ÁÑÉÈÌÇÔÉÊÇ ÏËÏÊËÇÑÙÓÇ

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1 ÌÜèçìá 7 ÁÑÉÈÌÇÔÉÊÇ ÏËÏÊËÇÑÙÓÇ ÅéóáãùãÞ ¼ìïéá, üðùò êáé óôï ÌÜèçìá ÐñïóÝããéóç Ðáñáãþãùí, ç ðñïóåããéóôéêþ ôéìþ ôïõ ïñéóìýíïõ ïëïêëçñþìáôïò ñçóéìïðïéåßôáé êõñßùò, üôáí I(f) = f(x) dx i) ëüãù ôçò ðïëýðëïêçò ìïñöþò ôïõ ôýðïõ ìéáò óõíüñôçóçò åßíáé áäýíáôïò ï èåùñçôéêüò õðïëïãéóìüò ôïõ, êáé ii) äåí åßíáé ãíùóôüò ï ôýðïò ôçò óõíüñôçóçò, áëëü ìüíïí ïé ôéìýò ôçò óå ïñéóìýíá óçìåßá. Ïé ðñïóåããßóåéò ðïõ èá åîåôáóôïýí óôï ìüèçìá áõôü âáóßæïíôáé óôïí ôýðï ðáñåìâïëþò ôïõ Newton. Óýìöùíá ìå ôïí áñéèìü êáé ôïí ôñüðï ðïõ 7

2 8 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò óõíäõüæïíôáé ôá óçìåßá ðáñåìâïëþò ðñïêýðôïõí ïé ìýèïäïé õðïëïãéóìïý Þ üðùò óõíþèùò ëýãïíôáé ïé êáíüíåò ïëïêëþñùóçò (qudrture rules) Áðëïß êáíüíåò ïëïêëþñùóçò ÁíÜëïãá ìå ôïí èåùñïýìåíï áñéèìü ôùí óçìåßùí ðáñåìâïëþò Ý ïõìå ôïõò ðáñáêüôù êáíüíåò: Êáíüíáò ôïõ ïñèïãùíßïõ ÅéóáãùãéêÝò Ýííïéåò óôù ôï ïñéóìýíï ïëïêëþñùìá I(f) = f(x) dx; ( ) üðïõ ç f(x) èåùñåßôáé üôé åßíáé ìßá óõíå Þò óõíüñôçóç óôï [; ] Þ ãåíéêüôåñá óôçí ðåñßðôùóç ðïõ äåí åßíáé ãíùóôüò ï ôýðïò ôçò üôé åßíáé ãíùóôýò ïé ôéìýò ôçò óôá n + 1 äéáöïñåôéêü óçìåßá x, x 1, : : :, x n ôïõ [; ]. Ôüôå, üðùò åßíáé Þäç ãíùóôü, éó ýåé ï ðáñáêüôù ôýðïò ðáñåìâïëþò ôïõ Newton: f(x) P n (x) = f [x ] + f [x ; x 1 ] (x x ) + : : : ( ) +f [x ; x 1 ; : : : ; x n ] (x x ) (x x n 1 ) : ÕðïèÝôïíôáò üôé f(x) > ãéá êüèå x [; ], ôï ïëïêëþñùìá (7:1:1 1) ãåùìåôñéêü éóïýôáé ìå ôï åìâáäüí ôïõ ó Þìáôïò ðïõ ïñßæåôáé áðü ôïí x- Üîïíá, ôéò åõèåßåò x =, x = êáé ôï äéüãñáììá ôçò y = f(x) (Ó ). Óôïí êáíüíá ðïõ áêïëïõèåß ñçóéìïðïéåßôáé ìüíïí Ýíá óçìåßï ðáñåìâïëþò. óôù üôé Ý ïõìå ôï 1 ÂëÝðå âéâëéïãñáößá [1,, ] êáé: https : ==en:wikipedi:org=wiki=numericl integrtion

3 Áðëïß êáíüíåò ïëïêëþñùóçò 9 fx x Ó Þìá : ÃåùìåôñéêÞ åñìçíåßá ïñéóìýíïõ ïëïêëçñþìáôïò f(x) dx. óçìåßï ðáñåìâïëþò : x Ôüôå áðü ôçí (7:1:1 ) ðñïêýðôåé ôüôå üôé ïðüôå I(f) = f(x) P (x) = f [x ] = f (x ) ; f(x) dx f (x ) dx = ( )f (x ) ( ) ðïõ åßíáé ãíùóôüò ùò ï êáíüíáò ôïõ ïñèïãùíßïõ (rectngle rule). ÁíÜëïãá ìå ôéò èýóåéò ôïõ óçìåßïõ x äéáêñßíïõìå ôþñá ôéò åîþò ðåñéðôþóåéò: áí x = (Ó ), áíôßóôïé á x = (Ó ), ôüôå áðü ôçí (7:1:1 ) Ý ïõìå I(f) = f(x) dx ( )f(); ( ) ÅðåéäÞ õðüñ åé Ýíá óçìåßï ðáñåìâïëþò ðñýðåé n + 1 = + 1, ïðüôå n = êáé åðïìýíùò ôï ðïëõþíõìï ðáñåìâïëþò èá åßíáé ìçäåíéêïý âáèìïý - âëýðå ÌÜèçìá: ÐïëõùíõìéêÞ ÐáñåìâïëÞ - Èåþñçìá ôïõ Lgrnge.

4 4 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò áíôßóôïé á I(f) = f(x) dx ( )f(): ( ) fx 4 fx 4 y y x (i) 1 4 x (ii) Ó Þìá : (i) ÐñïóÝããéóç (7:1:1 4) üðïõ x =, y = f() (ðñüóéíç åðéöüíåéá) êáé (ii) ðñïóýããéóç (7:1:1 5) üðïõ x =, y = f(). Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x). óôù ôþñá üôé x = ( + )= (Ó ). Ôüôå ç (7:1:1 ) ãñüöåôáé I(f) = f(x) dx ( )f ( ) + ( ) ðïõ åßíáé ãíùóôüò ùò ï êáíüíáò ôïõ ìýóïõ óçìåßïõ (midpoint rule). ÐáñáôÞñçóç Ïé ôýðïé (7:1:1 4) - (7:1:1 5) åßíáé áðü ôïõò ðñþôïõò ðïõ ñçóéìïðïéþèçêáí óôçí ðñïóýããéóç ôïõ ïëïêëçñþìáôïò (7:1:1 1). ïõí éóôïñéêü åíäéáöýñïí, áëëü ëüãù ôçò ìéêñþò áêñßâåéáò äåí ñçóéìïðïéïýíôáé óôéò åöáñìïãýò.

5 Áðëïß êáíüíåò ïëïêëþñùóçò 41 fx 4 fx 4 y c 1 y c 1 1 c 4 x () 1 c 4 x () Ó Þìá : () ÃåùìåôñéêÞ åñìçíåßá ïñéóìýíïõ ïëïêëçñþìáôïò f(x) dx. () Êáíüíáò ôïõ ìýóïõ óçìåßïõ: ôýðïò (7:1:1 6) üðïõ c = x = + êáé y c = f(c) (êüêêéíç åõèåßá). Ç ðñüóéíç åðéöüíåéá ðñïóåããßæåé ôï ïëïêëþñùìá (7:1:1 1). ÐáñÜäåéãìá Æçôåßôáé íá õðïëïãéóôåß ìå ôïõò ðáñáðüíù êáíüíåò ôï ïëïêëþñùìá (Ó ) I = 1: dx ; ( ) 1 + x üôáí ç èåùñçôéêþ ôéìþ åßíáé ( I = ln x + ) x 1: : Ëýóç. óôù f(x) = 1= 1 + x. Ôüôå äéáäï éêü Ý ïõìå: Ôýðïò (7:1:1 4), üðïõ f() = f() = 1 êáé = 1: = 1: (Ó ) I 1: 1 = 1. ìå áðüëõôï óöüëìá e = 1: 1:15 97 = :184 7: Ôýðïò (7:1:1 5), üðïõ f() = f(1:) = :64 184, üìïéá = 1: (Ó ) I 1: : = ìå áðüëõôï óöüëìá e = : :15 97 = :47 75:

6 4 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx 1. fx 1. D C x (). A x B () Ó Þìá : () Ç ãåùìåôñéêþ åñìçíåßá ôïõ ïñéóìýíïõ ïëïêëçñþìáôïò (7:1:1 7) êáé () ôï ïñèïãþíéï ABCD ðïõ ôï ðñïóåããßæåé óýìöùíá ìå ôïí ôýðï (7:1:1 4), üôáí = AB = 1: êáé f() = AD = 1:, üðïõ ôï AB óõìâïëßæåé ôï ìýôñï ôïõ AB. fx 1..8 D.6 C fx 1. D.8.6 C.4.4. A x B. A. B c x Ó Þìá : () ôï ïñèïãþíéï ABCD ðïõ ðñïóåããßæåé ôï ïëïêëþñùìá (7:1:1 7) óýìöùíá ìå ôïí ôýðï (7:1:1 5), üôáí = AB = 1:, f() = AD = : êáé () ìå ôïí ôýðï (7:1:1 6), üôáí üìïéá = AB = 1: êáé f(c) = ( ) + f = AD = : Ôýðïò (7:1:1 6), üðïõ ( ) + f = f(:6) = : ìå = 1: (Ó ) I 1: : = ìå áðüëõôï óöüëìá e = 1:8 99 1:15 97 = :1 19:

7 7.1. Êáíüíáò ôïõ ôñáðåæßïõ óôù üôé Ý ïõìå ôá Áðëïß êáíüíåò ïëïêëþñùóçò 4 óçìåßá ðáñåìâïëþò : x, x 1 Ôüôå, åðåéäþ ôá óçìåßá åßíáé ôï ðïëõþíõìï ðáñåìâïëþò P (x) èá åßíáé 1ïõ âáèìïý, äçëáäþ óýìöùíá ìå ôçí (7:1:1 ) f(x) P 1 (x) = f (x ) + f [x ; x 1 ] (x x ) ; äçëáäþ ðñüêåéôáé ãéá åõèåßá ãñáììþ. ÈÝôïíôáò (Ó ) x = ; x 1 = êáé h = ; áðü ôçí (7:1:1 1) ðñïêýðôåé üôé I(f) P 1 (x) dx = {f [x ] + f [x ; x 1 ] (x x )} dx = f (x ) dx + f [x ; x 1 ] (x x ) dx = h {f() + f()}: ( ) Ï ôýðïò (7:1: 1) åßíáé ãíùóôüò ùò ï êáíüíáò ôïõ ôñáðåæßïõ (trpezoidl rule). ÐáñÜäåéãìá Æçôåßôáé íá õðïëïãéóôåß ìå ôïí êáíüíá ôïõ ôñáðåæßïõ ôï ïëïêëþñùìá (7:1:1 7) ôïõ Ðáñáäåßãìáôïò Ëýóç. ¼ìïéá åßíáé f(x) = 1= 1 + x, åíþ ç èåùñçôéêþ ôéìþ I

8 44 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx 4 fx 4 y y y 1 y x () 1 4 x () Ó Þìá : Áðëüò êáíüíáò ôïõ ôñáðåæßïõ ôýðïò (7:1: 1): () Óçìåßá ðáñåìâïëþò: x =, x 1 = ìå ôéìýò y = f(), y = f() áíôßóôïé á. Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x), ç êüêêéíç ôïõ ðïëõùíýìïõ P 1 (x) êáé ç êßôñéíç åðéöüíåéá ôï åìâáäüí I(f). () Ç ðñüóéíç åðéöüíåéá ïñßæåé ôï ôñáðýæéï ðïõ ðñïóåããßæåé ôï ïëïêëþñùìá I(f). fx fx D C x (). A x B () Ó Þìá : () Ç ãåùìåôñéêþ åñìçíåßá ôïõ ïñéóìýíïõ ïëïêëçñþìáôïò (7:1:1 7) êáé () ôï ôñáðýæéï ABCD ðïõ ôï ðñïóåããßæåé óýìöùíá ìå ôïí ôýðï (7:1: 1). Ôüôå óýìöùíá ìå ôïí ôýðï (7:1: 1) Ý ïõìå (Ó ): AB = = h = 1:; f() = AD = 1 êáé f() = BC = :64 184; ïðüôå É 1: (1: + :64 184) = ìå óöüëìá e = 1:15 97 : = :1 86:

9 7.1. Êáíüíáò ôïõ Simpson ÁíÜëïãá, Ýóôù üôé Ý ïõìå ôá Áðëïß êáíüíåò ïëïêëþñùóçò 45 óçìåßá ðáñåìâïëþò : x, x 1, x Ôüôå, åðåéäþ ôá óçìåßá åßíáé, ôï ðïëõþíõìï ðáñåìâïëþò P (x) èá åßíáé ïõ âáèìïý (ðáñáâïëþ), äçëáäþ f(x) P (x) = f [x ] + f [x ; x 1 ] (x x ) ÈÝôïíôáò (Ó ) +f [x ; x 1 ; x ] (x x ) (x x 1 ) : x = ; x 1 = + ; x = ìå h = êáé áíôéêáèéóôþíôáò óôçí (7:1:1 1) ôåëéêü ìå ðáñüìïéïõò õðïëïãéóìïýò åêåßíùí ôçò ÐáñáãñÜöïõ 7.1. Ý ïõìå ôïí ðáñáêüôù êáíüíá ïëïêëþñùóçò: I(f) P (x) dx = 6 { f() + 4f ( + ) } + f() = h {f (x ) + 4f (x 1 ) + f (x )} ( ) ðïõ åßíáé ãíùóôüò ùò ï êáíüíáò ôïõ Simpson (Simpson's rule). ÐáñÜäåéãìá Íá õðïëïãéóôåß ìå ôïí êáíüíá ôïõ Simpson ôï ïëïêëþñùìá (7:1:1 7) ôïõ Ðáñáäåßãìáôïò ÁêñéâÝóôåñá êáíüíáò ôïõ Simpson ìå ïõ âáèìïý ðïëõþíõìï ðáñåìâïëþò (Simpson's rule with qudrtic interpolting polynomil).

10 46 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx 4 fx 4 y y y y c 1 y y c 1 1 c 4 x () 1 c 4 x () Ó Þìá : Áðëüò êáíüíáò ôïõ Simpson ôýðïò (7:1: 1): () Óçìåßá ðáñåìâïëþò: x =, x 1 = c = +, x = ìå áíôßóôïé åò ôéìýò y = f(), y c = f(c) êáé y = f(). Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x) êáé ç êüêêéíç ôïõ ðïëõùíýìïõ P (x), åíþ ç êßôñéíç åðéöüíåéá ôï åìâáäüí I(f). () Ç ðñüóéíç åðéöüíåéá ðñïóåããßæåé ôï ïëïêëþñùìá I(f). Ëýóç. ¼ìïéá åßíáé f(x) = 1= 1 + x, åíþ ç èåùñçôéêþ ôéìþ I Ôüôå óýìöùíá ìå ôïí ôýðï (7:1: 1) Ý ïõìå (Ó ): h = = 1: = :6; f() = f (x ) = AD = 1; f(c) = ( ) + f = f (x 1 ) = :857 49; êáé f() = f (x ) = BC = :64 184; ïðüôå É :6 (1: + 4 : :64 184) = ìå óöüëìá e = 1:14 1 1:15 97 = :1 94:

11 Áðëïß êáíüíåò ïëïêëþñùóçò 47 fx x () fx 1. D G.8 C.6.4. A B x. c () Ó Þìá : () Ôï ïëïêëþñùìá (7:1:1 7) êáé () ç åðéöüíåéá ABCGD ðïõ ôï ðñïóåããßæåé óýìöùíá ìå ôïí ôýðï (7:1: 1) Êáíüíáò ôùí /8 óôù üôé Ý ïõìå ôá óçìåßá ðáñåìâïëþò : x, x 1, x, x Ôüôå åðåéäþ ôá óçìåßá åßíáé 4, ôï ðïëõþíõìï ðáñåìâïëþò P (x) èá åßíáé ïõ âáèìïý, äçëáäþ f(x) P (x) = f (x ) + f [x ; x 1 ] (x x ) +f [x ; x 1 ; x ] (x x ) (x x 1 ) +f [x ; x 1 ; x ; x ] (x x ) (x x 1 ) (x x ) : Ìå õðïëïãéóìïýò üìïéïõò åêåßíùí ôçò ÐáñáãñÜöïõ 7.1. èåùñþíôáò üôé ôï äéüóôçìá [; ] õðïäéáéñåßôáé óå éóáðý ïíôá äéáóôþìáôá áðü ôá óçìåßá (Ó ) x = ; x 1 = x + + x = ìå h = ; x = x + ( + ) áðïäåéêíýåôáé üôé ï ôýðïò õðïëïãéóìïý ôïõ ïëïêëçñþìáôïò (7:1:1 1) ôåëéêü ;

12 48 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò y fx 4 y fx 4 y 1 y c d x () 1 4 c d x () Ó Þìá : Áðëüò êáíüíáò ôùí =8 ôïõ Simpson ìå ôýðï (7:1:4 1). () Óçìåßá ðáñåìâïëþò x =, x 1 = c = +, x = d = (+), x = ìå ôéìýò óôá Üêñá y = f() êáé y = f(). Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x) êáé ç êüêêéíç ôïõ ðïëõùíýìïõ P (x), åíþ ç êßôñéíç åðéöüíåéá ôï åìâáäüí I(f). () Ç ðñüóéíç åðéöüíåéá ðñïóåããßæåé ôï ïëïêëþñùìá I(f). ãñüöåôáé ùò åîþò: I(f) P (x) dx = h 8 {f (x ) + f (x 1 ) + f (x ) + f (x )} : ( ) Ï ôýðïò (7:1:4 1) åßíáé ãíùóôüò ùò ï êáíüíáò ôùí =8 ôïõ Simpson (Simpson's =8 rule). 4 ÐáñÜäåéãìá ¼ìïéá ìå ôïí êáíüíá ôùí =8 ôïõ Simpson ôï ïëïêëþñùìá (7:1:1 7) ôïõ Ðáñáäåßãìáôïò Ëýóç. Åßíáé f(x) = 1= 1 + x êáé ç èåùñçôéêþ ôéìþ I , ïðüôå 4 Åðßóçò åßíáé ãíùóôüò êáé ùò êáíüíáò ôïõ Simpson ìå ïõ âáèìïý ðïëõþíõìï ðáñåìâïëþò (Simpson's rule with cuic interpolting polynomil).

13 Áðëïß êáíüíåò ïëïêëþñùóçò 49 fx x () fx D A. H B..4 c.6.8 d x G C () Ó Þìá : () Ôï ïëïêëþñùìá (7:1:1 7) êáé () ç åðéöüíåéá ABCGHD ðïõ ôï ðñïóåããßæåé óýìöùíá ìå ôïí ôýðï (7:1:4 1). óýìöùíá ìå ôïí ôýðï (7:1:4 1) Ý ïõìå (Ó ): h = = 1: = :4; f() = f (x ) = AD = 1; f(c) = ( ) + f = f (x 1 ) = :98 477; f(d) = ( ) ( + ) f = f (x ) = :78 869; êáé f() = f (x ) = BC = :64 184; ïðüôå É :4 8 (1: + : : :64 184) = 1.15 ìå óöüëìá e = 1:15 1:14 1 = : 74: Óôïí Ðßíáêá äßíïíôáé óõãêåíôñùôéêü ïé ðáñáðüíù õðïëïãéóèåßóåò ðñïóåããßóåéò ôïõ ïëïêëçñþìáôïò (7:1:1 7) ôïõ Ðáñáäåßãìáôïò êáé ôá áíôßóôïé á óöüëìáôá óå ó Ýóç ìå ôç èåùñçôéêþ ôéìþ I 1: ìåóá ðñïêýðôåé üôé ï êáíüíáò ôùí =8 ôïõ Simpson äßíåé ôá áêñéâýóôåñá áðïôåëýóìáôá. ¼ëïé ïé ðáñáðüíù êáíüíåò ïëïêëþñùóçò åßíáé ãíùóôïß åðßóçò êáé ùò êáíüíåò ïëïêëþñùóçò ôùí Newton-Cotes (Newton-Cotes rules) êáé áñáêôç-

14 5 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò Ðßíáêáò : Áðëïß êáíüíåò ïëïêëþñùóçò: õðïëïãéóìüò ôïõ ïëïêëçñþìáôïò (7:1:1 7) ôïõ Ðáñáäåßãìáôïò ìå èåùñçôéêþ ôéìþ I Êáíüíáò ÐñïóÝããéóç I ÓöÜëìá õðïëïãéóìïý ÌÝóïõ óçìåßïõ 1:8 99 :1 19 Ôñáðåæßïõ : :1 86 Simpson 1:14 1 :1 94 Simpson =8 1:15 : 74 ñéóôéêü ôïõò åßíáé ç ðñïóýããéóç ôïõ ïëïêëçñþìáôïò (7:1:1 1) ìå éóáðý ïíôá óçìåßá. Ïé áíôßóôïé ïé ôýðïé ëýãïíôáé ôüôå êáé ôýðïé ïëïêëþñùóçò ôùí Newton-Cotes (Newton-Cotes formuls) Óýíèåôïé êáíüíåò Ç áêñßâåéá ôùí ôýðùí óôïõò áðëïýò êáíüíåò ïëïêëþñùóçò åßíáé ðåñéïñéóìýíç, êõñßùò üôáí ôï äéüóôçìá ïëïêëþñùóçò åßíáé ìåãüëï. íáò ôñüðïò ãéá íá Ý ïõìå êáëýôåñç áêñßâåéá åßíáé íá áõîçèåß ï áñéèìüò ôùí óçìåßùí ðáñåìâïëþò, ðïõ üìùò, üðùò åßíáé öõóéêü, èá äõóêïëýøåé ðåñéóóüôåñï ôïí õðïëïãéóìü ôïõ ôýðïõ. íáò Üëëïò ôñüðïò ôüôå, ðïõ ðáñáêüìðôåé ôéò äõóêïëßåò áõôýò, åßíáé íá õðïäéáéñåèåß êáôüëëçëá ôï äéüóôçìá ïëïêëþñùóçò óå åðéìýñïõò õðïäéáóôþìáôá êáé íá åöáñìïóôåß Ýíáò áðü ôïõò ðáñáðüíù êáíüíåò óå êáèýíá áðü ôá õðïäéáóôþìáôá áõôü. Ç ðáñáðüíù äéáäéêáóßá, üôáí ãåíéêåõôåß, ïäçãåß óôïõò ëåãüìåíïõò óýíèåôïõò êáíüíåò ïëïêëþñùóçò (composite qudrture rules). Óôç óõíý åéá äßíïíôáé ïé êõñéüôåñïé óýíèåôïé êáíüíåò ïëïêëþñùóçò, ðïõ êõñßùò óõíáíôþíôáé óôéò äéüöïñåò åöáñìïãýò: 5 Ï áíáãíþóôçò ãéá ìéá åêôåíýóôåñç ìåëýôç ðáñáðýìðåôáé óôç âéâëéïãñáößá [1,, ].

15 Óýíèåôïò êáíüíáò ôïõ ôñáðåæßïõ Óýíèåôïò êáíüíáò ôïõ ôñáðåæßïõ óôù ôï ïëïêëþñùìá (Ó ) f(x) dx; üðïõ ôï äéüóôçìá ïëïêëþñùóçò [; ] õðïäéáéñåßôáé óå N ôï ðëþèïò õðïäéáóôþìáôá ðëüôïõò (Ó ) h = N áðü ôá N + 1 óçìåßá x i = + ih; i = ; 1; : : : ; N: Ôüôå óýìöùíá ìå ãíùóôþ éäéüôçôá ôùí ïñéóìýíùí ïëïêëçñùìüôùí 6 êáé ôïí ôýðï (7:1:1 1) Ý ïõìå I(f) = =x N f(x) dx = =x x 1 x f(x) dx + x x 1 f(x) dx + : : : + x N x N 1 f(x) dx h {f (x ) + f (x 1 )} + h {f (x 1) + f (x )} + : : : + h {f (x N 1) + f (x N )} ; äçëáäþ 6 I(f) h {f (x ) + [f (x 1 ) + : : : + f (x N 1 )] + f (x N )} (7..1-1) Èåþñçìá óôù üôé ç óõíüñôçóç f åßíáé ïëïêëçñþóéìç óôï [á; â ]. Ôüôå, áí ã åßíáé Ýíá óçìåßï ìå á < ã < â, éó ýåé üôé â á f(x) dx = ã á f(x) dx + â ã f(x) dx:

16 5 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx x Ó Þìá : ÃåùìåôñéêÞ åñìçíåßá ïñéóìýíïõ ïëïêëçñþìáôïò Áëãüñéèìïò (óýíèåôïõ êáíüíá ôïõ ôñáðåæßïõ) ÄåäïìÝíá ; ; N; h = ( )=N óôù S = f () + f () ; S 1 = : Ãéá i = 1; ; : : : ; N 1 x = x + ih; S 1 := S 1 + f(x) ôýëïò i I = h (S + S1) f(x) dx. ðïõ åßíáé ãíùóôüò ùò ï óýíèåôïò êáíüíáò ôïõ ôñáðåæßïõ (composite trpezoidl rule). Ï õðïëïãéóìüò äßíåôáé óôïí Áëãüñéèìï Óýíèåôïò êáíüíáò ôïõ Simpson 7Õðïäéáéñþíôáò ôï äéüóôçìá ïëïêëþñùóçò [; ] óå N ôï ðëþèïò õðïäéáóôþìáôá ðëüôïõò h = ( )=N áðü ôá N + 1 óçìåßá x, x 1, : : :, x N, üìïéá 7 ÅðåéäÞ ï áðëüò êáíüíáò ôïõ Simpson áðáéôåß ãéá ôçí åöáñìïãþ ôïõ óçìåßá, äçëáäþ õðïäéáóôþìáôá, ç õðïäéáßñåóç ôïõ äéáóôþìáôïò ïëïêëþñùóçò [; ] ðñýðåé íá ãßíåé óå Üñôéï áñéèìü õðïäéáóôçìüôùí.

17 Óýíèåôïò êáíüíáò ôïõ Simpson 5 fx x Ó Þìá : Óýíèåôïò êáíüíáò ôïõ ôñáðåæßïõ. Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x). óýìöùíá ìå ôç ãíùóôþ éäéüôçôá ôùí ïñéóìýíùí ïëïêëçñùìüôùí êáé ôïí ôýðï (7:1:1 1) Ý ïõìå I(f) = =x N f(x) dx =x = x x f(x) dx + x 4 x f(x) dx + : : : + x N x N f(x) dx h {f (x ) + 4f (x 1 ) + f (x )} + h {f (x ) + 4f (x ) + f (x 4 )} + : : : + h {f (x N ) + 4f (x N 1 ) + f (x N )} ;

18 54 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx x Ó Þìá : óýíèåôïò êáíüíáò ôïõ Simpson. Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x) äçëáäþ I(f) h {f (x ) + 4 [f (x 1 ) + f (x ) + : : : + f (x N 1 )] + [f (x ) + f (x 4 ) + : : : + f (x N )] +f (x N )} ( ) ðïõ åßíáé ãíùóôüò ùò óýíèåôïò êáíüíáò ôïõ Simpson (composite Simpson's rule). Ï õðïëïãéóìüò äßíåôáé óôïí Áëãüñéèìï ÐáñÜäåéãìá Æçôåßôáé íá õðïëïãéóôåß ìå ôïí óýíèåôï êáíüíá ôïõ ôñáðåæßïõ, áíôßóôïé á ôïõ Simpson ôï ïëïêëþñùìá (7:1:1 7) ôïõ Ðáñáäåßãìáôïò , äçëáäþ ôï I = 1: Ç èåùñçôéêþ ôéìþ åßíáé I = ln dx ; üôáí h = :1: 1 + x ( x + ) x 1:

19 Óýíèåôïò êáíüíáò ôïõ Simpson 55 Áëãüñéèìïò (óýíèåôïõ êáíüíá ôïõ Simpson) ÄåäïìÝíá ; êáé n = N Üñôéïò óôù h = ( )=n; S = f () + f () ; S 1 = S = : Ãéá i = 1; ; : : : ; n 1 x = x + ih áí i Üñôéïò S = S + f(x); äéáöïñåôéêü S 1 = S 1 + f(x) ôýëïò i I = h (S + 4S 1 + S ) Ëýóç. Áí = x = êáé = x 1 = 1: áñ éêü äçìéïõñãåßôáé ï Ðßíáêáò ôùí ôéìþí (x i ; f (x i )). Ôüôå ï ôýðïò: (7::1 1) ãéá ôï óýíèåôï êáíüíá ôïõ ôñáðåæßïõ äßíåé I h {f (x ) + [f (x 1 ) + : : : + f (x 11 )] + f (x 1 )} = ; áíôßóôïé á ï (7:: 1) ãéá ôïí óýíèåôï êáíüíá ôïõ Simpson I h {f (x ) + 4 [f (x 1 ) + f (x ) + f (x 5 ) + f (x 7 ) + f (x 9 ) + f (x 11 )] + [f (x ) + f (x 4 ) + f (x 6 ) + f (x 8 ) + f (x 1 )] + f (x 1 )} = : Óõãêñßíïíôáò ôéò ðáñáðüíù ôéìýò ìå ôéò áíôßóôïé åò ôïõ Ðßíáêá åðáëçèåýåôáé áõôü ðïõ Ý åé Þäç ãñáöåß óôçí åéóáãùãþ ôçò ÐáñáãñÜöïõ 7., äçëáäþ üôé ïé óýíèåôïé êáíüíåò ïëïêëþñùóçò ïäçãïýí óå áýîçóç ôçò áêñßâåéáò ðñïóýããéóçò ôïõ ïëïêëçñþìáôïò f(x)dx. Óôïí Ðßíáêá ðáñáôßèåíôáé ïé ôéìýò ôùí óöáëìüôùí ôïõ ïëïêëçñþìáôïò I ãéá äéüöïñåò ôéìýò ôïõ h. Áðü ôá áðïôåëýóìáôá ôïõ Ðßíáêá ðñïêýðôåé üôé ï óýíèåôïò êáíüíáò ôïõ Simpson ãéá ôéìýò ôïõ h ìå h :1, ïðüôå ôï N åßíáé áñêåôü ìåãüëï, äßíåé áêñéâýóôåñá áðïôåëýóìáôá áðü ôïí

20 56 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò Ðßíáêáò : ÐáñÜäåéãìá : ïé ôéìýò (x i ; f (x i )). x i f (x i ) x i f (x i ) x = :.1 x 7 = : x 1 = : x 8 = : x = : x 9 = : x = : x 1 = 1: x 4 = : x 11 = 1: x 5 = : x 1 = 1: x 6 = : Ðßíáêáò : ÐáñÜäåéãìá : ôá óöüëìáôá ôçò ïëïêëþñùóçò ôïõ I ãéá ôéò äéüöïñåò ôéìýò ôïõ h (õðåíèõìßæåôáé üôé :6E 4 = :6 1 4 = :6 ê.ëð.). h ÓöÜëìá ôñáðåæßïõ ÓöÜëìá Simpson.1.6E-4 9.6E E-5 1.9E-9.1.6E E E E E E-1

21 Óýíèåôïò êáíüíáò ôùí /8 57 áíôßóôïé ï ôïõ ôñáðåæßïõ, åíþ, üôáí h = :1, ïðüôå Ý ïõìå ìåãáëýôåñç áýîçóç ôïõ N, áêñéâýóôåñá áðïôåëýóìáôá ðñïêýðôïõí áðü ôïí êáíüíá ôïõ ôñáðåæßïõ. ÈåùñçôéêÜ åßíáé áíáìåíüìåíï, üôáí ç ëåðôüôçôá ôçò äéáìýñéóçò ôåßíåé óôï ìçäýí Þ äéáöïñåôéêü üôáí ôï h ôåßíåé óôï ìçäýí, ç áñéèìçôéêþ ôéìþ ôïõ ïëïêëçñþìáôïò íá ôåßíåé óôç èåùñçôéêþ ôéìþ ôïõ. ÐïëëÝò öïñýò üìùò, üðùò êáé ðáñáðüíù, óõìâáßíåé ôï h íá åëáôôþíåôáé, ùñßò íá Ý ïõìå êáé áíôßóôïé ç ìåßùóç ôïõ óöüëìáôïò. Áõôü êýñéá ïöåßëåôáé áöåíüò ìåí óôï óöüëìá ðïõ ðáñïõóéüæåé ï êáíüíáò ïëïêëþñùóçò ðïõ ñçóéìïðïéåßôáé êáé áöåôýñïõ óôá óöüëìáôá óôñïããõëïðïßçóçò ðïõ õðåéóýñ ïíôáé óôïõò äéüöïñïõò õðïëïãéóìïýò. 7.. Óýíèåôïò êáíüíáò ôùí /8 8Õðïäéáéñþíôáò ôï äéüóôçìá ïëïêëþñùóçò óå N ôï ðëþèïò õðïäéáóôþìáôá ðëüôïõò h = ( )=N áðü ôá N + 1 óçìåßá x, x 1, : : :, x N üìïéá ìå ôïõò ðáñáðüíù óýíèåôïõò êáíüíåò ôïõ ôñáðåæßïõ êáé ôïõ Simpson Ý ïõìå I(f) = =x N f(x) dx = =x x x f(x) dx + x 6 x f(x) dx + : : : + x N x N f(x) dx h 8 {f (x ) + f (x 1 ) + f (x ) + f (x )} + h 8 {f (x ) + f (x 4 ) + f (x 5 ) + f (x 6 )} + : : : + h 8 {f (x N ) + f (x N ) + f (x N 1 ) + f (x N )} ; 8 ÅðåéäÞ ï áðëüò êáíüíáò ôùí =8 áðáéôåß ãéá ôçí åöáñìïãþ ôïõ 4 óçìåßá, äçëáäþ õðïäéáóôþìáôá, ç õðïäéáßñåóç ôïõ äéáóôþìáôïò ïëïêëþñùóçò [; ] ðñýðåé íá ãßíåé óå áñéèìü õðïäéáóôçìüôùí, ðïõ íá åßíáé ðïëëáðëüóéïò ôïõ.

22 58 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx x Ó Þìá : óýíèåôïò êáíüíáò ôùí =8 ôïõ Simpson. Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x). ïðüôå I(f) h 8 {f (x ) + [f (x 1 ) + f (x 4 ) + : : : + f (x N )] + [f (x ) + f (x 5 ) + : : : + f (x N 1 )] + [f (x ) + f (x 6 ) + : : : + f (x N )] +f (x N )} ( ) ðïõ åßíáé ãíùóôüò ùò óýíèåôïò êáíüíáò ïëïêëþñùóçò ôùí =8 ôïõ Simpson (composite =8 Simpson's rule). ÐáñÜäåéãìá óôù üôé æçôåßôáé íá õðïëïãéóôåß ìå ôïí óýíèåôï êáíüíá ôùí /8 ôï ïëïêëþñùìá I ôïõ Ðáñáäåßãìáôïò Ôüôå óýìöùíá ìå ôïí ôýðï (7:: 1),

23 Óýíèåôïò êáíüíáò ôùí /8 59 üôáí h = :1, Ý ïõìå I h 8 {f (x ) + [f (x 1 ) + f (x 4 ) + f (x 7 ) + f (x 1 )] + [f (x ) + f (x 5 ) + f (x 8 ) + f (x 11 )] + [f (x ) + f (x 6 ) + f (x 9 )] +f (x 1 )} = : ñá ç ðñïóåããéóôéêþ ôéìþ óõìðßðôåé ìå ôçí áíôßóôïé ç èåùñçôéêþ óôá 6 äåêáäéêü øçößá. ÁóêÞóåéò 1. óôù üôé ôï äéüóôçìá ïëïêëþñùóçò [; ] õðïäéáéñåßôáé óå N ôï ðëþèïò õðïäéáóôþìáôá ðëüôïõò (Ó ) h = N áðü ôá N + 1 óçìåßá x i = + ih; i = ; 1; : : : ; N: Åöáñìüæïíôáò ôïí ôýðï (7:1:1 6) ôïõ áðëïý êáíüíá ôïõ ìýóïõ óçìåßïõ óå êüèå Ýíá õðïäéüóôçìá, õðïëïãßóôå ôïí áíôßóôïé ï ôýðï ôïõ óýíèåôïõ êáíüíá. Óôç óõíý åéá åöáñìüóôå ôïí ôýðï áõôü óôïí õðïëïãéóìü ôïõ ïëïêëçñþìáôïò ôïõ Ðáñáäåßãìáôïò êáé óõãêñßíáôå ôá áðïôåëýóìáôá ìå ôá áíôßóôïé á ôùí Üëëùí ìåèüäùí.. Íá õðïëïãéóôåß ìå ôïõò ðáñáðüíù óýíèåôïõò êáíüíåò ôï ïëïêëþñùìá I = 1: e x dx; üôáí h = :1 êáé íá ãßíåé óýãêñéóç ôùí áðïôåëåóìüôùí ìå ôç èåùñçôéêþ ôéìþ I : Åßíáé ãíùóôü üôé 1 dx 1 + x = 4 : Íá õðïëïãéóôåß ôï ïëïêëþñùìá ìå ôïí óýíèåôï êáíüíá ôïõ ôñáðåæßïõ êáé ôïõ Simpson üôáí ôï h = :1; :1 êáé íá ãßíåé óýãêñéóç ôùí áðïôåëåóìüôùí ìå

24 6 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò fx x Ó Þìá : óýíèåôïò êáíüíáò ôïõ ìýóïõ óçìåßïõ. Ç ìðëå êáìðýëç äåß íåé ôï äéüãñáììá ôçò f(x). ôç èåùñçôéêþ ôéìþ. 4. Ìå ôïí óýíèåôï êáíüíá ôïõ ôñáðåæßïõ íá õðïëïãéóôåß ôï ïëïêëþñùìá 1 xe x dx; üôáí h = :1 êáé íá ãßíåé óýãêñéóç ôïõ áðïôåëýóìáôïò ìå ôç èåùñçôéêþ ôéìþ ôïõ ïëïêëçñþìáôïò. 5. ¼ìïéá ìå ôïí óýíèåôï êáíüíá ôïõ Simpson êáé ôùí /8 ôï ïëïêëþñùìá :6 ( I = 1 x ) = dx; üôáí h = :1, :5 êáé íá ãßíåé óýãêñéóç ôùí áðïôåëåóìüôùí ìå ôç èåùñçôéêþ ôéìþ ôïõ ïëïêëçñþìáôïò I

25 ÅéóáãùãéêÝò Ýííïéåò Êáíüíåò ïëïêëþñùóçò ôïõ Guss 7..1 ÅéóáãùãéêÝò Ýííïéåò ¼ëïé ïé óýíèåôïé êáíüíåò ïëïêëþñùóçò ôçò ÐáñáãñÜöïõ 7., ðïõ åßíáé ãíùóôïß åðßóçò êáé ùò óýíèåôïé êáíüíåò ïëïêëþñùóçò ôùí Newton-Cotes, åß áí ðñïêýøåé áðü ôï ðïëõþíõìï ðáñåìâïëþò ôïõ Newton èåùñþíôáò êüèå öïñü Ýíáí áñéèìü éóáðå üíôùí óçìåßùí. Ïé ôýðïé õðïëïãéóìïý ôùí óýíèåôùí êáíüíùí ðïõ åîåôüóôçêáí ôåëéêü ëáìâüíïõí ôç ìïñöþ: Ôñáðåæßïõ I(f) h f (x ) + h f (x 1 ) + : : : + h f (x N 1 ) + h f (x N) = w f (x ) + w 1 f (x 1 ) + : : : + w N 1 f (x N 1 ) + w N f (x N ) : Simpson I(f) h f (x ) + 4h f (x 1) + 4h f (x ) + : : : + 4h f (x N 1) + h f (x ) + h f (x 4) + : : : + h f (x N ) + h f (x N) = h f (x ) + 4h f (x 1) + h f (x ) + : : : + h f (x N ) + 4h f (x N 1) + h f (x N) = w f (x ) + w 1 f (x 1 ) + w f (x ) + : : : + w N f (x N ) +w N 1 f (x N 1 ) + w N f (x N ) :

26 6 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò =8 ôïõ Simpson I(f) h 8 f (x ) + 9h 8 f (x 1) + : : : + 9h 8 f (x N ) + 9h 8 f (x ) + : : : + 9h 8 f (x N 1) + 6h 8 f (x ) + : : : + 6h 8 f (x N ) + h 8 f (x N) = h 8 f (x ) + 9h 8 f (x 1) + 9h 8 f (x ) + 6h 8 f (x ) + : : : + 6h 8 f (x N ) + 9h 8 f (x N ) + 9h 8 f (x N 1) + h 8 f (x N) = w f (x ) + w 1 f (x 1 ) + w f (x ) + w f (x ) + : : : +w N f (x N ) + w N f (x N ) + w N 1 f (x N 1 ) +w N f (x N ) : ÅðïìÝíùò åßíáé äõíáôü íá ãñáöïýí ôåëéêü - ðáñáëåßðïíôáò ùñßò âëüâç ôçò ãåíéêüôçôáò êáé ãéá ëüãïõò åõêïëßáò ôïí üñï ìå äåßêôç - óôç ìïñöþ: I(f) = f(x)dx w 1 f (x 1 ) + w f (x ) + : : : + w n f (x n ) ; (7..1-1) üðïõ ïé óõíôåëåóôýò ôùí ôéìþí f (x i ); i = 1; ; : : : ; n Þ üðùò óõíþèùò ëýãïíôáé âüñç (weights) w i ; i = 1; ; : : : ; n äåí åîáñôþíôáé áðü ôç óõíüñôçóç f(x). Óôïõò êáíüíåò ïëïêëþñùóçò ðïõ èá åîåôáóôïýí óôç óõíý åéá ôïõ ìáèþìáôïò êáé ðïõ åßíáé ãíùóôïß ùò êáíüíåò ïëïêëþñùóçò ôïõ Guss (Gussin qudrture), ç ïëïêëçñùôýá óõíüñôçóç áíôéêáèßóôáôáé ìå Üëëá ðñïóåããéóôéêü ðïëõþíõìá, ïðüôå äçìéïõñãïýíôáé Üëëïé ôýðïé áñéèìçôéêþò ïëïêëþñùóçò ìå óõíôåëåóôýò êáôü êáíüíá ìç ñçôïýò áñéèìïýò. åé áðïäåé èåß ðåéñáìáôéêü üôé ïé ôýðïé ðïõ ðñïêýðôïõí áðü ôïõò êáíüíåò áõôïýò Ý ïõí ôï ðëåïíýêôçìá ìå êáôüëëçëç åêëïãþ ôùí óçìåßùí x i ; i = 1; ; : : : ; n íá äßíïõí ìåãáëýôåñç

27 ÅéóáãùãéêÝò Ýííïéåò 6 áêñßâåéá áõôþò ðïõ äßíåôáé áðü ôïõò ôýðïõò ôùí Newton-Cotes ìå ôïí ßäéï áñéèìü óçìåßùí. Ôá óçìåßá ðáñåìâïëþò óôéò ðåñéðôþóåéò áõôýò ãåíéêü äåí éóáðý ïõí. óôù ôï ïëïêëþñùìá I(f) = f(x)dx: Áðïäåéêíýåôáé óôá ÌáèçìáôéêÜ üôé ç ïëïêëçñùôýá óõíüñôçóç åßíáé äõíáôüí íá ãñáöåß óôç ìïñöþ f(x) = w(x) g(x); ( ) üôáí w(x) åßíáé ìßá ìç áñíçôéêþ ïëïêëçñþóéìç óõíüñôçóç óôï [; ] ðïõ ëýãåôáé óõíüñôçóç âüñïõò. Ôüôå ç (7::1 ) óýìöùíá ìå ôçí (7::1 1) ãñüöåôáé I(f) = w(x) g(x) dx ( ) w 1 g (x 1 ) + w g (x ) + : : : + w n g (x n ) ; üðïõ ôá âüñç w i ; i = 1; ; : : : ; n åîáñôþíôáé áðü ôá óçìåßá x i ; i = 1; ; : : : ; n êáé ôç óõíüñôçóç w(x) áëëü ü é áðü ôçí g(x). 9 ÅðåéäÞ ç ìåëýôç ôçò (7::1 ) óôç ãåíéêþ ðåñßðôùóç îåöåýãåé ôïõ óêïðïý ôïõ ìáèþìáôïò, åîåôüæåôáé ìüíïí ç ðåñßðôùóç üðïõ ç óõíüñôçóç g(x) åßíáé Ýíá ðïëõþíõìï âáèìïý Ýóôù m, äçëáäþ g(x) = P m (x) = + 1 x + : : : + m x m ; (7..1-4) üôáí i R ãéá êüèå i = ; 1; : : : ; m. Óôïí ôýðï (7::1 ) æçôåßôáé íá ðñïóäéïñéóôïýí ôá w i êáé ôá óçìåßá x i ; i = 1; ; : : : ; n, Ýôóé þóôå ôï óöüëìá ôçò ðñïóýããéóçò íá åßíáé åëü éóôï. 9 Ç áñßèìçóç ôùí óçìåßùí óôçí ðåñßðôùóç áõôþ ãßíåôáé áðü ôïí äåßêôç 1, óå áíôßèåóç ìå ôùí ôýðùí Newton-Cotes ðïõ ñçóéìïðïéïýí ðïëõþíõìá ðáñåìâïëþò êáé ãßíåôáé áðü ôïí äåßêôç.

28 64 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò Ç (7::1 ) äéáäï éêü ãñüöåôáé I(f) = w(x) [ + : : : + m x m ] dx = w 1 g (x 1 ) + w g (x ) + : : : + w n g (x n ) ; (7..1-5) üðïõ ðñïöáíþò g (x i ) = + 1 x i + : : : + m x m i ãéá êüèå i = 1; ; : : : ; n. Áðü ôçí (7::1 5) ðñïêýðôåé ôüôå üôé I(f) = w(x) dx + 1 x w(x) dx + : : : + m x m w(x) dx = w 1 ( + 1 x 1 + : : : + m x m 1 ) + w ( + 1 x + : : : + m x m ) + : : : + w n ( + 1 x n + : : : + m x m n ) Þ w(x) dx + 1 x w(x) dx + : : : + m x m w(x) dx = (w 1 + w + : : : + w n ) + 1 (w 1 x 1 + w x + : : : + w n x n ) + : : : + m (w 1 x m 1 + w x m + : : : + w nx m n ) : ÅðåéäÞ ç ôåëåõôáßá éóüôçôá ðñýðåé íá éó ýåé ãéá êüèå ðïëõþíõìï ôçò ìïñöþò (7::1 4), áðü ôçí åîßóùóç ôùí óõíôåëåóôþí i ; i = ; 1; : : : ; m Ý ïõìå w 1 + w + : : : + w n = w 1 x 1 + w x + : : : + w n x n = w 1 x 1 + w x + : : : + w n x n =. w 1 x m 1 + w x m + : : : + w n x m n = w(x) dx x w(x) dx x w(x) dx. x m w(x) dx: (7..1-6)

29 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 65 Ç (7::1 6) ïñßæåé Ýíá óýóôçìá m + 1 åîéóþóåùí ìå n áãíþóôïõò ôá âüñç w 1 ; w ; : : :, w n, êáé n áãíþóôïõò ôá óçìåßá x 1 ; x ; : : : ; x n. Ôï óýóôçìá áõôü èá ðñýðåé íá Ý åé ëýóç ãéá êüèå óõíüñôçóç w(x), ðïõ, üðùò áðïäåéêíýåôáé óôá ÌáèçìáôéêÜ, áðáéôåßôáé íá éó ýåé üôé m + 1 n, äçëáäþ m n 1: (7..1-7) Ôüôå, áí m = n 1, ôï óýóôçìá (7::1 7) Ý åé ðüíôïôå ëýóç ðïõ üìùò åßíáé áñêåôü ðïëýðëïêç áêüìá êáé üôáí ôï ðëþèïò ôùí óçìåßùí n åßíáé ðïëý ìéêñü. Áðü ôï óýíïëï ôùí ëýóåùí ôïõ ðáñáðüíù óõóôþìáôïò èá åîåôáóôåß óôç óõíý åéá ìüíï ìéá ðåñßðôùóç: 7.. Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre Óôçí ðåñßðôùóç áõôþ ç ëýóç ôïõ óõóôþìáôïò (7::1 6) õðïëïãßæåôáé, üôáí ç óõíüñôçóç âüñïõò åßíáé w(x) = 1 êáé ôï äéüóôçìá ïëïêëþñùóçò [; ] = [ 1; 1]. Ôüôå ôï óýóôçìá ãñüöåôáé w 1 + : : : + w n = w 1 x 1 + : : : + w n x n = w 1 x 1 + : : : + w n x n =. w 1 x m 1 + : : : + w n x m n = = dx = x dx = x dx = 1 1. x m dx ; áí m ðåñéôôüò, 1 ; 1 + m áí m Üñôéïò. ( )

30 66 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò Áðü ôç èåùñßá ôùí ïñèïãùíßùí ðïëõùíýìùí ðñïêýðôåé üôé ôá óçìåßá x i ; i = 1; ; : : : ; n ðïõ åðáëçèåýïõí ôï óýóôçìá (7:: ) åßíáé ïé ñßæåò ôùí ðïëõùíýìùí P n (x) ôïõ Legendre âáèìïý n, ðïõ äßíïíôáé áðü ôïí ôýðï 1 P n (x) = [( x 1 ) n] (n) n n! (7.. - ) ãéá êüèå n = 1; ; : : :, üôáí x [ 1; 1], åíþ åßíáé P (x) = 1. Ìå åöáñìïãþ ôïõ ôýðïõ (7:: ) ðñïêýðôåé ôüôå üôé P 1 (x) = x P (x) = 1 ( 5x ) x P (x) = 1 ( x 1 ) P 4 (x) = 1 ( 5x 4 x + ) 8 ê.ëð. Ïé ñßæåò x i ; i = 1; ; : : : ; n êáé ôá áíôßóôïé á âüñç w i, üðïõ w i = ( ) 1 x i [P ; i = 1; ; : : : ; n; n (x i )] ãéá êüèå ðåñßðôùóç äßíïíôáé áðü ðßíáêåò, åíþ ãéá ôçí ðåñßðôùóç üðïõ i = 1; ; : : : ; 6 áðü ôïí Ðßíáêá óôç óõíý åéá ôïõ ìáèþìáôïò. Áðïäåéêíýåôáé üôé: i) ôá âüñç w i åßíáé ðüíôïôå èåôéêïß áñéèìïß, ii) ïé ñßæåò x i åßíáé ðñáãìáôéêýò êáé áíü äýï óõììåôñéêýò ùò ðñïò ôï ìçäýí, iii) ôá âüñç w i, ðïõ áíôéóôïé ïýí óå óõììåôñéêýò ñßæåò, åßíáé ßóá ìåôáîý ôïõò. Áí ôï äéüóôçìá ïëïêëþñùóçò åßíáé äéüöïñï ôïõ [ 1; 1], ôüôå ìå ôïí ìåôáó çìáôéóìü 11 x = ( )t + + (7.. - ) 1 Ï ôýðïò (7:: ) åßíáé ãíùóôüò ùò ôýðïò ôïõ Rodrigues (Rodrigues formul). 11 Áí x =, ôüôå åýêïëá ðñïêýðôåé áðü ôçí (7:: ) üôé t = 1, åíþ, áí x = üôé t = 1.

31 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 67 ôï ïëïêëþñùìá ìå äéüóôçìá ïëïêëþñùóçò [; ] ìåôáó çìáôßæåôáé óå áíôßóôïé ï ìå Üêñá [ 1; 1], äçëáäþ I(f) = f(x) dx = ÅðïìÝíùò óôçí ðåñßðôùóç áõôþ 1 1 f [ ( )x + + ] dx: ( ) I(f) = f(x) dx n i=1 w i { f [ ( ) xi + + ] } üôáí = n w i g (x i ) ; ( ) i=1 g (x i ) = f [ ( ) xi + + ] ( ) üðïõ óôï äåîéü ìýëïò ñçóéìïðïéþèçêå ãéá ëüãïõò åõêïëßáò ôï x i áíôß ôïõ t i. Ç ðáñáðüíù ìýèïäïò ïëïêëþñùóçò åßíáé ãíùóôþ ùò ïëïêëþñùóç ôùí Guss-Legendre (Guss-Legendre qudrture), åíþ ï (7:: 5) ùò ï ôýðïò ïëïêëþñùóçò ôùí Guss-Legendre. 1 Óçìåßùóç Ç ìýèïäïò ïëïêëþñùóçò ôùí Guss-Legendre i) äßíåé ôç ìåãáëýôåñç áêñßâåéá áðü êüèå Üëëç ìýèïäï ðïõ ñçóéìïðïéåß ôïí ßäéï áñéèìü óçìåßùí, 1 ÂëÝðå âéâëéïãñáößá [1,, ] êáé: https : ==en:wikipedi:org=wiki=gussin qudrture

32 68 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò Ðßíáêáò : ñéæþí ðïëõùíýìùí ôùí P n ; n = 1; ; : : : ; 6 êáé ôùí áíôßóôïé ùí âáñþí. P n x i w i P 1 x 1 = w 1 = : P x 1 = x = :57757 w 1 = w = 1: P x 1 = x = : w 1 = w = : x = w = : P 4 x 1 = x 4 = : w 1 = w 4 = : x = x = :99814 w = w = : P 5 x 1 = x 5 = : w 1 = w 5 = :69689 x = x 4 = : w = w 4 = : x = w = : P 6 x 1 = x 6 = : w 1 = w 6 = : x = x 5 = :66199 w = w 5 = : x = x 4 = : w = w 4 = : ii) äåí åßíáé äõíáôüí íá ñçóéìïðïéçèåß ãéá ôéò ðåñéðôþóåéò åêåßíåò ðïõ ç óõíüñôçóç äßíåôáé ìå ôéò ôéìýò ôçò óå ïñéóìýíá óçìåßá x i, ôá ïðïßá äåí óõìðßðôïõí ìå ôéò ñßæåò ôùí ðïëõùíýìùí Legendre, åßíáé üìùò ç êáëýôåñç ãéá ôéò ðåñéðôþóåéò ðïõ ç óõíüñôçóç f(x) äßíåôáé ìå ôïí áíáëõôéêü ôçò ôýðï - ðåñßðôùóç (i).

33 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 69 Ðßíáêáò : ÐáñÜäåéãìá x i w i g (x i ) = w i e x i x 1 = :86114 w 1 = :4785 w 1 e x 1 = :4785 e ( :86114) = :1657 x = :998 w = :6514 w e x = :6514 e ( :998) = :589 x = :998 w = :6514 w e x = :6514 e (:998) = :589 x 4 = :86114 w 4 = :4785 w 1 e x 1 = :4785 e (:86114) = :1657 ÐáñÜäåéãìá Ìå ôïí ôýðï ôùí Guss-Legendre ãéá 4 óçìåßá íá õðïëïãéóôåß ôï ïëïêëþñùìá (Ó ) 1 1 e x dx: Ëýóç. ÅðåéäÞ ç ïëïêëþñùóç ãßíåôáé óôï äéüóôçìá [ 1; 1], äåí áðáéôåßôáé íá ãßíåé ï ìåôáó çìáôéóìüò (7:: ). Ôüôå Ý ïõìå 1 1 e x dx 4 g (x i ) = w 1 e x 1 + w e x + w e x + w4 e x 4; i=1 üôáí ïé ñßæåò x i êáé ôá âüñç w i ; i = 1; ; ; 4 äßíïíôáé áðü ôïí Ðßíáêá Áðü ôá áðïôåëýóìáôá ôïõ Ðßíáêá óýìöùíá êáé ìå ôïí ôýðï (7:: 5) ðñïêýðôåé üôé 1 1 e x dx 1.49 (áêñéâþò ôéìþ I = Erf(1) ): Óçìåßùóç ¼ôáí ôï äéüóôçìá ïëïêëþñùóçò äåí åßíáé ôï [ 1; 1], ôüôå ç ðáñáðüíù ëýóç áðáéôåß áñ éêü ôçí åöáñìïãþ ôïõ ìåôáó çìáôéóìïý (7:: ), üðùò áõôü ãßíåôáé óôï ðáñáêüôù ðáñüäåéãìá:

34 7 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò fx x () fx x 1.5 x x.5 x 4 1. x..4 () Ó Þìá : ÐáñÜäåéãìá () Ç ìðëå êáìðýëç ïñßæåé ôï äéüãñáììá ôçò óõíüñôçóçò e x, åíþ ôï åìâáäüí ôïõ ó Þìáôïò éóïýôáé ìå ôçí ôéìþ ôïõ ïëïêëçñþìáôïò 1 e x dx. () Ç êüêêéíç êáìðýëç ïñßæåé ôï 1 äéüãñáììá ôïõ ðïëõùíýìïõ P 4 (x) = 1 ( 8 x + 5x 4) ôïõ Legendre, åíþ ôá óçìåßá x 1 ; x ; x ; x 4 ôéò ñßæåò ôïõ. ÐáñÜäåéãìá ¼ìïéá ìå ôïí ôýðï ôùí Guss-Legendre ãéá 4 óçìåßá ôï ïëïêëþñùìá (Ó ) 1 I = e x dx; üôáí ç èåùñçôéêþ ôéìþ åßíáé : Ëýóç. Áñ éêü ìå ôïí ôýðï (7:: ) ãßíåôáé ìåôáó çìáôéóìüò ôïõ äéáóôþìáôïò ïëïêëþñùóçò óôï äéüóôçìá [ 1; 1] èýôïíôáò ñá x = (1 )t = :5t + :5; ïðüôå dx = d(:5t + :5) = (:5t + :5) dt = :5 dt: I = = 1 1 e x dx = 1 1 e (:5 t+:5) :5 dt 1 :5 e (:5 x+:5) dx = g(x) dx; 1 1

35 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 71 Ðßíáêáò : ÐáñÜäåéãìá x i w i w i g (x i ) = w i [ :5 e (:5 x i+:5) ] x 1 = :86114 w 1 = :4785 w 1 g (x 1 ) = : x = :998 w = :6514 w g (x ) = :9 44 x = :998 w = :6514 w g (x ) = :8 14 x 4 = :86114 w 4 = :4785 w 1 g (x 4 ) = : üôáí ôåèåß ãéá åõêïëßá x áíôß ôïõ t êáé g(x) = :5 e (:5 x+:5) : Ôüôå áðü ôá áðïôåëýóìáôá ôïõ Ðßíáêá 7.. -, üôáí w 1 g (x 1 ) = :4785 w g (x ) = :6514 w g (x ) = :6514 w 4 g (x 4 ) = :4785 [ :5 e [:5 ( :86114)+:5]] = : [ :5 e [:5 ( :998)+:5]] = :9 44 [ :5 e [:5 (:998)+:5]] = :8 14 [ :5 e [:5 (:86114)+:5]] = : ; óýìöùíá êáé ìå ôïí ôýðï (7:: 5) ðñïêýðôåé üôé 1 e x dx = 1 1 :5 e (:5 x+:5) dx = g(x) dx 1 1 w 1 g (x 1 ) + w g (x ) + w g (x ) + w 4 g (x 4 ) = : Ôüôå, åðåéäþ ç èåùñçôéêþ ôéìþ åßíáé :746 84, Ý ïõìå áðüëõôï óöüëìá e = 7:

36 7 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò fx fx x () x Ó Þìá : ÐáñÜäåéãìá () Ç ìðëå êáìðýëç ïñßæåé ôï äéüãñáììá ôçò óõíüñôçóçò e x, åíþ ôï åìâáäüí ôïõ ó Þìáôïò éóïýôáé ìå ôçí ôéìþ ôïõ ïëïêëçñþìáôïò 1 e x dx. () Ç êüêêéíç êáìðýëç ïñßæåé ôï äéüãñáììá ôçò óõíüñôçóçò :5 e (:5 x+:5), åíþ ôï åìâáäüí ôïõ ó Þìáôïò éóïýôáé ìå ôçí ôéìþ ôïõ ïëïêëçñþìáôïò 1 :5 e (:5 x+:5) dx. 1 () ÐáñÜäåéãìá óôù ôï ïëïêëþñùìá (Ó ) I = :6 x dx 1 + x 4 : Íá ëõèåß ìå ôïí êáíüíá ôùí Guss-Seidel ãéá, áíôßóôïé á 6 óçìåßá êáé íá ãßíåé óýãêñéóç ôùí áðïôåëåóìüôùí ìå ôç èåùñçôéêþ ôéìþ I = 1 rcsinh ( x ) : : Óôç óõíý åéá íá ãßíåé óýãêñéóç ìå ôçí áíôßóôïé ç ëýóç ðïõ ðñïêýðôåé áðü ôïõò óýíèåôïõò êáíüíåò ôïõ ôñáðåæßïõ, ôïõ Simpson êáé ôùí =8 ôïõ Simpson, üôáí h = :1. Ëýóç.

37 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 7 fx x Ó Þìá : ÐáñÜäåéãìá Ç ìðëå êáìðýëç ïñßæåé ôï äéüãñáììá ôçò óõíüñôçóçò, åíþ ôï åìâáäüí ôïõ ó Þìáôïò éóïýôáé ìå ôçí ôéìþ x 1+x 4 ôïõ ïëïêëçñþìáôïò :6 x dx 1+x 4 : Êáíüíáò ôùí Guss-Seidel ãéá óçìåßá (Ó ) Áñ éêü ìå ôïí ôýðï (7:: ) ãßíåôáé ìåôáó çìáôéóìüò ôïõ äéáóôþìáôïò ïëïêëþñùóçò óôï äéüóôçìá [ 1; 1] èýôïíôáò x = (:6 )t + :6 + = :t + :; ïðüôå dx = d(:t + :) = (:t + :) dt = : dt: ñá I = = : x dx = 1 + x 4 1 :t + : 1 + (:t + :) 4 : dt :9 (x + 1) 1 dx = g(x) dx; 1 + (:x + :) 4 1

38 74 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò Ðßíáêáò : ÐáñÜäåéãìá : êáíüíáò ôùí Guss-Seidel ãéá óçìåßá. :9(x x i w i w i g (x i ) = w i +1) i 1+(:x i +:) 4 x 1 = : w 1 = : w 1 g (x 1 ) = :11 75 x = w = : w g (x ) = : x = : w = : w g (x ) = : üôáí üìïéá Ý åé ôåèåß ãéá åõêïëßá x áíôß ôïõ t êáé g(x) = :9 (x + 1) 1 + (:x + :) 4 : Ôüôå áðü ôá áðïôåëýóìáôá ôïõ Ðßíáêá , óýìöùíá êáé ìå ôïí ôýðï (7:: 5) üôáí w 1 g (x 1 ) = : :9 ( : ) = : [:( : ) + :] 4 [ ] :9 ( + 1) w g (x ) = : = : (: + :]) 4 [ ] :9 (: ) w g (x ) = : = :85 674; 1 + (: : :]) 4 ðñïêýðôåé üôé :6 x dx 1 + x 4 = 1 1 :9 (x + 1) 1 dx = 1 + (:x + :) 4 1 g(x) dx w 1 g (x 1 ) + w g (x ) + w g (x ) = ; äçëáäþ õðüñ åé áðüëõôï óöüëìá e = 7:

39 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 75 gx x x 1 x x..4.6 () gx x x 1 x x x 4 x 5 x 6..4 () Ó Þìá : ÐáñÜäåéãìá : Ç ìðëå êáìðýëç ïñßæåé ôï :9(x+1) äéüãñáììá ôçò óõíüñôçóçò g(x) =. Ç êüêêéíç êáìðýëç ïñßæåé 4 1+(:x+:) ôç ãñáöéêþ ðáñüóôáóç ôïõ ðïëõùíýìïõ: () P (x) = 1 ( x + 5x ) ôïõ Legendre, åíþ ôá óçìåßá x 1 ; x ; x åßíáé ïé ñßæåò ôïõ êáé () P 6 (x) = ( x 15x 4 + 1x 6) ôïõ Legendre, åíþ ôá óçìåßá x 1 ; : : : ; x åßíáé ïé ñßæåò ôïõ. Êáíüíáò ôùí Guss-Seidel ãéá 6 óçìåßá (Ó ) ¼ìïéá áðü ôá áðïôåëýóìáôá ôïõ Ðßíáêá , óýìöùíá êáé ìå ôïí ôýðï (7:: 5), Ý ïõìå :6 x dx 1 + x 4 = 1 1 :9 (x + 1) 1 dx = 1 + (:x + :) 4 1 g(x) dx w 1 g (x 1 ) + w g (x ) + w g (x ) + w 4 g (x 4 ) +w 5 g (x 5 ) + w 6 g (x 6 ) = êáé óõãêñßíïíôáò ìå ôç èåùñçôéêþ ôéìþ ðñïêýðôåé áðüëõôï óöüëìá Óýíèåôïé êáíüíåò e = : : Óýìöùíá ìå ôéò ôéìýò ôïõ Ðßíáêá ðñïêýðôïõí ôá åîþò: Ôñáðåæßïõ I h {f (x ) + [f (x 1 ) + : : : + f (x 5 )] + f (x 6 )} =

40 76 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò Ðßíáêáò : ÐáñÜäåéãìá : êáíüíáò ôùí Guss-Seidel ãéá 6 óçìåßá. :9(x x i w i w i g (x i ) = w i +1) i 1+(:x i +:) 4 x 1 = :94695 w 1 = : w 1 g (x 1 ) = : x = :66199 w = : w g (x ) = : x = : w = : w g (x ) = : x 4 = : w 4 = : w 4 g (x 4 ) = : x 5 = :66199 w 5 = : w 5 g (x 5 ) = : x 6 = :94695 w 6 = : w 6 g (x 6 ) = : Ðßíáêáò : ÐáñÜäåéãìá Óýíèåôïé êáíüíåò: ïé ôéìýò ôùí óçìåßùí x i êáé ôùí ôéìþí f (x i ) = x i. 1+x 4 i x i f (x i ) x i f (x i ) x = : x 4 = :4 : x 1 = :1 : x 5 = :5 : x = : : x 6 = :6 :564 5 x = : :98 79

41 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 77 ìå áðüëõôï óöüëìá: e = : Simpson I h {f (x ) + 4 [f (x 1 ) + f (x ) + f (x 5 )] + [f (x ) + f (x 4 )] + f (x 6 )} =.176 ìå áðüëõôï óöüëìá: e = : =8 ôïõ Simpson I h 8 {f (x ) + [f (x 1 ) + f (x 4 )] + [f (x ) + f (x 5 )] + f (x ) + f (x 6 )} = ìå áðüëõôï óöüëìá: e = 6: Óõãêñßíïíôáò ôï åëü éóôï ôùí óöáëìüôùí ôùí ðáñáðüíù óýíèåôùí êáíüíùí ìå ôï áíôßóôïé ï óöüëìá e = : ôïõ êáíüíá ôùí Guss- Legendre ãéá 6 óçìåßá Ý ïõìå ìéá ðñïöáíþ ðåéñáìáôéêþ åðáëþèåõóç ôçò ÐáñáôÞñçóçò (i). ÁóêÞóåéò 1. óôù ôï ïëïêëþñùìá I = :6 x dx 1 + x : i) Íá ëõèåß ìå ôïí êáíüíá ôùí Guss-Legendre ãéá, áíôßóôïé á 6 óçìåßá êáé íá ãßíåé óýãêñéóç ôùí áðïôåëåóìüôùí ìå ôç èåùñçôéêþ ôéìþ I = 1 ( x ) 1 + x : êáé ôçí áíôßóôïé ç ëýóç ðïõ ðñïêýðôåé áðü ôïõò óýíèåôïõò êáíüíåò ôïõ ôñáðåæßïõ, Simpson êáé =8 ôïõ Simpson, üôáí h = :1.

42 78 Êáíüíåò ïëïêëþñùóçò ôïõ Guss Êáè. Á. ÌðñÜôóïò ii) Ôï ïëïêëþñùìá íá ãñáöåß ùò åîþò: I = :6 x :1 dx = x :6 dx x 1 + x : : : + x dx 1 + x :5 = I 1 + : : : + I 6 : Íá õðïëïãéóôåß ôï ïëïêëþñùìá I åöáñìüæïíôáò óå êáèýíá áðü ôá ðáñáðüíù ïëïêëçñþìáôá I i ; i = 1; : : : ; 6 ôïí êáíüíá ôùí Guss-Legendre ãéá 6 óçìåßá êáé íá ãßíåé óýãêñéóç ôïõ áðïôåëýóìáôïò ìå ôïí áíôßóôïé ï ôçò ðåñßðôùóçò (i). Óôç óõíý åéá íá ãñáöåß ôï ðñüãñáììá ëýóçò ôçò (ii) ìå ôï MATLAB.. ¼ìïéá ìå ôïí êáíüíá ôùí Guss-Legendre ãéá, áíôßóôïé á 6 óçìåßá ôï ïëïêëþñùìá üôáí h = :1 ìå èåùñçôéêþ ôéìþ I = 1: dx 1 + x ; ( I = ln x + ) x 1: + 1 1:15 97 (âëýðå ÐáñÜäåéãìá ) êáé íá ãßíåé óýãêñéóç ìå ôá áðïôåëýóìáôá ôïõ Ðßíáêá êáé ôïõ óýíèåôïõ êáíüíá ôùí =8 ôïõ Simpson (ÐáñÜäåéãìá ). ÁðáíôÞóåéò 1. ÔéìÞ ïëïêëçñþìáôïò ìå óýíèåôï êáíüíá ôñáðåæßïõ: : , Simpson: : , =8 Simpson: : , Guss-Legendre óçìåßá: : óçìåßá: : ¼ìïéá åßíáé ôñáðåæßïõ: 1:15 711, Simpson: 1:15 97, =8 Simpson: 1:15 97, Guss-Legendre óçìåßá: 1: óçìåßá: 1:15 97 ìå óöüëìá: 6:

43 Êáíüíåò ïëïêëþñùóçò ôùí Guss-Legendre 79 ÐáñáôÞñçóç Óå ïñéóìýíá áðïôåëýóìáôá õðüñ åé ãéá ôç ñçóéìïðïéïýìåíç áêñßâåéá ôáýôéóç ìå ôçí áíôßóôïé ç èåùñçôéêþ ôéìþ. Óôçí ðñáãìáôéêüôçôá, üôáí ç Üóêçóç ëõèåß ìå õðïëïãéóôþ, õðüñ åé óöüëìá óôçí ðñïóýããéóç, üðùò áõôü ôïõ êáíüíá ôùí 6 óçìåßùí ôùí Guss- Legendre ôçò óêçóçò.

44

45 7.4 Âéâëéïãñáößá [1] Aêñßâçò, Ã. & ÄïõãáëÞò, Â. (1995). ÅéóáãùãÞ óôçí ÁñéèìçôéêÞ ÁíÜëõóç. ÐáíåðéóôçìéáêÝò Åêäüóåéò ÊñÞôçò. ISBN 978{96{54{{6. [] ÌðñÜôóïò, Á. (11). ÅöáñìïóìÝíá ÌáèçìáôéêÜ. Åêäüóåéò Á. Óôáìïýëç. ISBN 978{96{51{874{7. [] Burden, R. L. & Fires, D. J. (1). Numericl Anlysis. Brooks/Cole (7th ed.). ISBN 978{{54{816{. Âéâëéïãñáößá ãéá ðåñáéôýñù ìåëýôç ÓôåöáíÜêïò,. (9). Ðñïãñáììáôéóìüò Ç/Õ ìå MATLAB. Ãêéïýñäáò ÅêäïôéêÞ. ISBN 978{96{87{856{8. Atkinson, K. E. (1989). An Introduction to Numericl Anlysis. John Wiley & Sons (nd ed.). ISBN {471{5{. Conte, S. D. & de Boor, C. (198). Elementry Numericl Anlysis: An Algorithmic Approch. McGrw{Hill Inc (rd ed.). ISBN 978{{ 7{1447{9. Don, E. (6). Schum's Outlines { Mthemtic. Åêäüóåéò ÊëåéäÜñéèìïò. ISBN 978{96{9{961{. Leder, L. J. (4). Numericl Anlysis nd Scientic Computtion. Addison{Wesley. ISBN 978{{1{7499{7. 81

46 8 ÐñïóÝããéóç ïëïêëçñùìüôùí Êáè. Á. ÌðñÜôóïò Schtzmn, M. (). Numericl Anlysis: A Mthemticl Introduction. Oxford: Clrendon Press. ISBN 978{{19{8579{1. Stoer, J. & Bulirsch, R. (). Introduction to Numericl Anlysis. Springer (rd ed.). ISBN 978{{87{9545{. Sli, E. & Myers, D. (). An Introduction to Numericl Anlysis. Cmridge University Press. ISBN 978{{51{794{8. ÌáèçìáôéêÝò âüóåéò äåäïìýíùí Pge