ÅÑÃÁÓÉÁ ÃÉÁ ÔÏ ÌÁÈÇÌÁ: ÅÉÓÁÃÙÃÇ ÓÔÇÍÁÑÉÈÌÇÔÉÊÇ ÁÍÁËÕÓÇ. ÅðéìïñöùôÞò: Â. Á. ÄÏÕÃÁËÇÓ

Åðéìïñöùôéêü Ðñüãñáììá Ãéá ôïõò Åêðáéäåõôéêïýò-Ìáèçìáôéêïýò óôï Ìáèçìáôéêü ôìþìá ôïõ Ðáíåðéóôçìßïõ Áèçíþí êáôü ôçí ðåñßïäï Äåêåìâñßïõ 2000-Éïõíßïõ 200 ìå Õðåýèõíï ôïí êáèçãçôþ Ð. ÓôñÜíôæáëï ÅÑÃÁÓÉÁ ÃÉÁ ÔÏ ÌÁÈÇÌÁ: ÅÉÓÁÃÙÃÇ ÓÔÇÍÁÑÉÈÌÇÔÉÊÇ ÁÍÁËÕÓÇ ÅðéìïñöùôÞò: Â. Á. ÄÏÕÃÁËÇÓ Åðéìïñöïýìåíïò: Ãåþñãéïò Ðáíüðïõëïò ÁÈÇÍÁ ÉÏÕÍÉÏÓ 200

ÐÑÏËÏÃÏÓ Ç åñãáóßá áõôþ Ýãéíå áðü ôïí åêðáéäåõôéêü ôçò ÄåõôåñïâÜèìéáò Åêðáßäåõóçò Ðáíüðïõëï Ãåþñãéï ãéá ôéò áíüãêåò ôïõ ìáèþìáôïò «ÅÉÓÁÃÙÃÇ ÓÔÇÍ ÁÑÉÈÌÇÔÉÊÇ ÁÍÁËÕÓÇ» ðïõ äßäáîå ï êáèçãçôþò ôïõ ÐáíåðéóôÞìéïõ ÁèÞíáò ê. Âáóßëåéïò ÄïõãáëÞò óôá ðëáßóéá ôïõ Åðéìïñöùôéêïý ÐñïãñÜììáôïò ãéá ôïõò Åêðáéäåõôéêïýò- Ìáèçìáôéêïýò óôï Ìáèçìáôéêü ôìþìá ôïõ Ðáíåðéóôçìßïõ Áèçíþí êáôü ôçí ðåñßïäï Äåêåìâñßïõ 2000-Éïõíßïõ 200 ìå Õðåýèõíï ôïí êáèçãçôþ Ð. ÓôñÜíôæáëï. ÁÈÇÍÁ ÉÏÕÍÉÏÓ 200 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 2 -

v Ï ìéêñüôåñïò äõíáôüò öïñýáò ìéáò (ìç ôåôñéììýíçò) êõâéêþò spline áðïôåëåßôáé áðü ôýóóåñá äéáäï éêü õðïäéáóôþìáôá åíüò äéáìåñéóìïý, åßíáé äçëáäþ ôçò ìïñöþò [x j-2, x j+2 ]. Áðüäåéîç: óôù [x ï, x n ] o ìéêñüôåñïò äõíáôüò öïñýáò ìéáò (ìç ôåôñéììýíçò) êõâéêþò spline S [ x ï, x n]. ÅðåéäÞ Ýîù áðü áõôü ôï äéüóôçìá, áõôþ ìçäåíßæåôáé ôáõôïôéêü êé åðåéäþ S º C 2 [x ï, x n ], èá ðñýðåé ôüóï ïé ôéìýò ôçò æçôïýìåíçò spline óôá óçìåßá x ï êáé x n üóï êáé ïé ôéìýò ôçò ðñþôçò êáé äåýôåñçò ðáñáãþãïõ ôçò íá ìçäåíßæïíôáé. ïõìå ëïéðüí ãéá ôéò äéüöïñåò ðåñéðôþóåéò: ) Ôï äéüóôçìá [x ï, x n ] áðïôåëåßôáé áðü Ýíá õðïäéüóôçìá [x ï, x ]. Èá Ýðñåðå íá éó ýåé: S(x 0 )=0=S(x ) áëëü êáé S (x 0 )=0=S (x ), S (x 0 )=0=S (x ), ïðüôå ç S èá åß å ôç ìïñöþ: S(x)=á(x- x 0 ) 3 (x- x ) 3 êáé Ýîé ñßæåò, Üñá äåí èá Þôáí êõâéêü ðïëõþíõìï. 2) Ôï äéüóôçìá [x ï, x n ] íá ðåñéåß å äýï äéáäï éêü õðïäéáóôþìáôá, íá Þôáí äçëáäþ ôçò ìïñöþò [x ï, x 2 ]. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 3 -

Ãéá íá åßíáé ç æçôïýìåíç S êõâéêü ðïëõþíõìï óôï äéüóôçìá [x ï, x ], ëüãù ôùí ðñïûðïèýóåùí ðïõ ìíçìïíåýóáìå ðáñáðüíù, èá ðñýðåé íá Ý åé ôç ìïñöþ : S [ x ï, x ] (x)= á(x- x 0 ) 3 êáé ãéá íá åßíáé êõâéêü ðïëõþíõìï óôï äéüóôçìá [x, x 2 ], èá ðñýðåé íá Ý åé ôç ìïñöþ: S [ x, x 2](x)= â(x- x 2 ) 3 x ï x x 2 ÅðåéäÞ S º C 2 [x ï, x 2 ] èá ðñýðåé S [ x ï, x ](x )= S [ x, x 2](x ) á(x - x 0 ) 3 = â(x - x 2 ) 3 á = â (x - x 2 ) 3 (x - x 0 ) 3 () Åðßóçò èá ðñýðåé lims [ x ï, x ](x)=lim S [ x, x 2](x) x x - x x + ÄçëáäÞ 3á(x - x 0 ) 2 =3â(x - x 2 ) 2 á = â (x - x 2 ) 2 (2) (x - x 0 ) 2 ÅðåéäÞ x 0 x x 2 ïé ó Ýóåéò () êáé (2) äåí åßíáé ãåíéêü óõìâéâáóôýò ïðüôå äåí õðüñ åé ç æçôïýìåíç (ìç ôåôñéììýíç) êõâéêþ spline. Ïé () êáé (2) óõíáëçèåýïõí ãéá á=â=0, ôüôå üìùò Ý ïõìå ôåôñéììýíç ðåñßðôùóç. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 4 -

3) Ôï äéüóôçìá [x ï, x n ] íá ðåñéåß å ôñßá äéáäï éêü õðïäéáóôþìáôá, íá Þôáí äçëáäþ ôçò ìïñöþò [x ï, x 3 ]. Ãéá ôïõò ëüãïõò ðïõ áíáöýñáìå óôçí ðñïçãïýìåíç ðåñßðôùóç óôá áêñáßá. x ï x x 2 x 3 èá Ý åé áíôßóôïé á ôéò ìïñöýò: S [ x ï, x ](x)= á(x- x 0 ) 3 êáé S [ x 2, x 3](x)=â(x- x 3 ) 3. óôù üôé ç ìïñöþ ôçò óôï åíäéüìåóï äéüóôçìá [x,x 2 ], åßíáé ç: S [ x, x 2](x)=ãx 3 +äx 2 +åx+æ. õðïäéáóôþìáôá [x ï, x ] êáé [x 2, x 3 ], ç æ ç ô ï ý ì å í ç s p l i n e ÅðåéäÞ SºC 2 [x ï,x 3 ] ðñýðåé íá Ý ïõìå ïìáëþ ìåôüâáóç óôá óçìåßá x êáé x 2. ÐñÝðåé äçëáäþ S [ x ï, x ](x )= S [ x, x 2](x ) êáé S [ x 2, x 3](x 2 )= S [ x, x 2](x 2 ) äçëáäþ á(x - x 0 ) 3 = ã(x ) 3 +ä(x ) 2 +åx +æ â(x 2 - x 3 ) 3 = ã(x 2 ) 3 +ä(x 2 ) 2 +åx 2 +æ (3) áêüìá èá ðñýðåé êáèþò åðßóçò êáé lims [ x ï, x ](x)=lim S [ x, x 2](x) x x - x x + lims [ x 2, x 3](x)=lim S [ x, x 2](x) x x - x x + äçëáäþ: 3á(x - x 0 ) 2 = 3ã(x ) 2 +2äx +å 3â(x 2 - x 3 ) 2 = 3ã(x 2 ) 2 +2äx 2 +å (4) Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 5 -

ÅðåéäÞ x 0 x x 2 x 3, ïé (3) êáé (4) óõíáëçèåýïõí ìüíï üôáí á=â=ã=ä=å=æ=0 êáé åðïìýíùò ïýôå óå áõôþ ôçí ðåñßðôùóç õðüñ åé ìç ôåôñéììýíç êõâéêþ spline. 4) Ôï äéüóôçìá [x ï,x n ] íá ðåñéý åé ôýóóåñá äéáäï éêü õðïäéáóôþìáôá íá åßíáé äçëáäþ ôçò ìïñöþò [x ï,x 4 ] (óôç ãåíéêþ ðåñßðôùóç åßíáé ôçò ìïñöþò [x j,x j+2 ]). Óôçí ðåñßðôùóç áõôþ åßíáé äõíáôüí íá ðñïóäéïñßóïõìå 6 ðáñáìýôñïõò áðü 5 åîéóþóåéò ðïõ ðñïêýðôïõí áðü ôéò óõíèþêåò ðáñåìâïëþò óôïõò êüìâïõò x i, 0 i 4 êáèþò êáé ôéò óõíèþêåò óõíý åéáò ôçò S êáé ôçò S óôïõò åóùôåñéêïýò êüìâïõò x i, 0 i 3 êáé äýï åðéðëýïí óõíïñéáêýò óõíèþêåò óôïõò óõíïñéáêïýò êüìâïõò x 0 êáé x 4. Óôç ãåíéêþ ðåñßðôùóç ìåôü ôïí ðñïóäéïñéóìü ôùí 6 ðáñáìýôñùí áðü ôéò 5 åîéóþóåéò ðïõ ðñïêýðôïõí áðü ôéò óõíèþêåò ðáñåìâïëþò óôïõò êüìâïõò x i, 0 i n êáèþò êáé ôéò óõíèþêåò óõíý åéáò ôçò S êáé ôçò S óôïõò åóùôåñéêïýò êüìâïõò x i, 0 i n- êáé äýï åðéðëýïí óõíïñéáêýò óõíèþêåò óôïõò óõíïñéáêïýò êüìâïõò x 0 =a êáé x n =b, êáôáëþãïõìå óôïí õðïëïãéóìü ìéáò ìç ôåôñéììýíçò ôìçìáôéêü êõâéêþò óõíüñôçóçò ôïõ C 2 [á,b] (á=x 0,x n =b) ç ïðïßá ìçäåíßæåôáé ôáõôïôéêü Ýîù áðü ôï äéüóôçìá [x j-2, x j+2 ]. (¼ëá ôá óôáèåñü ðïëëáðëüóéü ôçò èá Ý ïõí ðñïöáíþò ôéò ßäéåò éäéüôçôåò.) Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 6 -

Óôçí ðåñßðôùóç ïìïéüìïñöïõ äéáìåñéóìïý Ä:= a= x 0 < x < < x n =b ôïõ [a,b] ìå âþìá h=(b-a)/n, ïé (ìç ôåôñéììýíåò) êõâéêýò spline ìå ôï ìéêñüôåñï äõíáôü öïñýá ïíïìüæïíôáé B-splines. Ãéá ïìïéüìïñöï ëïéðüí äéáìåñéóìü Ä h ìå h=x j+ -x j ç B-spline s j ìå êýíôñï x j (2 j n-2,) äßíåôáé áðü ôïí ôýðï: s j (x)= (x-x j-2 ) 3, x j-2 x x j- 6h 3 2 (x-x j ) 2 (x-x j ) 3, x j- x x j 3 h 2 2h 3 + x j+ -x + (x j+ -x) 2 _ (x j+ -x) 3, x j x x j+ 6 2h 2h 2 2h 3 ( x j+2 -x) 3, x j+ x x j+2 6h 3 0, x º [x j-2, x j+2 ] 2 3. s j (x) 6 6 x j-2 x j- x j x j+ x j+2 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 7 -

v Ïé óõíáñôþóåéò {ö j }, - j n+, ïé ïðïßåò ïñßæïíôáé áðü ôéò ó Ýóåéò, ö j := s j [a,b] := ðåñéïñéóìüò ôçò s j óôï [a,b], áðïôåëïýí âüóç ôïõ þñïõ S 3 (Ä h ) ôùí êõâéêþí splines ùò ðñïò ôïí ïìïéüìïñöï äéáìåñéóìü Ä:= a= x 0 < x < < x n =b ôïõ [a,b] ìå âþìá h=(ba)/n, üðïõ s j åßíáé ç B-spline ìå êýíôñï x j (2 j n-2,) ðïõ äßíåôáé áðü ôïí ôýðï: s j (x)= (x-x j-2 ) 3, x j-2 x x j- 6h 3 2 (x-x j ) 2 (x-x j ) 3 3 h 2 2h 3, x j- x x j + x j+ -x + (x j+ -x) 2 _ (x j+ -x) 3, x j x x j+ 6 2h 2h 2 2h 3 ( x j+2 -x) 3, x j+ x x j+2 6h 3 0, x º [x j-2, x j+2 ] Áðüäåéîç: Èåùñïýìå ôïõò åðéðëýïí êüìâïõò x - =a-h êáé x n+ =b+h, êáé ïñßæïõìå ôéò n+3 óõíáñôþóåéò ö -, ö 0, ö,, ö n-, ö n, ö n+ áðü ôéò ó Ýóåéò, ö j := s j [a,b] := ðåñéïñéóìüò ôçò s j óôï [a,b],. Ïé óõíáñôþóåéò ö -, ö 0, ö,, ö n-, ö n, ö n+, å ß í á é ã ñ á ì ì é ê Ü á í å î Ü ñ ô ç ô å ò ã é á ô ß ã é á C j, j=-,,n+ Ý ïõìå: Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 8 -

. Åñãáóßá óôçí..áñéèìçôéêç ÁÍÁËÕÓÇ n + Ó C j ö j = 0 j =- n + Ó C j ö j (x k ) = 0, j =- k=-,,n+ c - ö - (x ê )+ c 0 ö 0 (x ê )+ c ö (x ê )+ +c n ö n (x ê )+ c n+ ö n+ (x ê )=0, k=-,,n+ () ÄçëáäÞ áíôéêáèéóôþíôáò äéáäï éêü ôéò ôéìýò x -, x 0,,x n,x n+, óôçí () ðáßñíïõìå ôï ðáñáêüôù ïìïãåíýò ãñáììéêü óýóôçìá (n+3) x (n+3): c - ö - (x - )+ c 0 ö 0 (x - )+ c ö (x - )+ +c n ö n (x - )+ c n+ ö n+ (x - )=0 c - ö - (x 0 )+ c 0 ö 0 (x 0 )+ c ö (x 0 )+ +c n ö n (x 0 )+ c n+ ö n+ (x 0 )=0 c - ö - (x )+ c 0 ö 0 (x )+ c ö (x )+ +c n ö n (x )+ c n+ ö n+ (x )=0.. c - ö - (x n-)+ c 0 ö 0 (x n-)+ c ö (x n-)+ +c n ö n (x n-)+ c n+ ö n+ (x n-)=0 c - ö - (x n )+ c 0 ö 0 (x n )+ c ö (x n ) + +c n ö n (x n ) + c n+ ö n+ (x n ) =0 c - ö - (x n+ )+ c 0 ö 0 (x n+ )+ c ö (x n+ ) +c n ö n (x n+ )+ c n+ ö n+ (x n+ )=0 Áíôéêáèéóôþíôáò óôï óýóôçìá, ôéò ôéìýò ôùí ó õ í á ñ ô Þ ó å ù í ö -, ö 0,, ö n, ö n + ó ô á ó ç ì å ß á x -, x 0,,x n,x n+ Ý ïõìå: 2 3 c - + 6 c 0 =0 6 c - + 2 3 c 0 + 6 c =0 6 c 0 + 2 3 c + 6 c 2 =0.. 6 c n-2 + 2 3 c n- + 6 c n =0 6 c n- + 2 3 c n + 6 c n+ =0 6 c n + 2 3 c n+ =0 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 9 -

Ôï ïìïãåíýò áõôü óýóôçìá ðéíüêùí: Á C = Ï, üðïõ : ãñüöåôáé õðü ìïñöþ Á= 2 3 2 6 3 0 6 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 6 2 3 0 0 0 0 0 0 6 0 0 0 0 0 0 2 6 3 0 0 0 0 0 0 0 2 6 3 0 0 0 0 0 0 0 0 6 0 6 6 2 3 = 6 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0... 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 = 6 B Á, Â ôñé-äéáãþíéïé ðßíáêåò (n+3)x(n+3), c - c 0 c 0 0 0 C = c n- c n c n+ (n+3)x ðßíáêáò êáé Ï = 0 0 0 ï ìçäåíéêüò ðßíáêáò (n+3)x. ÅðåéäÞ Á = 6 Â èá åßíáé detá = ( 6 ) n+3. detâ(). Á ê ü ì á, ó ô ï í ð ß í á ê á Â = [ â i j ] ì å i, j =,, n + 3 ð ï ë ë á ð ë á ó é Ü æ ï í ô á ò ô ç í ç ó ô Þ ë ç ì å - 4 ( = - b ) ê á é ð ñ ï ó è Ý ô ï í ô á ò ó ô ç ä å ý ô å ñ ç ó ô Þ ë ç ê á é å ð á í á ë á ì â Ü í ï í ô á ò ô ç ä é á ä é ê á ó ß á á õ ô Þ ð ï ë ë á ð ë á ó é Ü æ ï í ô á ò ô ç í ê ó ô Þ ë ç ì å - b k k ê á é Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 0 -

ð ñ ï ó è Ý ô ï í ô á ò ó ô ç í ê + ó ô Þ ë ç ( ã é á ê = 2,, n + 2 ) ðñïêýðôåé ï ôñéãùíéêüò ðßíáêáò  =[â ij ] ìå äéáãþíéá óôïé åßá â = 4, â 22 = - b + 4, â 33 = - b 22 + 4,, â (ê+)(ê+) = - b k k + 4,, â (n+3)(n+3) = - bhn+2lhn+2l + 4. Áðü ôïí ôñüðï ðïõ ðñïýêõøå ï ðßíáêáò  =[â ij ] (i, j =,, n + 3 ) Ý ïõìå üôé detâ = detâ (2). éó ýåé: ÐñÜãìáôé: Ãéá ôá äéáãþíéá óôïé åßá ôïõ ðßíáêá  = [ â ij ] 3<â mm 4, m=,2,,n+3. Ñ (ãéá m=): 3<â 4 3<44 ðïõ éó ýåé Ñ ê (ãéá m=ê): 3<â êê 4 (3) Ýóôù üôé éó ýåé. Ñ ê+ (ãéá m=ê+): 3<â (ê+)(ê+) 4 èá äåßîù üôé éó ýåé: (3) 4 b kk < - < - 3 3 b kk - 4 3<- +4< - 3 b kk +4- +44 3<â (ê+)(ê+)4 4 ÅðïìÝíùò ç ó Ýóç 3< â mm 4, éó ýåé ãéá êüèå m üðïõ m=,2,,n+3 êáé èá Ý ïõìå: 3 < â 4 3 < â 22 4 3 < â 33 4 3<â (n+3)(n+3) 4 êáé ìå ðïëëáðëáóéáóìü êáôü ìýëç åßíáé: 3 n+3 < â. â 22. â 33.. â (n+3)(n+3) 4 n+3 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - -

ê á é å ð ï ì Ý í ù ò ã é á ô ï í ð ß í á ê á  = [ â i j ] å ß í á é : 3 n+3 <detâ 4 n+3 êáé áðü ôçí (2) èá Ý ïõìå 3 n+3 <detâ4 n+3 ïðüôå ( 6 ) n+3. 3 n+3 <( 6 ) n+3. detâ( 6 ) n+3. 4 n+3 êáé ôåëéêü áðü ôçí () ðñïêýðôåé üôé ( 2 )n+3 <detá( 2 3 )n+3 ñá åßíáé deta 0, êáé åðïìýíùò ôï ïìïãåíýò óýóôçìá Á C =Ï Ý åé ìïíáäéêþ ëýóç ôç ìçäåíéêþ äçëáäþ C j =0, ãéá êüèå j ìå j= -,,n+. Å ð é ð ë Ý ï í å ð å é ä Þ ö j º S 3 ( Ä ) ä ç ë á ä Þ ï é n+3 ãñáììéêü áíåîüñôçôåò óõíáñôþóåéò {ö j }, - j n+ áíþêïõí êáé óôï äéáíõóìáôéêü þñï S 3 ( Ä ) ð ï õ Ý å é äéüóôáóç (n+3) èá åßíáé ãéá êüèå t º S 3 (Ä h ) (ïðüôå êáé t º S 3 (Ä)) ö -, ö 0, ö,, ö n-, ö n, ö n+, t ãñáììéêü å î á ñ ô ç ì Ý í å ò. ÅðïìÝíùò èá õðüñ ïõí óôáèåñýò C i, i=-,,n+ü é üëåò ìçäýí þóôå n + íá Ý ïõìå t = Ó C i ö i. i =- Ïé óõíáñôþóåéò ëïéðüí {ö j }, - j n+ áðïôåëïýí âüóç ôïõ þñïõ S 3 (Ä h ) ôùí êõâéêþí splines ùò ðñïò ôïí ïìïéüìïñöï äéáìåñéóìü Ä h. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 2 -

v Ãéá k,nº èåùñþóôå ôç óõíüñôçóç ƒ, ƒ(x)cos(kðx), xº[0,], êáé ôïí ïìïéüìïñöï äéáìåñéóìü ôïõ [0,], ðïõ ïñßæåôáé áðü ôá óçìåßá x i ih, i=0,,2,,n, h=/n. a) óôù º n ôï ðïëõþíõìï ðáñåìâïëþò Lagrange ôçò ƒ ãéá ôá óçìåßá x i. Õðïëïãßóôå Ýíá Üíù öñüãìá ôïõ ìåãßóôïõ áðïëýôïõ óöüëìáôïò max 0 x ƒ(x) - (x) óõíáñôþóåé ôùí k êáé n êáé äéáôõðþóôå ó Ýóç ðïõ ðñýðåé íá ðëçñïýí ôá k êáé n, Ýôóé þóôå 0 êáèþò k êáé n. b) óôù s ç ãñáììéêþ spline óôï [0,] ðïõ ðáñåìâüëëåôáé óôéò ôéìýò ôçò ƒ óôá x i. Õðïëïãßóôå Ýíá Üíù öñüãìá ôïõ ìåãßóôïõ áðïëýôïõ óöüëìáôïò s max 0 x ƒ(x) - s(x) óõíáñôþóåé ôùí k êáé n êáé äéáôõðþóôå ó Ýóç ð ï õ ð ñ Ý ð å é í á ð ë ç ñ ï ý í ô á k ê á é n, Ý ô ó é þ ó ô å s 0 êáèþò k êáé n. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 3 -

Ãéá ôï a) Ëýóç: ÅðåéäÞ ç ƒ(x)cos(kðx), xº[0,], Ý åé ðáñáãþãïõò Ýùò ôçí n+ ôüîç (ƒºc n+ [0,]), êé åðåéäþ ï äéáìåñéóìüò ôïõ äéáóôþìáôïò [0,] åßíáé ïìïéüìïñöïò, óýìöùíá ìå ôçí èá éó ýåé: ƒ n h n+ 4Hn + Lƒ (n+) () max 0 x ƒ (x) (x) hn+ 4Hn + Lmax 0 x ƒ (n+) (x) (2) üìùò ƒ (n+) (x) = (kð) n+ sin(kðx), n=2ë ëº (kð) n+ cos(kðx), n=2ë+ êáé óå êüèå ðåñßðôùóç: max 0 x ƒ (n+) (x) =(kð) n+. ôóé ëïéðüí áðü ôç (2) Ý ïõìå: max 0 x ƒ (x) (x) hn+ 4Hn + L (kð)n+ = I nmn+.hkpln+ 4Hn + L= HkpLn+ 4Hn + Ln n+ = 4Hn + L.Jkp nnn+ ñá Ýíá Üíù öñüãìá ôïõ ìåãßóôïõ áðïëýôïõ óöüëìáôïò óõíáñôþóåé ôùí k êáé n åßíáé ç ðïóüôçôá : 4Hn + L.Jkp nnn+ (3) Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 4 -

Ãéá íá ôåßíåé ç ðïóüôçôá 4Hn + L.Jkp nnn+ óôï ìçäýí êáèþò k êáé n èá ðñýðåé n>kð. Ãéá ôï b) ÅðåéäÞ ƒ (x) =(kð) 2 cos(kðx), ôï öñüãìá ãéá ôï ìýãéóôï áðüëõôï óöüëìá èá åßíáé: s max 0 x ƒ (x) s (x) h2 8 max 0 x ƒ (x) = nl2 =H.HkpL2 =HkpL2 8 8 n = 8Jkp nn2 2 ñá Ýíá Üíù öñüãìá ôïõ ìåãßóôïõ áðïëýôïõ óöüëìáôïò s óõíáñôþóåé ôùí k êáé n åßíáé ç ðïóüôçôá: 8Jkp nn2, (4) Ãéá íá ôåßíåé ç ðïóüôçôá 8Jkp k êáé n èá ðñýðåé nk 2. nn2 óôï ìçäýí êáèþò Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 5 -

v Äßíåôáé ç óõíüñôçóç 0, 0 x, ƒ(x) (x-) 4, < x 2. a) Ðñïóåããßóôå ôçí ƒ óôï äéüóôçìá [0,2] ìå ìéá ôìçìáôéêü ðïëõùíõìéêþ óõíüñôçóç ôçò ìïñöþò 0, 0 x, (x) á + â(x-) + ã(x-) 2 + ä(x-) 3 < x 2. Ðñïóäéïñßóôå ôá á,â,ã,ä, õðïèýôïíôáò üôé ºC [0,2] êáé üôé (0)ƒ(0), (0)ƒ (0), ()ƒ(), (2)ƒ(2), (2)ƒ (2). b) Óõìðßðôåé ç óõíüñôçóç ìå ôçí ðáñåìâüëëïõóá êõâéêþ spline s ôçò ƒ óôá óçìåßá {0,,2} ìå óõíïñéáêýò óõíèþêåò s (0)ƒ (0), s (2)ƒ (2); Ëýóç: ÅðåéäÞ ºC [0,2], ç óõíüñôçóç èá åßíáé óõíå Þò óôï óçìåßï ïðüôå: áëëü Üñá : á = 0 PHxL PHxL lim lim PHxL x - PHxL lim lim x - = phl= 0 & = x + x + = a Åßíáé êáé åðïìýíùò (x) â+2ãx-2ã+3äx 2-6äx+3ä ( + ) â+2ã-2ã+3ä 2-6ä+3ä = â Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 6 -

Áðü ôçí Üëëç ìåñéü ( - ) 0 êáé åðåéäþ ºC [0,2] Ýðåôáé üôé ( + ) ( - ) Üñá : â = 0 Áêüìç (2) á + â(2-) + ã(2-) 2 + ä(2-) 3 =ã+ä êáé ƒ(2)=(2-) 4 = ïðüôå ã+ä= () Ãéá < x 2 åßíáé ƒ(x) (x-) 4 = x 4-4x 3 +6 x 2-4 x+ Ïðüôå ƒ (x) 4x 3-2 x 2 +2 x-4 åðïìýíùò ƒ (2)= 42 3-2 2 2 +2 2-4= 4 Åðßóçò ãéá < x 2 åßíáé (x)= 2ãx-2ã+3äx 2-6äx+3ä êáé åðïìýíùò (2)=2ã2-2ã+3ä2 2-6ä2+3ä =4ã-2ã+2ä-2ä+3ä = 2ã+3ä Á ð ü (2)ƒ (2) Ý ï õ ì å 2 ã + 3 ä = 4 ( 2 ) () ã+ä = -2ã-2ä= -2 (2) 2ã+3ä=4 2ã+3ä = 4 ã+ä = ã+2= ã= - ä=2 ä=2 ä= 2 ñá á=0, â=0, ã=- êáé ä=2 ïé ôéìýò ôùí ðáñáìýôñùí. ôóé ç æçôïýìåíç ðïëõùíõìéêþ óõíüñôçóç åßíáé ç: Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 7 -

0, 0 x, (x) -(x-) 2 + 2(x-) 3, < x 2. Þ áëëéþò: (x) 0, 0 x, 3x 3-7x 2 + 8x-3, < x 2. Ãéá ôï åñþôçìá b) ðáñáôçñïýìå üôé: Ç ðïëõùíõìéêþ óõíüñôçóç éêáíïðïéåß ôéò ßäéåò óõíïñéáêýò óõíèþêåò ìå ôçí ðáñåìâüëëïõóá êõâéêþ spline s ôçò ƒ áëëü ºC [0,2] êáé üðùò äéáðéóôþíïõìå äåí Ý åé óõíå Þ äåýôåñç ðáñüãùãï óôï óçìåßï ãéáôß: ãéá 0 x < (x)0 ïðüôå ( - )0 åíþ ãéá < x 2 (x)2x-4 ïðüôå ( + )2-4=-2 äçëáäþ ( - ) ( + ). ñá ç ðïëõùíõìéêþ óõíüñôçóç äåí óõìðßðôåé ìå ôçí ðáñåìâüëëïõóá êõâéêþ spline s ôçò ƒ óôá óçìåßá {0,,2}. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 8 -

v óôù ƒºc 2 [a,b] êáé s ç ðáñåìâüëëïõóá êõâéêþ spline ðïõ ðáñåìâüëëåôáé óôéò ôéìýò ôçò ƒ óôá óçìåßá x i åíüò äéáìåñéóìïý a= x 0 < x < < x n =b ôïõ [a,b] êáé ðïõ éêáíïðïéåß ôéò óõíïñéáêýò óõíèþêåò s (x 0 )=ƒ (x 0 ), s (x n )=ƒ (x n ). Äåßîôå üôé àabhf "HxḺS"HxLS "HxLâx=0, êáé óõìðåñüíåôå üôé àa bjs "HxL2 âx àa bhf "HxLN2 âx. Áðüäåéîç: n- x bhf "HxL- S"HxL.S "HxLâx i+hf "HxL- S "HxL. S"HxLâ x àa =âi=0 àxi (êáé ïëïêëçñþíïíôáò êáôü ðáñüãïíôåò ) n- Hf 'HxL- S'HxL.S"HxLx i+ n- àxi x i+hf 'HxL- S'HxL.S n '''HxLâ x =âi=0 x i -âi=0 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 9 -

ÅðåéäÞ ïé óõíáñôþóåéò f ', S' êáé S " åßíáé óõíå åßò Ý ïõìå: n- Hf 'HxL- S'HxL.S"HxLx i+ =@f 'HbL- S'HbLD.S"HbL-@f 'HaL- S'HaLD.S"HaL=0 x i âi=0 ÁëëÜ êáé ôï äåýôåñï Üèñïéóìá åßíáé ìçäýí ìéá êáé ç S n '''HxLåßíáé óôáèåñþ áöïý seé 3 ( S n '''HxL=C m ) êáé: x i+hf 'HxL- S'HxLâ x=@fhx i+l- SHx àx i+ld-@fhx il- SHx ild= i 0-0= 0 Åßíáé: (ƒ - s ) 2 = ƒ 2-2 ƒ s +s 2 ƒ 2 =(ƒ -s ) 2 +2ƒ s -s 2 = (ƒ -s ) 2 +2ƒ s -2s 2 + s 2 ƒ 2 =(ƒ -s ) 2 +2(ƒ -s )s +s 2 ñá : àabjf "HxL2 âx= àa bis"hxl2 bhf "HxL- S"HxL2 bhf "HxL- S"HxL. S"HxLâ x âx= âx+2àa +àa àa b@f "HxL- S"HxLD2 â x +àa bbs "HxLD2 b@s"hxlf2 âx ³ â x àa. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 20 -

v óôù ƒ(x)x 5 +x 4, {-2,-,0,,2}Ýíáò äéáìåñéóìüò ôïõ äéáóôþìáôïò [-2,2]. Ðñïóäéïñßóôå, ãéá ôïí äéáìåñéóìü áõôüí, ôç öõóéêþ êõâéêþ ðáñåìâüëëåôáé óôçí ƒ óôá óçìåßá -2,-,0,,2. Ãíùñßæïõìå üôé: Ëýóç: s(x)= 6h [ s i " (x-x i- ) 3 - s " i- (x-x i ) 3 ]+( y i h - s i " " h -( y i- h - s i- " êáé s i- +4 " si + s i+ spline, ç ïðïßá h 6 )(x- x i-) 6 )( x- x i), x e [ x i-, x i ]. () " 6 = h 2 ( y i- -2 y i+ y i+),i=,,n- (2) ÅðïìÝíùò ãéá ôçí æçôïýìåíç öõóéêþ êõâéêþ s p l i n e è á Ý ï õ ì å ë ü ã ù ô ç ò ( ) ã é á ô á ô Ý ó ó å ñ á õðïäéáóôþìáôá: Ãéá x e [ fh-l -2, -] s(x) = 6 [s (-) 2L (x-(-2))3 -s (-2) (x-(-)) 3 ]+ +( - s (-) 6 ) (x -(-2)) - ( fh- - s (-2) 6 ) (x-(-)), Ãéá x e [ -, 0] s(x) = 6 [s (0) L (x-(-))3 -s (-) (x-0) 3 ]+ - s (0) 6 ) (x -(-)) - ( fh- - s (-) 6 ) (x-0), +( fh0l Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 2 -

Ãéá x e [ fhl 0, ] s(x) = 6 [s () fh0l (x-0)3 -s (0) (x-) 3 ]+ +( - s () ) (x -0) - ( - s (0) 6 6 ) (x-), fh2l êáé ãéá x e [, 2] s(x) = fhl 6 [s (2) (x-)3 -s () (x-2) 3 ]+ +( - s (2) ) (x -) - ( - s () 6 6 ) (x-2). üðïõ ôá s (-2), s (-), s (0), s () êáé s (2) éêáíïðïéïýí óýìöùíá ìå ôçí (2), ôéò ó Ýóåéò: s (-2) + 4 s (-) + s (0) = 6 2 [ ƒ(-2) -2 ƒ(-)+ ƒ(0)] s (-) + 4 s (0) + s () = 6 2 [ ƒ(-) -2 ƒ(0) + ƒ()] (3) s (0) + 4 s () + s (2) = 6 2 [ƒ(0) -2 ƒ() + ƒ(2)] ÅðåéäÞ: ƒ(-2) = (-2) 5 +(-2) 4 =-32+6=-6 ƒ(-) = (-) 5 +(-) 4 =-+=0 ƒ(0) = 0 5 +0 4 =0 ƒ() = 5 + 4 =2 Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 22 -

ƒ(2) = 2 5 +2 4 =32+6=48, Ôï óýóôçìá (3) ãñüöåôáé: s (-2)+ 4 s (-) + s (0) = -96 s (-) +4 s (0) + s () = 2 s (0) + 4 s () + s (2) = 264 ¼ìùò åðåéäþ ðñüêåéôáé ãéá ôç öõóéêþ êõâéêþ spline åßíáé s (-2) = 0 = s (2) êáé Ýôóé ôï óýóôçìá ãñüöåôáé: 4 s (-) + s (0) = -96 s (-) +4 s (0) + s () = 2 s (0) + 4 s () = 264 èýôù: s (-)=á, s (0)=â, s ()=ã êáé ôï óýóôçìá ãñüöåôáé: 4 á + â = -96 á +4 â + ã = 2 (4) ïõìå: 4 0 D= 4 0 4 â + 4 ã = 264-96 0 = 56, D á = +2 4 264 0 =-224. Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 23 -

4-96 0 4-96 D â = +2 =-480 êáé D ã = 4 +2 =386 0 264 4 0 264 Ïðüôå á= D a D = - 224 56 = - 53 7 â= D b D = - 480 56 = - 60 7 ã= D g D = 386 56 = 477 7 ÅðïìÝíùò: s (-)= - 53 7, s (0)= - 60 7, s ()= 477 7 ôóé ãéá ôïí äéáìåñéóìü áõôü èá Ý ïõìå: Ãéá x e [ -2, -] s(x) = - 53 [( 6 7 )(x-(-2))3 -s (-2) (x-(-)) 3 ]+.(0-( - 53 7 ) ) (x -(-2))- 6 (-6 0. - 5 ) (x-(-)) = 6 4 [(x+2)3 -(x+2)] +6(x+), Ãéá x e [ -, 0] s(x) = 6 [ - 60 7 (x-(-))3 -( - 53 7 - (0- ( - 53 7 ) 6 ) (x-0)= - )(x-0) 3 ]+(0- ( - 60 7 ) ) (x -(-)) - 6 4 [20(x+3)3-5x 3-20(x+)+5x], Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 24 -

Ãéá x e [ 0, ] s(x) = 6 [ 477 7 (x-0)3 - ( - 60 7 ) (x-)3 ] + (2-477 7 ) (x 0) 6 -(0 - ( - 60 7 ) 6 ) (x-)= 4 [59x3 +20(x-) 3-3x-20(x-)] Êáé ãéá x e [, 2] s(x) = 6 [0.(x-)3-477 7 (x-2)3 ]+(48-0. 477 ) (x -) (2-6 7 = - 4 [59(x-2)3-672(x-)-3(x-2)] 6 ) (x-2) ñá ï ôýðïò ôçò öõóéêþò êõâéêþò spline, ç ïðïßá ðáñåìâüëëåôáé óôçí ƒ óôá óçìåßá -2,-,0,,2 èá åßíáé: S(x)= - 4 [5(x+2)3-5(x+2) 224(x+)], ãéá x e [-2,-] 4 [-20(x+3)3 +5x 3 +20(x+)-5x], ãéá x e [-, 0] 4 [59x3 +20(x-) 3-3x-20(x-)], ãéá x e [ 0, ] - 4 [59(x-2)3-672(x-)-3(x-2)], ãéá x e [, 2] Ã.Ðáíüðïõëïò Éïýíéïò 200 óåëßäá - 25 -