≤ 10 − 5 Ÿ k ≥ log 2 10 5 ≈ 16 . 6 YDOXWD]LRQL GL IXQ]LRQH GRYH RYYHUR VRQR QHFHVVDULH DOPHQR LWHUD]LRQL SHU RWWHQHUH ªx º = min{ n ∈ : n ≥ x} OD SUHFLVLRQH ULFKLHVWD
  5. 5. 3UR H FRQWUR DOFROR GHO SXQWR ILVVR ,O PHWRGR GL ELVH]LRQH KD LO YDQWDJJLR GL ULFKLHGHUH SRFKH RQVLGHULDPR XQD IXQ]LRQH D E@ 4 ? ^ ` 6H SHU RJQL LSRWHVL VXOOD IXQ]LRQH I SHU DYHUH OD FRQYHUJHQ]D [ LQ D E@ GHILQLDPR FRQWLQXLWj H VHJQR RSSRVWR QHJOL HVWUHPL GHOO·LQWHUYDOOR J[ [ [I[ DOORUD 7XWWDYLD OD FRQYHUJHQ]D YHUVR OR ]HUR GHOOD IXQ]LRQH q P PROWR OHQWD DSSURVVLPDWLYDPHQWH VL JXDGDJQD XQD FLIUD SS P P J J I I[ GHFLPDOH RJQL WUH LWHUD]LRQL (· QHFHVVDULR GLVSRUUH GL PHWRGL FKH FRQYHUJDQR DOOD VROX]LRQH GHOO·HTXD]LRQH SL UDSLGDPHQWH [ J[ ,Q SDUWLFRODUH GHWHUPLQDUH XQR ]HUR GHOOD IXQ]LRQH I HTXLYDOH D GHWHUPLQDUH XQ SXQWR ILVVR GHOOD IXQ]LRQH J ,QWHUSUHWD]LRQH JHRPHWULFD (VLVWHQ]D GHO SXQWR ILVVR 7HRUHPD 6H J D E@ 4 q FRQWLQXD H D ” J [ ” E SHU RJQL [ LQ D E@ DOORUD J DPPHWWH DOPHQR XQ SXQWR ILVVR 'LP VH GHILQLDPR K [ P I P J [ ² [ DOORUD OD IXQ]LRQH K q I FRQWLQXD H VL KD KD • KE ” ,Q SDUWLFRODUH SHU LO 7HRUHPD GHJOL =HUL OD IXQ]LRQH K VL DQQXOOD LQ DOPHQR XQ SXQWR GL D E@
  6. 6. 8QLFLWj GHO SXQWR ILVVR ,SRWHVL QRQ VRGGLVIDWWH 7HRUHPD RQWLQXLWj 6H J D E@ 4 q FRQWLQXD GHULYDELOH H WDOH FKH D ” J [ ” E SHU RJQL [ LQ D E@ _J· [ _ ” / SHU RJQL [ LQ D E@ $OORUD J DPPHWWH XQ XQLFR SXQWR ILVVR 'LP VH [ 'L VRQR G GXH SXQWL IL L GL L L GL J L E@ L ILVVL GLVWLQWL LQ D DOORUD _[ _ _J [ ²J _ 3HU LO 7HRUHPD GL /DJUDQJH HVLVWH LQ D E@ WDOH FKH J[ ²J J· [ 4XLQGL _[ _ _J· _Ã_[ _ ” /_[ _ _[ _ FKH q DVVXUGR ,SRWHVL QRQ VRGGLVIDWWH ,SRWHVL QRQ VRGGLVIDWWH 9DORUL LQ D E@ 1RQ WURSSR RVFLOODQWH
  7. 7. 0HWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH (VHPSL 0 g ' ( x) 1 − 1 g ' ( x) 0 6XSSRQLDPR FKH O·HTXD]LRQH DVVHJQDWD I [ VLD VWDWD VFULWWD QHOOD IRUPD [ J[ H VLD [ XQ YDORUH DSSURVVLPDWR GHOOD UDGLFH ,O SURFHVVR LWHUDWLYR [N J [N N « SUHQGH LO QRPH GL PHWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH g ' ( x) 1 g ' ( x ) −1 6H J q FRQWLQXD H OD VXFFHVVLRQH ^[ N ` FRQYHUJH DG XQ FHUWR SXQWR [ SHU N ’ DOORUD [ q XQ SXQWR ILVVR GL J 2VVHUYD]LRQH 7HRUHPD GL FRQYHUJHQ]D JOREDOH 'DL JUDILFL SUHFHGHQWL SRVVLDPR LQWXLUH FKH OD SHQGHQ]D 7HRUHPD GHOOD IXQ]LRQH J LQIOXLVFH VXOOD FRQYHUJHQ]D GHO PHWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH DO SXQWR ILVVR 6LD J D E@ 4 FRQWLQXD GHULYDELOH H WDOH FKH D ” J [ ” E SHU RJQL [ LQ D E@ ,Q SDUWLFRODUH SHU YDORUL GL _J· [ _ PDJJLRUL GL VL KD FKH _J· [ _ ” / SHU RJQL [ LQ D E@ LO PHWRGR QRQ FRQYHUJH P J 6H [ q XQ SXQWR GL D E@ H [ q O·XQLFR SXQWR ILVVR GL J (QXQFLDPR H GLPRVWULDPR RUD LO WHRUHPD GL FRQYHUJHQ]D DOORUD OD VXFFHVVLRQH ^[ N ` GHILQLWD PHGLDQWH LO PHWRGR JOREDOH SHU LO PHWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH GHOOH DSSURVVLPD]LRQL VXFFHVVLYH FRQYHUJH D [ SHU N ’ ,QROWUH VL KD Lk x (k ) − x* ≤ x ( 0 ) − x (1 ) 1− L
  8. 8. 7HRUHPD GL FRQYHUJHQ]D JOREDOH 7HRUHPD GL FRQYHUJHQ]D JOREDOH 'LP GDO 7HRUHPD GL /DJUDQJH VHJXH FKH ,QILQH _[ N [ _ _J [ N J[ _ _J· _Ã_[ N [ _ x ( 0 ) − x * = x ( 0 ) − x (1 ) + x (1 ) − x * ≤ x ( 0 ) − x (1 ) + x (1 ) − x * FRQ LQ , [ N [ 4XLQGL ≤ x ( 0 ) − x (1 ) + L x ( 0 ) − x * _[ N [ _ _J· _Ã_[ N [ _ ” / Ã_[ N [ _ ” / Ã_[ _ N [ _”«” /N Ã_[ _ [ _ 1 3RLFKp lim L = 0 VL KD FKH lim x k (k ) − x * = 0 RYYHUR x (0) − x * ≤ x ( 0 ) − x (1 ) k→∞ k→∞ 1− L lim x ( k ) = x * k→∞ Lk x ( k ) − x * ≤ Lk x ( 0 ) − x * ≤ x ( 0 ) − x (1 ) 1− L ­[ , ] se 'HQRWLDPR I ( , ) = ® ¯[ , ] se 2UGLQH GL FRQYHUJHQ]D 2UGLQH GL FRQYHUJHQ]D RQVLGHULDPR LO PHWRGR LWHUDWLYR GHILQLWR GD XQD FHUWD RPH FDVL SDUWLFRODUL GLUHPR FKH LO PHWRGR KD YHORFLWj GL IXQ]LRQH J H GDOOD VXFFHVVLRQH FRQYHUJHQ]D [N J [N N « [ ILVVDWR TXDGUDWLFD VH S 6XSSRQLDPR FKH OD VXFFHVVLRQH ^[ ` FRQYHUJD DG XQ FHUWR N YDORUH [ SHU N ’ x (k ) − x * lim 2 = C, con C 0 k→∞ x ( k −1 ) − x * 'LFLDPR FKH LO PHWRGR KD RUGLQH S • R KD YHORFLWj GL FRQYHUJHQ]D SDUL D S • VH OLQHDUH VH S x (k ) − x * ­ 0 C ≤ 1 se p = 1 lim = C, con ® x (k ) − x * ¯C 0 se p 1 k→∞ ( k −1 ) p x − x* lim = C, con 0 C 1 k→∞ x ( k −1 ) − x * ,O YDORUH q GHWWR FRVWDQWH DVLQWRWLFD GHOO·HUURUH
  9. 9. (VHPSLR RQYHUJHQ]D OLQHDUH 3HU LO PHWRGR GL ELVH]LRQH VL SXz GLPRVWUDUH FKH YDOH 7HRUHPD x (k ) − x * 1 6LD J D E@ 4 XQD IXQ]LRQH GL FODVVH WDOH FKH lim = k→∞ x ( k −1 ) − x * 2 D ” J [ ” E SHU RJQL [ LQ D E@ _J· [ _ ” / SHU RJQL [ LQ D E@ J J· [  SHU RJQL [ LQ D E@ ,Q SDUWLFRODUH LO PHWRGR GL ELVH]LRQH KD YHORFLWj GL $OORUD SHU RJQL [ LQ D E@ OD VXFFHVVLRQH ^[ N ` GHILQLWD FRQYHUJHQ]D OLQHDUH PHGLDQWH LO PHWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH FRQYHUJH D [ SHU N ’ FRQ YHORFLWj OLQHDUH 9HGLDPR FRPH FRVWUXLUH PHWRGL LWHUDWLYL FRQ YHORFLWj GL ,QROWUH VL KD FRQYHUJHQ]D PDJJLRUH C = g ' ( x *) RQYHUJHQ]D OLQHDUH RQYHUJHQ]D TXDGUDWLFD 'LP OD FRQYHUJHQ]D q DVVLFXUDWD GDO 7HRUHPD GL 7HRUHPD FRQYHUJHQ]D JOREDOH YLVWR LQ SUHFHGHQ]D 6LD J D E@ 4 XQD IXQ]LRQH GL FODVVH WDOH FKH 3HU TXDQWR ULJXDUGD O·RUGLQH VFULYLDPR D ” J [ ” E SHU RJQL [ LQ D E@ _J· [ _ ” / SHU RJQL [ LQ D E@ g ( x ( k −1) ) = g ( x *) + g ' ( ) k )( x ( k −1) − x *) ), k ∈ I ( x ( k −1) , x *) ) J J· [ J·· [  SHU RJQL [ LQ D E@ x (k ) x* $OORUD SHU RJQL [ LQ D E@ OD VXFFHVVLRQH ^[ N ` GHILQLWD PHGLDQWH LO PHWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH x (k ) − x * x ( k −1 ) − x * FRQYHUJH D [ SHU N ’ FRQ YHORFLWj TXDGUDWLFD lim = lim g ' ( k) = g ' ( x *) ,QROWUH VL KD k→∞ x ( k −1 ) − x * k→∞ x ( k −1 ) − x * 1 C = g ' ' ( x *) 2
  10. 10. RQYHUJHQ]D TXDGUDWLFD RQYHUJHQ]D TXDGUDWLFD 'LP OD FRQYHUJHQ]D q DVVLFXUDWD GDO 7HRUHPD GL 'DO WHRUHPD SUHFHGHQWH VHJXH FKH VL SXz GLVSRUUH GL XQ FRQYHUJHQ]D JOREDOH YLVWR LQ SUHFHGHQ]D PHWRGR LWHUDWLYR [ N J [N FKH DEELD YHORFLWj GL FRQYHUJHQ]D TXDGUDWLFD VH J· [ 3HU TXDQWR ULJXDUGD O·RUGLQH VFULYLDPR 3RLFKp J [ [ ² [ I [ FHUFKLDPR GXQTXH XQD IXQ]LRQH g ' ' ( k ) ( k −1 ) g ( x ( k −1 ) ) = g ( x *) + g ' ( x *)( x ( k −1) − x *) + ) )( ) (x − x *) 2 , ) @ D E@ 4?^ ` WDOH FKH J· [ ^ J 0D 2 ( k −1 ) k ∈ I (x , x *) 0 = g ' ( x *) = 1 − ' ( x *) f ( x *) − ( x *) f ' ( x *) x (k ) x* 0 2 =0 ( k −1 ) x (k ) − x* g''( ) x − x* g ' ' ( x *) lim = lim k = 1 k→∞ ( k −1 ) 2 k→∞ 2 ( k −1 ) 2 2 ( x *) = x − x* x − x* f ' ( x *) ,O PHWRGR GL 1HZWRQ ,QWHUSUHWD]LRQH JHRPHWULFD f ( x ( k −1 ) ) 5HWWD WDQJHQWH D I LQ [ N I [ N 8QD SRVVLELOLWj SHU VRGGLVIDUH TXHVWD FRQGL]LRQH q SRUUH x ( k ) = x ( k −1 ) − I [N I· [ N [ [N f ' ( x ( k −1 ) ) 1 ( x) = f ' ( x) SHU RJQL [ LQ D E@ H VFULYHUH LO PHWRGR LWHUDWLYR FRPH f ( x ( k −1 ) ) x ( k ) = x ( k −1 ) − , k = 1, 2 ,..., x (0) ∈ [a, b] f ' ( x ( k −1 ) ) 4XHVWR PHWRGR LWHUDWLYR SUHQGH LO QRPH GL PHWRGR GL 1HZWRQ
  11. 11. RQYHUJHQ]D GHO PHWRGR GL 1HZWRQ DVL SDUWLFRODUL 7HRUHPD I· [ N 6LD I D E@ 4 XQD IXQ]LRQH GL FODVVH WDOH FKH I· [  SHU RJQL [ LQ D E@ H VLD [ LQ D E@ WDOH FKH I [ $OORUD HVLVWH XQ LQWRUQR , GL [ FRQWHQXWR LQ D E@ WDOH FKH S SHU RJQL [ LQ , OD VXFFHVVLRQH JHQHUDWD GDO PHWRGR GL J J P 1HZWRQ FRQYHUJH TXDGUDWLFDPHQWH D [ ,Q SDUWLFRODUH LO WHRUHPD DVVLFXUD FKH LO PHWRGR GL 1HZWRQ FRQYHUJH VH VL VFHJOLH O·LWHUDWD LQL]LDOH ´VXIILFLHQWHPHQWH YLFLQRµ D [ ,Q FDVR FRQWUDULR LO PHWRGR GL 1HZWRQ SXz JHQHUDUH XQD VXFFHVVLRQH GLYHUJHQWH R XQD VXFFHVVLRQH FRQYHUJHQWH D XQD UDGLFH O·DOJRULWPR VL DUUHVWD GLYHUVD GD [ RQYHUJHQ]D JOREDOH DVL SDUWLFRODUL GHO PHWRGR GL 1HZWRQ UDGLFL GLYHUVH 7HRUHPD 6LD I D E@ 4 XQD IXQ]LRQH GL FODVVH WDOH FKH ID IE ! I· [  SHU RJQL [ LQ D E@ I I·· [ ” SHU RJQL [ LQ D E@ [ _I E _ ” E D Ã_I· E _ [ $OORUD LO PHWRGR GL 1HZWRQ JHQHUD XQD VXFFHVVLRQH GL LWHUDWH FKH FRQYHUJRQR DOO·XQLFR ]HUR GL I LQ D E@ D SDUWLUH GD TXDOVLDVL [ LQ D E@ FRQYHUJHQ]D GLSHQGHQWH GDOOD VFHOWD GL [
  12. 12. RQYHUJHQ]D JOREDOH RQYHUJHQ]D JOREDOH GHO PHWRGR GL 1HZWRQ GHO PHWRGR GL 1HZWRQ /·XOWLPD LSRWHVL VHUYH VROR SHU HYLWDUH FKH XQ·LWHUDWD HVFD GDOO·LQWHUYDOOR D E@ GRYH YDOJRQR OH LSRWHVL ´EXRQHµ 6H TXHVWD FRQGL]LRQH QRQ q YHULILFDWD SXz FDSLWDUH FKH XQD FHUWD LWHUDWD [ N HVFD GDOO·LQWHUYDOOR I· [  O·DOJRULWPR QRQ VL DUUHVWD I·· [ ” IXQ]LRQH FRQFDYD _I E _ ” E D Ã_I· E _ WXWWH OH LWHUDWH VRQR LQ D E@ RQYHUJHQ]D JOREDOH 0HWRGL TXDVL 1HZWRQ GHO PHWRGR GL 1HZWRQ /D VWHVVD WHVL YDOH SHU DOWUL WUH JUXSSL GL FRQGL]LRQL ,O PHWRGR GL 1HZWRQ ULFKLHGH DG RJQL SDVVR OD YDOXWD]LRQH ID IE ! GHOOD GHULYDWD SULPD GHOOD IXQ]LRQH I LQ XQ GHWHUPLQDWR SXQWR I· [  SHU RJQL [ LQ D E@ I·· [ • SHU RJQL [ LQ D E@ 1HO FDVR LQ FXL TXHVWR FDOFROR ULVXOWDVVH GLIILFLOH R _I D _ ” E D Ã_I· D _ DGGLULWWXUD LPSRVVLELOH q XWLOH DYHUH D GLVSRVL]LRQH PS S ID ! IE YDULDQWL GHO PHWRGR GL 1HZWRQ FKH QRQ ULFKLHGDQR OD I· [  SHU RJQL [ LQ D E@ YDOXWD]LRQH GL WDOH GHULYDWD I·· [ • SHU RJQL [ LQ D E@ _I E _ ” E D Ã_I· E _ ,O ORUR VFKHPD JHQHUDOH q LO VHJXHQWH ID ! IE f ( x ( k −1 ) ) I· [  SHU RJQL [ LQ D E@ x ( k ) = x ( k −1 ) − , k = 1, 2 ,..., m k ≈ f ' ( x ( k −1) ) I·· [ ” SHU RJQL [ LQ D E@ mk _I D _ ” E D Ã_I· D _
  13. 13. ,O PHWRGR GHOOH FRUGH ,QWHUSUHWD]LRQH JHRPHWULFD $VVXPHQGR FKH I VLD VXIILFLHQWHPHQWH UHJRODUH VL SXz DVVXPHUH FKH LQ SURVVLPLWj GHOOD UDGLFH [ OD VXD GHULYDWD SULPD YDULL ´SRFRµ 3HUWDQWR VH [ q SURVVLPR DOOD UDGLFH DOORUD SS P O·DSSURVVLPD]LRQH f ' ( x ( k −1 ) ) ≈ f ' ( x ( 0 ) ) ≡ m k SXz HVVHUH FRQYHQLHQWHPHQWH XWLOL]]DWD ,Q TXHVWR PRGR VL RWWLHQH O·LWHUD]LRQH [ [ [ ( k −1 ) f ( x ( k −1 ) ) x (k ) = x − , k = 1, 2 ,... f ' ( x (0) ) , VHJPHQWL VRQR WXWWH FRUGH SDUDOOHOH WUD ORUR FKH GHILQLVFH LO PHWRGR GHOOH FRUGH ,O PHWRGR GHOOH FRUGH ,O PHWRGR GHOOH VHFDQWL ,Q JHQHUDOH VL SDUOD GL PHWRGR GHOOH FRUGH TXDQGR PN q 8QD VFHOWD DOWHUQDWLYD DO PHWRGR GHOOH FRUGH FRQVLVWH FRVWDQWH QRQ YDULD DO YDULDUH GL N QHOO·DSSURVVLPDUH OD GHULYDWD GL I FRQ LO VXR UDSSRUWR LQFUHPHQWDOH 8Q·DOWHUQDWLYD DOOD VFHOWD PN I· [ FRQVLVWH QHO GHILQLUH f ( x ( k −1 ) ) − f ( x ( k − 2 ) ) f ' ( x ( k −1 ) ) ≈ ≡ mk x ( k −1 ) − x ( k − 2 ) f (b ) − f ( a ) mk = ,Q , TXHVWR PRGR VL RWWLHQH O·LW W G L WWL O·LWHUD]LRQH L b−a f ( x ( k − 2 ) ) x ( k − 1 ) − f ( x ( k −1 ) ) x ( k − 2 ) ,O FRVWR SHU LWHUD]LRQH GHO PHWRGR GHOOH FRUGH q SDUL D XQD x (k ) = , k = 2 ,3,... YDOXWD]LRQH GL IXQ]LRQH RYYHUR q DQDORJR D TXHOOR GHO f ( x ( k −1 ) ) − f ( x ( k − 2 ) ) PHWRGR GL ELVH]LRQH $O SDUL GL TXHVW·XOWLPR WXWWDYLD VL FKH GHILQLVFH LO PHWRGR GHOOH VHFDQWL GLPRVWUD FKH LO VXR RUGLQH GL FRQYHUJHQ]D q VROR OLQHDUH 2VVHUYLDPR FKH SHU LQQHVFDUH LO PHWRGR VHUYRQR GXH LWHUDWH LQL]LDOL [ H [
  14. 14. ,QWHUSUHWD]LRQH JHRPHWULFD ,O PHWRGR GHOOH VHFDQWL 6L GLPRVWUD FKH VH [ q XQD UDGLFH VHPSOLFH DOORUD LO PHWRGR GHOOH VHFDQWL KD YHORFLWj GL FRQYHUJHQ]D 5 +1 p= ≈ 1 . 618 2 ,O FRUULVSRQGHQWH FRVWR SHU LWHUD]LRQH q GDWR GD XQD VROD YDOXWD]LRQH GL IXQ]LRQH HG q TXLQGL DQDORJR D TXHOOR GHO PHWRGR GL ELVH]LRQH H GHO PHWRGR GHOOH FRUGH $QFKH VH O·RUGLQH GL FRQYHUJHQ]D GHO PHWRGR GHOOH VHFDQWL [ [ [ [ q LQIHULRUH ULVSHWWR D TXHOOR GHO PHWRGR GL 1HZWRQ HVVR ULVXOWD WXWWDYLD HVVHUH VSHVVR SUHIHULELOH D TXHVW·XOWLPR 5HWWD VHFDQWH OD FXUYD SDVVDQWH SHU SHU LO IDWWR GL QRQ ULFKLHGHUH OD YDOXWD]LRQH GHOOD GHULYDWD [N I [N H [N I [N SULPD GL I DG RJQL LWHUD]LRQH ULWHUL GL DUUHVWR DVL SDUWLFRODUL 3Xz HVVHUH XWLOH FRQVLGHUDUH FRPH FULWHULR GL DUUHVWR SHU LO PHWRGR LWHUDWLYR XQD FRSSLD GL FRQGL]LRQL f ( x (k ) ) ≤ 1 , x ( k ) − x ( k −1 ) ≤ 2 FULWHUL GL DUUHVWR DVVROXWL I [N RSSXUH [ I [N ( k −1 ) f (x (k ) ) ≤ 1 f (x (0) ) , x (k ) −x ≤ 2 x (k ) [N [N [ FULWHUL GL DUUHVWR UHODWLYL [ N q ´VXIILFLHQWHPHQWH [ N q DQFRUD ´ORQWDQDµ GRYH VRQR VRJOLH GL WROOHUDQ]D SUHILVVDWH YLFLQDµ D [ PD _I [ N _ GD [ DQFKH VH _I [ N _ q DQFRUD ´JUDQGHµ q ´SLFFRORµ

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