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0HWRGR GHOOH DSSURVVLPD]LRQL VXFFHVVLYH                                                         (VHPSL
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'LP GDO 7HRUHPD GL /DJUDQJH VHJXH FKH                                                              ,QILQH
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(VHPSLR                                                                      RQYHUJHQ]D OLQHDUH
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FRQYHUJHQ]D PDJJLRUH                                                                                                         C = g ' ( x *)




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RQYHUJHQ]D TXDGUDWLFD                                                                                           RQYHUJHQ]D TXDGUDWLFD
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7HRUHPD                                                                                  I· [ N

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  • 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. 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. (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. 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. 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 [
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  • 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 [
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