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Calc Num Prato 04

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Calc Num Prato 04

  1. 1. $XWRYDORUL HG DXWRYHWWRUL &DOFROR GHJOL DXWRYDORUL 6LDQR $ XQD PDWULFH Q[Q XQ QXPHUR FRPSOHVVR H [ XQ YHWWRUH GL Q FRPSRQHQWL QRQ WXWWH QXOOH 6H GL XQD PDWULFH Ax = x DOORUD VL GLFH DXWRYDORUH GL $ H [ q LO FRUULVSRQGHQWH DXWRYHWWRUH /XFD =DQQL 0DUFR 3UDWR 2JQL DXWRYHWWRUH VRGGLVID O·HTXD]LRQH ( A − I )x = 0 &DOFROR 1XPHULFR H GXQTXH HVVHQGR [ O·DXWRYDORUH GHYH VRGGLVIDUH &RUVL GL /DXUHD LQ 0DWHPDWLFD H ,QIRUPDWLFD O·HTXD]LRQH det( A − I ) = 0 6SHWWUR GL XQD PDWULFH (VHPSL (VVHQGR GHW $ , XQ SROLQRPLR GL JUDGR Q LQ VL SXz § 2 3· FRQFOXGHUH FKH OD PDWULFH $ DPPHWWD Q DXWRYDORUL A=¨ ¨ 1 4 ¸, det( A − I ) = ¸ 2 −6 +5 = 0 HYHQWXDOPHQWH FRQWDWL FRQ OH ULVSHWWLYH PROWHSOLFLWj © ¹ DOJHEULFKH Ÿ 1 = 5, 2 = 1 /·LQVLHPH §1 0 0· $ ^ % q XQ DXWRYDORUH GL $ ` ¨ ¸ A = ¨ 3 − 2 1 ¸, det( A − I ) = (1 − ) 2 = 0 YLHQH FKLDPDWR VSHWWUR GL $ ¨ 6 − 6 3¸ © ¹ Ÿ 1 = 2 = 1, 3 = 0
  2. 2. ,PSRUWDQ]D GHJOL DXWRYDORUL , ,PSRUWDQ]D GHJOL DXWRYDORUL , 0HWRGL LWHUDWLYL SHU OD ULVROX]LRQH GL XQ VLVWHPD OLQHDUH $[ E VH FRQVLGHULDPR OD GHFRPSRVL]LRQH $ 0 1 FRQ 0 6L GLPRVWUD FKH LO PHWRGR LWHUDWLYR PDWULFH LQYHUWLELOH DOORUD x ( k +1) = M −1 Nx ( k ) + M −1b, k = 0,1,... Ax = b ⇔ ( M − N ) x = b ⇔ FRQYHUJH LQGLSHQGHQWHPHQWH GDOOD VFHOWD GHO YDORUH J S P Mx = Nx + b ⇔ x = M −1 Nx + M −1b LQL]LDOH [ VH H VROR VH 0 1 GRYH LO UDJJLR VSHWWUDOH 0 1 q GHILQLWR FRPH 6H GHILQLDPR OD VXFFHVVLRQH GL YHWWRUL x ( k +1) = M −1 Nx ( k ) + M −1b, k = 0,1,... ( M −1 N ) = max i i =1,...,n VL KD FKH ^[ N ` FRQYHUJH D XQ YHWWRUH [ VH H VROR VH [ q OD FRQ « Q DXWRYDORUL GHOOD PDWULFH 0 1 VROX]LRQH GHO VLVWHPD OLQHDUH GL SDUWHQ]D ,PSRUWDQ]D GHJOL DXWRYDORUL ,, ,PSRUWDQ]D GHJOL DXWRYDORUL ,,, 1XPHUR GL FRQGL]LRQDPHQWR GL XQD PDWULFH VH $ q XQD /D FRQRVFHQ]D GHJOL DXWRYDORUL GL XQD PDWULFH q XWLOH PDWULFH QRQ VLQJRODUH YDOH OD IRUPXOD DQFKH LQ PROWH DOWUH DSSOLFD]LRQL WUD OH TXDOL K ( A) = A−1 A OR VWXGLR GHOOD VWDELOLWj GL FHUWH VWUXWWXUH GRYH __ __ q XQD TXDOVLDVL QRUPD PDWULFLDOH G __Ã__ O O DOFXQH TXHVWLRQL GL VWDWLVWLFD 6H VFHJOLDPR OD QRUPD LO QXPHUR GL FRQGL]LRQDPHQWR GL VWXGLR GHJOL VWDWL GHOO·HQHUJLD $ DVVXPH OD IRUPD VWXGLR GL VLVWHPL GLQDPLFL K 2 ( A) = 1 n SURSDJD]LRQH GHL VHJQDOL GRYH • « • Q VRQR L YDORUL VLQJRODUL GL $ RYYHUR OH UDGLFL GHJOL DXWRYDORUL GHOOD PDWULFH $7$
  3. 3. ,PSRUWDQ]D GHJOL DXWRYDORUL DOFROR GL DXWRYDORUL SDUWLFRODUL 'DOOH RVVHUYD]LRQL SUHFHGHQWL SRVVLDPR GHGXUUH FKH LQ 6LD $ XQD PDWULFH Q[Q GLDJRQDOL]]DELOH HVLVWH XQD PDWULFH WDOXQH FLUFRVWDQ]H QRQ q QHFHVVDULR FRQRVFHUH 3 WDOH FKH $ 3'3 FRQ ' GLDJ « Q PDWULFH GLDJRQDOH LQWHUDPHQWH OR VSHWWUR GL XQD PDWULFH PD SXz HVVHUH GHJOL DXWRYDORUL GL $ $VVXPLDPR FKH VXIILFLHQWH DYHUH GHOOH EXRQH VWLPH VRODPHQWH GHOO·DXWRYDORUH GL PRGXOR PDVVLPR H R GHOO·DXWRYDORUH GL PRGXOR PLQLPR 1 2 ≥ ... ≥ n ,Q DOWUH FLUFRVWDQ]H LQYHFH LO SUREOHPD ULFKLHGH XQ PHWRGR QXPHULFR LQ JUDGR GL IRUQLUH XQ·DSSURVVLPD]LRQH RYYHUR FKH HVLVWD XQ XQLFR DXWRYDORUH GL PRGXOR PDVVLPR H DFFXUDWD GL WXWWL JOL DXWRYDORUL GHOOD PDWULFH FKH WDOH DXWRYDORUH DEELD PROWHSOLFLWj FRPH UDGLFH GHO SROLQRPLR FDUDWWHULVWLFR ,O PHWRGR GHOOH SRWHQ]H ,O PHWRGR GHOOH SRWHQ]H 6LD q ( 0 ) ∈ % n WDOH FKH q (0) =1 HUFKLDPR GL YHULILFDUH WDOH FRQYHUJHQ]D 2 RQVLGHULDPR LO VHJXHQWH SURFHGLPHQWR LWHUDWLYR 3URYLDPR SUHOLPLQDUPHQWH OD UHOD]LRQH ( k −1) (1) z (k ) = Aq Ak q ( 0 ) q (k ) = , k = 1,2,... , ( 2) (k ) = ( q ( k −1) ) H z ( k ) k = 1,2,... Ak q ( 0 ) 2 z (k ) (3) q (k ) = (k ) FRQ LO SULQFLSLR GL LQGX]LRQH z 2 z (1) Aq ( 0 ) %DVH q (1) = = 4XHVWR DOJRULWPR YD VRWWR LO QRPH GL PHWRGR GHOOH z (1) Aq ( 0 ) SRWHQ]H H IRUQLVFH O·DXWRYDORUH GL PRGXOR PDVVLPR GL XQD 2 2 PDWULFH H LO FRUULVSRQGHQWH DXWRYHWWRUH Ak q ( 0 ) ,SRWHVL LQGXWWLYD q (k ) = k (0) , k = 1,2,... Aq 2
  4. 4. ,O PHWRGR GHOOH SRWHQ]H ,O PHWRGR GHOOH SRWHQ]H § n · n n 3DVVR 4XLQGL Aqk (0) = A ¨ ¦ i xi ¸ = ¦ i A xi = ¦ k k k x i i i z ( k +1) Aq (k ) AA q k (0) A q k +1 ( 0 ) © i =1 ¹ i =1 i =1 q ( k +1) = = = = z ( k +1) Aq ( k ) AAk q ( 0 ) Ak +1q ( 0 ) § § k · ¸ · (x + y ) n ¨x + ¨ 1 ¦ 2 2 2 2 = k 1 1 i ¨ ¨ i ¸ xi = ¸ ¸ k 1 1 1 (k ) 1 © 1¹ ,QROWUH SRLFKp $ q GL , O L Kp GLDJRQDOL]]DELOH L VXRL DXWRYHWWRUL OL ELO L L © i =2 ¹ ^[ « [Q` FRVWLWXLVFRQR XQD EDVH GL % Q DYHQGR DVVXQWR  1H VHJXH FKH ,Q SDUWLFRODUH SRVVLDPR HVSULPHUH T FRPH FRPELQD]LRQH q (k ) Ak q ( 0 ) = k (0) = (x + y ) 1 1 k 1 (k ) OLQHDUH GHJOL HOHPHQWL GL WDOH EDVH Aq 2 (x + y ) k 1 1 1 (k ) 2 (x + y ) → n = ¦ i xi , (k ) q (0) ,..., ∈% x1 1 n = sign ( k ) 1 ± i =1 1 1 (x + y ) 1 (k ) 2 k →∞ x1 2 ,O PHWRGR GHOOH SRWHQ]H ULWHULR G·DUUHVWR ,O YHWWRUH T N WHQGH TXLQGL DOO·DXWRYHWWRUH [ QRUPDOL]]DWR RPH FULWHULR G·DUUHVWR VL SXz FRQVLGHUDUH DG HVHPSLR LO HG HYHQWXDOPHQWH FDPELDWR GL VHJQR FULWHULR UHODWLYR VXOOD GLIIHUHQ]D WUD GXH LWHUDWH 3HU TXDQWR ULJXDUGD OD VXFFHVVLRQH N VL KD VXFFHVVLYH ( k −1) ( x1 ) H Ax1 ( x1 ) H x1 (k ) − (k ) = (q ( k −1) ) H Aq ( k −1) → 2 = 1 2 = 1 ≤ toll , k = 1,2,... k →∞ (k ) x1 2 x1 2 RYYHUR LO YDORUH N FRQYHUJH DOO·DXWRYDORUH GL PRGXOR GRYH WROO q XQD VRJOLD SUHILVVDWD PDVVLPR
  5. 5. RPPHQWL (VHPSLR 9DOJRQR OH VHJXHQWL RVVHUYD]LRQL § 3 0 0 · 1 = 3, x1 = (1,0,2)T ¨ ¸ SHU FRPH q GHILQLWR LO YHWWRUH N OD FRQYHUJHQ]D q A = ¨− 4 6 2 ¸ 2 = 1, x2 = (0,2,−5)T WDQWR SL UDSLGD TXDQWR q SL SLFFROR LO UDSSRUWR _ _ ¨ 16 − 15 − 5 ¸ © ¹ 3 = 0, x3 = (0,1,−3)T VH $ q VLPPHWULFD LO PHWRGR GHOOH SRWHQ]H FRQYHUJH SL UDSLGDPHQWH 6L SXz I YHGHUH L I WWL FKH LO W LG W z IDU G LQIDWWL K WHUPLQH N L § 3 / 3· § 1 732050 · 1.732050 ¨ ¸ ¨ ¸ FRQYHUJH D FRPH _ _ N LQYHFH GL _ _N q (0) = ¨ 3 / 3 ¸, z = Aq = ¨ 2.309401 ¸, (1) (0) (1) = 1.000000 VH q UHDOH H KD PROWHSOLFLWj U LO PHWRGR q DQFRUD ¨ ¨ 3 / 3¸¸ ¨ − 2.309401¸ FRQYHUJHQWH D H D XQ VXR DXWRYHWWRUH PD LQ JHQHUDOH OD © ¹ © ¹ FRQYHUJHQ]D q PROWR SL OHQWD § 0.468521 · § 1.405563 · VH HVLVWRQR SL DXWRYDORUL GL PRGXOR PDVVLPR GLYHUVL WUD ¨ ¸ ( 2) ¨ ¸ ORUR LO PHWRGR GHOOH SRWHQ]H SXz QRQ FRQYHUJHUH q (1) = ¨ 0.624695 ¸, z = Aq = ¨ 0.624695 ¸, (1) ( 2) = 0.268292 ¨ − 0.624695 ¸ ¨ 1.249390 ¸ © ¹ © ¹ (VHPSLR ,O PHWRGR GHOOH SRWHQ]H LQYHUVH § 0.709299 · § 2.127898 · ¨ ¸ ( 3) ¨ ¸ 6LD $ XQD PDWULFH Q[Q QRQ VLQJRODUH H GLDJRQDOL]]DELOH H q ( 2) = ¨ 0.315244 ¸, z = Aq = ¨ 0.315244 ¸, ( 2) ( 3) = 3.795031 VLDQR « Q JOL DXWRYDORUL QRQQXOOL GL $ WDOL FKH ¨ 0.630488 ¸ ¨ 3.467685 ¸ © ¹ © ¹ ≥ ... ≥ 0 § 0.521453 · 1 n −1 n ¨ ¸ q ( 3) = ¨ 0.077252 ¸, ( 4 ) = 3.316299 ««« 2VVHUYLDPR FKH OD PDWULFH LQYHUVD $ KD DXWRYDORUL ¨ ¸ © 0.849775 ¹ 1 1 1 ≥ ... ≥ 0 § 0.447213 · WHVW G·DUUHVWR ¨ ¸ YHULILFDWR n n −1 1 q (15) = ¨ 0.000000 ¸, (16 ) = 3.000000 ¨ 0.894427 ¸ ,Q SDUWLFRODUH SHU GHWHUPLQDUH O·DXWRYDORUH GL PRGXOR © ¹ PLQLPR Q GL XQD PDWULFH $ q SRVVLELOH DSSOLFDUH LO PHWRGR GHOOH SRWHQ]H DOOD PDWULFH LQYHUVD 2VVHUYLDPR FKH (1,0,2)T / 5 = (0.447213,0,0.894427)T
  6. 6. $SSURVVLPD]LRQH GL WXWWL JOL ,O PHWRGR GHOOH SRWHQ]H LQYHUVH DXWRYDORUL GL XQD PDWULFH $QDORJDPHQWH VH q QRWD XQD VWLPD GL XQ DXWRYDORUH M $EELDPR YLVWR FRPH GHILQLUH XQ PHWRGR LWHUDWLYR FKH GHOOD PDWULFH $ RSSXUH VH VL YXROH FDOFRODUH O·DXWRYDORUH SHUPHWWD GL FDOFRODUH GHOOH EXRQH VWLPH GHOO·DXWRYDORUH GL SL YLFLQR D XQ FHUWR QXPHUR RYYHUR PRGXOR PDVVLPR H R GHOO·DXWRYDORUH GL PRGXOR PLQLPR GL XQD PDWULFH − i − j 0 ∀i = 1,..., n, i ≠ j 9HGLDPR RUD XQ DOWUR DOJRULWPR LWHUDWLYR FKH IRUQLVFH XQ·DSSURVVLPD]LRQH GL WXWWL JOL DXWRYDORUL GHOOD PDWULFH DOORUD VL SXz DSSOLFDUH LO PHWRGR GHOOH SRWHQ]H DOOD PDWULFH /D VWUDWHJLD FRQVLVWH QHO ULGXUUH OD PDWULFH GHOOD TXDOH VL $ , LQ TXDQWR YRJOLRQR FRQRVFHUH JOL DXWRYDORUL DG XQD IRUPD SHU OD TXDOH WDOH FDOFROR ULVXOWD VHPSOLFH PHGLDQWH WUDVIRUPD]LRQL GL 1 1 VLPLOLWXGLQH 7DOH WHFQLFD LWHUDWLYD q EDVDWD VXOOD 0 ∀i = 1,..., n, i ≠ j − j − i IDWWRUL]]D]LRQH 45 GHOOD PDWULFH 7UDVIRUPD]LRQL HOHPHQWDUL GL /HPPD GL +RXVHKROGHU +RXVHKROGHU 7HRUHPD 6LD X XQ YHWWRUH QRQ QXOOR H VLD 6LD [ XQ YHWWRUH QRQ QXOOR H VLD 1 1 2 = sign ( x1 ) x = u tu = u 2 2 2 2 6H 1 1 2 u = x + e1( n ) e = utu = u /D PDWULFH RUWRJRQDOH H VLPPHWULFD 2 2 2 1 FRQ H Q « W 4Q DOORUD OD PDWULFH U = I − uu t 1 U = I − uu t q GHWWD WUDVIRUPD]LRQH HOHPHQWDUH GL +RXVHKROGHU q XQD WUDVIRUPD]LRQH HOHPHQWDUH GL +RXVHKROGHU H VL KD Ux = − e1( n )
  7. 7. /HPPD GL +RXVHKROGHU )DWWRUL]]D]LRQH 45 %DVWD RVVHUYDUH LQIDWWL FKH 1 t 1 1 = u u = ( x + e1( n ) ) t ( x + e1( n ) ) = ( x t x + 2 x t e1( n ) + 2 ) 2 2 2 7HRUHPD 1 2 2 6LD $ XQD PDWULFH QRQ VLQJRODUH $OORUD $ SXz HVVHUH = ( 2 x 2 + 2 x t e1( n ) ) = x 2 + x1 IDWWRUL]]DWD QHO SURGRWWR GL XQD PDWULFH RUWRJRQDOH 4 H 2 GL XQD PDWULFH WULDQJRODUH VXSHULRUH QRQ VLQJRODUH 5 L L O L L O H TXLQGL § 1 t· ut x ( x + e1( n ) ) t x 7DOH IDWWRUL]]D]LRQH SXz HVVHUH RWWHQXWD PROWLSOLFDQGR $ Ux = ¨ I − uu ¸ x = x − u = x− ( x + e1( n ) ) © ¹ SHU Q WUDVIRUPD]LRQL HOHPHQWDUL GL +RXVHKROGHU 2 x 2 + x1 = x− 2 ( x + e1( n ) ) = − e1( n ) x 2 + x1 3DVVR 3DVVR § − sign(a11) ) a1(1) (1 a12 ) (2 a13 ) (2 a1(n ) · 2 ¨ ¸ a1n · § a a · 2 § a11 (1) (1) ¨ − sign(a22) ) a 22 ) ( 3) ¸ ¸ ¨ ¸ (2 ( ¨ 11 1n 0 a23) (3 a2 n ¨ 2 ¸ A=¨ ¸=¨ ¸ = [a1(1) ... a n1) ] ( U 2 A( 2 ) =¨ 0 0 a33) (3 a33) ¸ = A ( ( 3) ann ¸ ¨ an1) ann) ¸ n ¨a ( (1 ¨ ¸ © n1 ¹ © 1 ¹ ¨ ¸ ¨ 0 0 an3n −1 ( ) ann) ¸ (3 © , ¹ § − sign(a ) a(1) (1) a ( 2) a · ( 2) ¨ 11 1 2 12 ¸ 1n ¨ 0 a ( 2) a ¸ ( 2) x2 = a 22 ) = (a22) ,..., an2 ) ) t , u2 = x2 + ( (2 ( ( n −1) e U1 A = [U1a1(1) ... U1a n1) ] = ¨ ( 22 ¸= A 2n ( 2) 2 2 1 ¨ ¸ ¨ 0 ( 2) an ,n −1 ( 2) ¸ ann ¹ § I1 0 · © ¨ ¸ U2 = ¨ 1 1 ¨0 I n −1 − u2u 2 ¸ t ¸ x1 = a1(1) = (a11) ,..., an1) ) t , u1 = x1 + 1e1( n ) , U1 = I n − u1u1t (1 ( 1 © 2 ¹ 1
  8. 8. 3DVVR Q )DWWRUL]]D]LRQH 45 'RSR Q SDVVL VL DUULYHUj DG DYHUH 5LDVVXPHQGR VL KD § − sign(a11) ) a1(1) (1 a1(,2 )−1 a1(n ) · 2 U n −1U n − 2 ...U 2U1 A = R ¨ 2 n ¸ ¨ 0 ¸ U n −1 A( n −1) =¨ (n) ¸ =R FRQ 8 « 8Q WUDVIRUPD]LRQL HOHPHQWDUL GL +RXVHKROGHU H ¨ − sign(ann −,1n)−1 ) a nn −1) ( −1 ( −1 an −1,n ¸ 5 PDWULFH WULDQJRODUH VXSHULRUH QRQ VLQJRODUH 2 ¨ 0 0 annn) ¸ ( © , ¹ 6LD ( n −1) ( n −1) ( n −1) t Q = U1−1U 2 1...U n − 2U n −1 − −1 −1 xn −1 = a n −1 = (a n −1, n −1 ,a ) , un −1 = xn −1 + n , n −1 e( 2) n −1 1 $OORUD 4 q XQD PDWULFH RUWRJRQDOH LQ TXDQWR SURGRWWR GL § I n−2 0 · PDWULFL RUWRJRQDOL H YDOH OD UHOD]LRQH ¨ ¸ U n −1 = ¨ 1 ¨ 0 I2 − u n −1u n −1 ¸ t ¸ A = QR © n −1 ¹ 2VVHUYD]LRQL ,O PHWRGR 45 LWHUDWLYR /D IDWWRUL]]D]LRQH 45 GL XQD PDWULFH SXz HVVHUH RQGLGHULDPR LO VHJXHQWH DOJRULWPR DSSOLFDWD DOOD ULVROX]LRQH GL XQ VLVWHPD OLQHDUH $[ E ,Q WDO FDVR VL KD LQIDWWL 45 [ E RYYHUR LO VLVWHPD VLD $ XQD PDWULFH Q[Q H VLD $ $ OLQHDUH GD ULVROYHUH DVVXPH OD IRUPD 5[ 4WE 4 q RUWRJRQDOH TXLQGL 4 4 W VLVWHPD WULDQJRODUH VXSHULRUH SHU N « ULVROYLELOH FRQ LO PHWRGR GL VRVWLWX]LRQH DOO·LQGLHWUR DOO LQGLHWUR $N 4N5N /D FRPSOHVVLWj FRPSXWD]LRQDOH GHOOD IDWWRUL]]D]LRQH 45 PHGLDQWH WUDVIRUPD]LRQL HOHPHQWDUL GL +RXVHKROGHU q GL $N 5N4N Q RSHUD]LRQL PROWLSOLFDWLYH RYYHUR FLUFD LO GRSSLR ULVSHWWR DOOD IDWWRUL]]D]LRQH /8 GL *DXVV H FLUFD OD PHWj GRYH $N 4N5N q OD GHFRPSRVL]LRQH 45 GL $N ULVSHWWR DOOD IDWWRUL]]D]LRQH 45 RWWHQXWD PHGLDQWH WUDVIRUPD]LRQL GL *LYHQV 4XHVWR DOJRULWPR SUHQGH LO QRPH GL PHWRGR 45 LWHUDWLYR
  9. 9. 7UDVIRUPD]LRQL GL VLPLOLWXGLQH 7HRUHPD GL FRQYHUJHQ]D 2VVHUYLDPR FKH OH PDWULFL $N H $N VRQR VLPLOL LQ TXDQWR 7HRUHPD 6LD $ XQD PDWULFH Q[Q WDOH FKH L VXRL DXWRYDORUL DEELDQR Ak +1 = Rk Qk = Qk−1 Ak Qk WXWWL PRGXOL GLVWLQWL (VVHQGR 1 2 ... n 0 det( Ak +1 − I ) = det(Qk−1 Ak Qk − Qk−1Qk ) $OORUD OD VXFFHVVLRQH GL PDWULFL ^$N`N « ’ FRQYHUJH DG XQD −1 = det(Q ( Ak − I )Qk ) PDWULFH WULDQJRODUH VXSHULRUH 7 DYHQWH VXOOD GLDJRQDOH JOL k DXWRYDORUL GL $ = det(Qk−1 ) det( Ak − I ) det(Qk ) 6H LQROWUH $ q VLPPHWULFD DOORUD 7 q XQD PDWULFH GLDJRQDOH = det( Ak − I ) VL KD FKH WXWWH OH PDWULFL $N KDQQR JOL VWHVVL DXWRYDORUL (VHPSLR (VHPSLR §1 2· A = A1 = ¨¨ ¸ = Q1 R1 ¸ ©2 1¹ §1 2· ¨2 1¸ A = A1 = ¨ ¸ (Ÿ = 3, = −1) § − 0.4472 − 0.8944 · § − 2.2361 − 1.7889 · 1 2 Q1 = ¨ © ¹ ¨ − 0.8944 0.4472 ¸, R1 = ¨ ¸ ¨ ¸ − 1.3416 ¸ © ¹ © 0 ¹ ...... § 2 6000 1 2000 · 2.6000 1.2000 § 2 9512 0 4390 · 2.9512 0.4390 ¨ 1.2000 − 0.6000 ¸ = R1Q1 = Q2 R2 A2 = ¨ ¸ A3 = ¨¨ 0.4390 − 0.9512 ¸ ¸ © ¹ © ¹ § − 0.9080 − 0.4191· § − 2.8636 − 0.8381· ...... Q2 = ¨ ¨ − 0.4191 0.9080 ¸, R2 = ¨ ¸ ¨ ¸ © ¹ © 0 − 1.0476 ¸ ¹ § 3.0000 0.0001 · A11 = ¨ ¨ 0.0001 − 1.0000 ¸ ¸ § 2.9512 0.4390 · © ¹ A3 = ¨ ¨ ¸ = R2Q2 = Q3 R3 ¸ © 0.4390 − 0.9512 ¹
  10. 10. 2VVHUYD]LRQL 2VVHUYD]LRQL 6H OD PDWULFH $ KD SL DXWRYDORUL GL PRGXOR XJXDOH DOORUD 6XSSRQLDPR FKH LO PHWRGR 45 LWHUDWLYR JHQHUD PDWULFL 5N FRQ VWUXWWXUD OD PDWULFH $ VLD GLDJRQDOL]]DELOH RYYHUR FKH VL SRVVD WULDQJRODUH D EORFFKL GLDJRQDOH D EORFFKL VH OD PDWULFH $ VFULYHUH QHOOD IRUPD $ ;'; FRQ ' GLDJ « Q q VLPPHWULFD LQ FXL JOL DXWRYDORUL GHL EORFFKL GLDJRQDOL OD PDWULFH ; DPPHWWD IDWWRUL]]D]LRQH /8 RYYHUR L FRQYHUJRQR DJOL DXWRYDORUL GL $ PLQRUL SULQFLSDOL GL RJQL RUGLQH GL ; VRQR QRQ QXOOL § R11 R1 p · $OORUD VL GLPRVWUD FKH JOL HOHPHQWL GLDJRQDOL GL 7 VRQR ¨ ¸ QHOO·RUGLQH « Q Rk = ¨ ¸ ¨ R pp ¸ 6H ; QRQ DPPHWWH IDWWRUL]]D]LRQH /8 DOORUD LO PHWRGR q © ¹ DQFRUD FRQYHUJHQWH JOL HOHPHQWL GLDJRQDOL GL 7 FRLQFLGRQR 6H WXWWL JOL DXWRYDORUL GL $ VRQR VHPSOLFL DOORUD WDOL DQFRUD FRQ JOL DXWRYDORUL GL $ PD TXHVWL XOWLPL QRQ VRQR SL EORFFKL KDQQR RUGLQH R LQ RUGLQH GL PRGXOR GHFUHVFHQWH RVWR FRPSXWD]LRQDOH 0DWULFL GL +HVVHQEHUJ ,O PHWRGR 45 LWHUDWLYR DSSOLFDWR DG XQD PDWULFH Q[Q KD DG /D ULGX]LRQH GHOOD PDWULFH $ LQ IRUPD GL +HVVHQEHUJ RJQL SDVVR XQ FRVWR FRPSXWD]LRQDOH GHOO·RUGLQH GL Q VXSHULRUH SXz HVVHUH RWWHQXWD PHGLDQWH WUDVIRUPD]LRQL RSHUD]LRQL PROWLSOLFDWLYH SDUL DO FDOFROR GHOOD HOHPHQWDUL GL +RXVHKROGHU IDWWRUL]]D]LRQH 4N5N H DOOD PROWLSOLFD]LRQH GHOOD PDWULFH WULDQJRODUH 5N SHU OH PDWULFL HOHPHQWDUL GHOOD § a11 a1n · § a11) (1 a1(1) · ¨ ¸ ¨ (1) n ¸ IDWWRUL]]D]LRQH ¨a a2 n ¸ ¨ a21 a2 n ¸ § a11) (1) (1 a1(1) · 3HU DEEDVVDUH LO FRVWR FRPSXWD]LRQDOH FRPSOHVVLYR A = ¨ 21 ¸=¨ = ¨ (1) ¸ ¨a n ¸ (1) ¸ ¸ © 1 an ¹ FRQYLHQH WUDVIRUPDUH OD PDWULFH $ LQ IRUPD GL ¨ ¸ ¨ ¨a ann ¸ © an1) ¨ ( (1) ¸ +HVVHQEHUJ VXSHULRUH DLM SHU RJQL L ! M © n1 ¹ 1 ann ¹ §* * * * *· 6LD 8 ¶ OD WUDVIRUPD]LRQH HOHPHQWDUH GL +RXVHKROGHU WDOH ¨ ¸ ¨* * * * *¸ FKH ¨0 * * * *¸ ¨ ¸ ¨0 0 * * *¸ U1 ' a1(1) = 1e1( n −1) ¨0 0 0 * *¸ © ¹
  11. 11. 0DWULFL GL +HVVHQEHUJ 0DWULFL GL +HVVHQEHUJ 3RVWD ,WHUDQGR LO UDJLRQDPHQWR DO SDVVR N HVLPR VL FHUFKHUj OD §I 0 · WUDVIRUPD]LRQH HOHPHQWDUH GL +RXVHKROGHU 8N· WDOH FKH ¨ 0 U '¸ U1 = ¨ 1 ¸ © 1 ¹ U k ' a kk ) = k e1( n − k ) , a kk ) = (ak k )1,k ,..., ank ) ) t ( ( ( + (k §I 0 · H SRVWD U k = ¨ k VL KD FKH ¨ 0 U ' ¸ VL DYUj ¸ © k ¹ § a11) (1 a12 ) (2 a1(n ) · 2 § a11) a12 ) (1 (2 a1(n +1) · k ¨ ( 2) ¸ ¨ ( 2) ¸ ¨ a21 a22) (2 a22 ) ¸ ( ¨ a21 ¸ n ¨ 0 ( k +1) ¸ A2 = U1 AU1 = ¨ 0 ¸ (k ) ( k +1) −1 ¨ akk ak ,k +1 akn ¸ ¨ ¸ −1 Ak +1 = U k AkU k = ¨ ( k +1) ( k +1) ak +1,k ak +1,k +1 ak k +,1n) ¸ ( ¨ ¸ ¨ +1 ¸ ¨ (2 ¸ ¨ 0 ¸ © 0 an2) ( 2 ann) ¹ ¨ ¸ ¨ ¨ 0 ¸ © 0 a1(n +1) k ann +1) ¸ (k ¹ 0DWULFL GL +HVVHQEHUJ 0DWULFL GL +HVVHQEHUJ /D WUDVIRUPD]LRQH GHOOD PDWULFH LQ IRUPD GL +HVVHQEHUJ 'RSR Q SDVVL VL DUULYHUj DG DYHUH VXSHULRUH YLHQH HVHJXLWD XQD YROWD VROD SHUFKp LO PHWRGR 45 LWHUDWLYR DSSOLFDWR D PDWULFL LQ IRUPD GL +HVVHQEHUJ VXSHULRUH SURGXFH PDWULFL $N FKH VRQR DQFRUD LQ IRUPD GL An −1 = U n − 2 An − 2U n − 2 = U n − 2 ...U1 AU1−1...U n − 2 = Q −1 AQ = S −1 −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
  12. 12. 0DWULFL VLPPHWULFKH $SSOLFD]LRQH 69' GL XQD PDWULFH $EELDPR YLVWR FKH RJQL PDWULFH P[Q GL UDQJR N VL SXz 6H OD PDWULFH $ q VLPPHWULFD OD FRUULVSRQGHQWH PDWULFH IDWWRUL]]DUH FRPH LQ IRUPD GL +HVVHQEHUJ VXSHULRUH RWWHQXWD DSSOLFDQGR DG $ 8 97 $ LO PHWRGR GL +RXVHKROGHU q DQFRUD VLPPHWULFD H TXLQGL FRQ ULVXOWD WULGLDJRQDOH FRVu FRPH WXWWH OH PDWULFL $N 8 PDWULFH RUWRJRQDOH P[P JHQHUDWH GDO PHWRGR 45 LWHUDWLYR 9 PDWULFH RUWRJRQDOH Q[Q P J § 0 0 0 0· /D ULGX]LRQH GL XQD PDWULFH VLPPHWULFD $ LQ IRUPD ¨ ¸ ¨0 ¸ GRYH L QXPHUL SRVLWLYL WULGLDJRQDOH ULFKLHGH FLUFD Q RSHUD]LRQL PROWLSOLFDWLYH ¨ H LO PHWRGR 45 LWHUDWLYR DSSOLFDWR DG XQD PDWULFH 0 0¸ « N VRQR OH UDGLFL GHJOL =¨ k −1 ¸ DXWRYDORUL GHOOD PDWULFH $7$ WULGLDJRQDOH KD DG RJQL SDVVR XQ FRVWR FRPSXWD]LRQDOH ¨0 0 0 0¸ GHOO·RUGLQH GL Q RSHUD]LRQL ¨ ¸ H VRQR GHWWL YDORUL VLQJRODUL ¨ ¸ GL $ ¨0 0¸ © 0 0 ¹ $SSOLFD]LRQH 69' GL XQD PDWULFH $SSOLFD]LRQH 69' GL XQD PDWULFH 9HGLDPR FRPH VL FDOFRODQR L YDORUL VLQJRODUL GL XQD PDWULFH §* * * *· §* * 0 0· ¨ ¸ ¨ ¸ XWLOL]]DQGR LO PHWRGR 45 LWHUDWLYR ¨0 * * *¸ ¨0 * * *¸ U1 A = ¨ 0 * * *¸ U1 AV1 = ¨ 0 * * *¸ 3HU SULPD FRVD VL WUDVIRUPD OD PDWULFH LQ IRUPD ¨ ¸ ¨ ¸ ¨0 * * *¸ ¨0 * * *¸ ELGLDJRQDOH VXSHULRUH PHGLDQWH PROWLSOLFD]LRQH D GHVWUD H ¨0 D VLQLVWUD SHU WUDVIRUPD]LRQL HOHPHQWDUL GL +RXVHKROGHU S I P P ¨ ©0 * * *¸ ¹ © * * *¸ ¹ §* * * *· §* * * *· §* * 0 0· §* * 0 0· ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨* * * *¸ ¨0 * * *¸ ¨0 * * *¸ ¨0 * * *¸ A = ¨* * * *¸ U1 A = ¨ 0 * * *¸ U1 AV1 = ¨ 0 * * *¸ U 2U1 AV1 = ¨ 0 0 * *¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨* * * *¸ ¨0 * * *¸ ¨0 * * *¸ ¨0 0 * *¸ ¨* ¨0 0 * *¸ © * * *¸ ¹ ¨0 © * * *¸ ¹ ¨0 © * * *¸ ¹ © ¹
  13. 13. $SSOLFD]LRQH 69' GL XQD PDWULFH $SSOLFD]LRQH 69' GL XQD PDWULFH §* * 0 0· §* * 0 0· ¨ ¸ ¨ ¸ ¨0 * * *¸ ¨0 * * 0¸ 4XLQGL 8$9W % FRQ 8 9 PDWULFL RUWRJRQDOL H % PDWULFH U 2U1 AV1 = ¨ 0 0 * *¸ U 2U1 AV1V2 = ¨ 0 0 * *¸ ELGLDJRQDOH VXSHULRUH ,Q SDUWLFRODUH ¨ ¸ ¨ ¸ ¨0 0 * *¸ ¨0 0 * *¸ ¨ B t B = (UAV t )t UAV t = VAtU tUAV t = VAt AV t ¨ ©0 ¸ 0 * *¹ ©0 0 * *¸ ¹ RYYHUR OH PDWULFL %W% H $W$ KDQQR JOL VWHVVL DXWRYDORUL §* * 0 0· §* * 0 0· ¨ ¸ ¨ ¸ ¨0 * * 0¸ ¨0 * * 0¸ (VVHQGR L YDORUL VLQJRODUL GL $ H % OH UDGLFL GHJOL DXWRYDORUL U 2U1 AV1V2 = ¨ 0 0 * *¸ U 4U 3U 2U1 AV1V2 = ¨ 0 0 * *¸ SRVLWLYL GL $W$ H %W% ULVSHWWLYDPHQWH OH PDWULFL $ H % FRVu ¨ ¸ ¨ ¸ FRVWUXLWH DYUDQQR JOL VWHVVL YDORUL VLQJRODUL ¨0 0 * *¸ ¨0 0 0 *¸ ¨0 0 * *¸ ¨0 © ¹ © 0 0 0¸ ¹ $SSOLFD]LRQH 69' GL XQD PDWULFH /D PDWULFH %W% q WULGLDJRQDOH H VLPPHWULFD (· SRVVLELOH GXQTXH DSSOLFDUH LO PHWRGR 45 LWHUDWLYR D %W% DO FRVWR GL Q RSHUD]LRQL PROWLSOLFDWLYH DG LWHUD]LRQH H RWWHQHUH FRVu O·DSSURVVLPD]LRQH GHL VXRL DXWRYDORUL 3HU DYHUH L YDORUL VLQJRODUL GL $ VDUj GXQTXH VXIILFLHQWH FRQVLGHUDUH OH UDGLFL GHJOL DXWRYDORUL SRVLWLYL GHOOD PDWULFH %W% DSSHQD WURYDWL

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