ÁñéèìçôéêÞ ÁíÜëõóç É - ÓÅÌÖÅ Åñãáóßá 2 ìåóåò êáé åðáíáëçðôéêýò ìýèïäïé

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1 ÁñéèìçôéêÞ ÁíÜëõóç É - ÓÅÌÖÅ Åñãáóßá 2 ìåóåò êáé åðáíáëçðôéêýò ìýèïäïé Íéêüëáò ÊÜñáëçò Á/Ì : ÔìÞìá 1ï 28 Óåðôåìâñßïõ, 26 1 ìåóåò ÌÝèïäïé 1.1 Åñþôçìá 1 ñçóéìïðïéþíôáò ôçí gauss.m êáé ôçí herm5.m, êáé åêôåëþíôáò ôçí åíôïëþ question1a ðñïêýðôåé ôï ðáñáêüôù ãñüöçìá ôçò sin(x), ôçò ðáñåìâïëþò ôçò sin(x) êáèþò êáé ôï óöüëìá ôçò, óôï äéüóôçìá [ 3 2 ; 3 2 ]. Ôï áðïôýëåóìá ðïõ ðñïêýðôåé ãéá ôïõò óõíôåëåóôýò ôïõ ðïëõùíýìïõ ðáñåìâïëþò åßíáé ôï áêüëïõèï : Üñá p(s) = : s 5 : s 3 + s 1

2 Åêôåëþíôáò ôçí åíôïëþ question1b ðñïêýðôåé ôï ðáñáêüôù ãñüöçìá ôçò sin(x), ôçò ðáñåìâïëþò ôçò sin(x) êáèþò êáé ôï óöüëìá ôçò, óôï äéüóôçìá [ 2 ; 2 ]. Ôï áðïôýëåóìá ðïõ ðñïêýðôåé ãéá ôïõò óõíôåëåóôýò ôïõ ðïëõùíýìïõ ðáñåìâïëþò åßíáé ôï áêüëïõèï : Üñá p(s) = : s 5 : s 3 + s 2

3 1.2 Åñþôçìá 2 ñçóéìïðïéþíôáò ôçí gauss.m êáé ôçí herm8.m, êáé åêôåëþíôáò ôçí åíôïëþ question2a ðñïêýðôåé ôï ðáñáêüôù ãñüöçìá ôçò sin(x), ôçò ðáñåìâïëþò ôçò sin(x) êáèþò êáé ôï óöüëìá ôçò, óôï äéüóôçìá [ 3 2 ; 3 2 ]. Ôï áðïôýëåóìá ðïõ ðñïêýðôåé ãéá ôïõò óõíôåëåóôýò ôïõ ðïëõùíýìïõ ðáñåìâïëþò åßíáé ôï áêüëïõèï : Üñá p(s) = : s 7 + : s 3 : s 3 + s 3

4 Åêôåëþíôáò ôçí åíôïëþ question1b ðñïêýðôåé ôï ðáñáêüôù ãñüöçìá ôçò sin(x), ôçò ðáñåìâïëþò ôçò sin(x) êáèþò êáé ôï óöüëìá ôçò, óôï äéüóôçìá [ 2 ; 2 ]. Ôï áðïôýëåóìá ðïõ ðñïêýðôåé ãéá ôïõò óõíôåëåóôýò ôïõ ðïëõùíýìïõ ðáñåìâïëþò åßíáé ôï áêüëïõèï : Üñá p(s) = : : s 3 : s 3 + s 4

5 1.3 Åñþôçìá 3 Ðáñáôçñïýìå ïôé óôïõò óõíôåëåóôýò ìçäåíßæïíôáé üëïé ïé üñïé Üñôéáò ôüîçò. Ç áéôßá ãéá áõôü ìðïñåß íá ãßíåé êáôáíïçôþ áí ðáñáôçñþóïõìå ïôé êáé ôï áíüðôõãìá ôçò óåéñüò Taylor ãéá ôçí óõíüñôçóç sin(x) ðåñéý åé ìüíï üñïõò ðåñéôôþò ôüîçò. 2 ÅðáíáëçðôéêÝò ÌÝèïäïé 2.1 Åñþôçìá 1 Ìå ñþóç ôïõ ôýðïõ ãéá ôçí åðßëõóç óõóôþìáôïò ìå ôçí ìýèïäï SOR, äçìéïõñãïýìå ôï sortri.m. Ç âýëôéóôç åðéëïãþ ôïõ ù äßíåôáé áðï ôïí ôýðï : ù = ñ(â) Åñþôçìá 2 Ìå ñþóç ôïõ script question3a.m ðñïêýðôåé ç ðáñáêüôù ãñáöéêþ ðáñüóôáóç : 5

6 Ðáñáôçñïýìå ïôé ï åëü éóôïò áñéèìüò åðáíáëþøåùí åßíáé 1 êáé åìöáíßæïíôáé ãéá ôéò ôéìýò ôïõ ù 1, 1.5, 1.1 Ç èåùñçôéêþ ôéìþ ôïõ ù ãéá ôïí åëü éóôï áñéèìü åðáíáëþøåùí, ðñïêýðôåé ìå ñþóç ôïõ script question3b.m êáé åßíáé : ù= ÐáñÜñôçìá Åäþ ðáñáôßèåôáé ï ðçãáßïò êþäéêáò ãéá ôéò óõíáñôþóåéò ðïõ ñçóéìïðïéþèçêáí ðáñáðüíù. 3.1 Gauss.m function x=gauss ( a, b ) ; n=length (b ) ; for i =1:n 1, [ amax, imax]=max( abs ( a ( i : n, i ) ) ) ; i f amax<eps, disp ( ' S i n g u l a r Matrix ' ) ; break ; end imax=imax+i 1; i f imax~=i, sa=a ( imax, i : n ) ; sb=b( imax ) ; a ( imax, i : n)=a ( i, i : n ) ; b( imax)=b( i ) ; a ( i, i : n)=sa ; b( i )=sb ; 6

7 end b( i +1:n)=b( i +1:n) b( i ) a ( i +1:n, i )/ a ( i, i ) ; a ( i +1:n, i +1:n)=a ( i +1:n, i +1:n) a ( i +1:n, i ) a ( i, i +1:n)/ a ( i, i ) ; i f abs ( a (n, n))<eps, disp ( ' S i n g u l a r Matrix ' ) ; break ; end x (n,1)=b(n)/ a (n, n ) ; for i=n 1: 1:1, x ( i,1)=( b( i ) a ( i, i +1:n) x ( i +1:n, 1 ) ) / a ( i, i ) ; 3.2 herm5.m function c=herm5 ( x, f, df ) ; i f nargin <3, error ( ' Not enough arguments. " ' ) ; i f ( ( size ( x)~= size ( f ) ) ( size ( f )~= size ( df ) ) ( size ( x)~= size ( df ) ) ), error ( 'Wrong s i z e o f input ' ) ; i f size ( x)~=3 error ( ' Not r i g h t # o f p o i n t s. ' ) ; a =[[ x. ^ 5 ] ' [ x. ^ 4 ] ' [ x. ^ 3 ] ' [ x. ^ 2 ] ' x ' [ ones ( size ( x ) ) ] ' [ 5 x. ^ 4 ] ' [ 4 x. ^ 3 ] ' [ 3 x. ^ 2 ] ' [ 2 x ] ' [ ones ( size ( x ) ) ] ' [ zeros ( size ( x ) ) b=[ f ' df ' ] ; c=gauss ( a, b) 3.3 herm8.m function c=herm5 ( x, f, df, ddf ) ; i f nargin <3, error ( ' Not enough arguments. ' ) ; i f ( ( size ( x)~= size ( f ) ) ( size ( f )~= size ( df ) ) ( size ( x)~= size ( df ) ) ), error ( 'Wrong s i z e o f input ' ) ; i f size ( x)~=3 error ( ' Not r i g h t # o f p o i n t s. ' ) ; a =[[ x. ^ 8 ] ' [ x. ^ 7 ] ' [ x. ^ 6 ] ' [ x. ^ 5 ] ' [ x. ^ 4 ] ' [ x. ^ 3 ] ' [ x. ^ 2 ] ' x ' [ ones ( siz [ 8 x. ^ 7 ] ' [ 7 x. ^ 6 ] ' [ 6 x. ^ 5 ] ' [ 5 x. ^ 4 ] ' [ 4 x. ^ 3 ] ' [ 3 x. ^ 2 ] ' [ 2 x ] ' [ [8 7 x. ^ 6 ] ' [7 6 x. ^ 5 ] ' [3 x. ^ 4 ] ' [2 x. ^ 3 ] ' [12 x. ^ 2 ] ' [ 6 x ] ' [ 2 [ o b=[ f ' df ' ddf ' ] ; c=gauss ( a, b) 7

8 3.4 sortri.m function [ x, k]= s o r t r i ( a, b, tol,w) ; i f nargin <2, error ( 'Wrong input ' ) ; i f nargin <3, t e l=eps ; i f nargin <4, d=diag ( a ) ; D=diag (d, ) ; L=t r i l ( a) D; U=triu ( a) D; B= inv (D) (L+U) ; y=eig (B) ; p=max( abs ( y ) ) ; l e s s=min( y ) ; i f l e s s <, w=1; else w=2/(1+ sqrt(1 p ^ 2 ) ) ; n=length (b ) ; x=ones (n, 1 ) ; xnew=zeros (n, 1 ) ; k=; while max( abs ( xnew x))> t o l & k<2, x=xnew ; k=k+1; xnew(1)=(1 w) x(1)+w/a ( 1, 1 ) ( b(1) a ( 1, 2 ) x ( 2 ) ) ; for i =2:n 1, xnew ( i )=(1 w) x ( i )+w/a ( i, i ) ( b( i ) a ( i, i 1) xnew ( i 1) a ( i, i +1) x ( i +1)); xnew (n)=(1 w) x (n)+w/a (n, n ) ( b(n) a (1, n 1) xnew (n 1)); x=xnew ; 3.5 question1a.m clear a l l ; c l f ; format long ; x=[ 3 pi /2 3 pi / 2 ] ; f =sin ( x ) ; df=cos ( x ) ; c=herm5 ( x, f, df ) ; x= 3 pi / 2 :. 1 : 3 pi / 2 ; y=polyval ( c, x ) ; 8

9 z=sin ( x ) ; error=z y ; plot ( x, y, ' g ', x, z, ' b ', x, error, ' r ' ) legend ( ' Hermite Polynomial ', ' Sin ( x ) ', ' Error ', 4 ) ; grid on ; 3.6 question1b.m clear a l l ; c l f ; format long ; x=[ pi /2 pi / 2 ] ; f =sin ( x ) ; df=cos ( x ) ; c=herm5 ( x, f, df ) ; x= pi / 2 :. 1 : pi / 2 ; y=polyval ( c, x ) ; z=sin ( x ) ; error=z y ; plot ( x, y, ' g ', x, z, ' r ', x, error, ' b ' ) legend ( ' Hermite Polynomial ', ' Sin ( x ) ', ' Error ', 4 ) ; grid on ; 3.7 question2a.m clear a l l ; format long ; x=[ 3 pi /2 3 pi / 2 ] ; f =sin ( x ) ; df=cos ( x ) ; ddf= sin ( x ) ; c=herm8 ( x, f, df, ddf ) ; x= 3 pi / 2 :. 1 : 3 pi / 2 ; y=polyval ( c, x ) ; z=sin ( x ) ; error=z y ; plot ( x, y, ' g ', x, z, ' b ', x, error, ' r ' ) legend ( ' Hermite Polynomial ', ' Sin ( x ) ', ' Error ', 4 ) ; grid on ; 3.8 question2b.m clear a l l ; format long ; x=[ pi /2 pi / 2 ] ; f =sin ( x ) ; df=cos ( x ) ; ddf= sin ( x ) ; c=herm8 ( x, f, df, ddf ) ; x= pi / 2 :. 1 : pi / 2 ; 9

10 y=polyval ( c, x ) ; z=sin ( x ) ; error=z y ; plot ( x, y, ' r ', x, z, ' b ', x, error, ' g ' ) legend ( ' Hermite Polynomial ', ' Sin ( x ) ', ' Error ', 4 ) ; grid on ; 3.9 question3a.m clear a l l ; c l f ; format long ; v=[3 3 ] ; u=1; for i =1: 197, v=[v 3 ] ; u=[u 1 ] ; a=diag ( v)+diag (u,1)+ diag (u, 1); for i =1: 199, b( i )= i /1; r =; for w=. 1 :. 5 : 1. 9, [ x, k]= s o r t r i ( a, b, 1 e 5,w) w r =[ r k ] ; w=. 1 :. 5 : 1. 9 ; w=[ w ] ; plot (w, r, ' r ' ) ; XLABEL( 'w ' ) ; YLABEL( '# o f times ' ) ; grid on ; 3.1 question3b.m d=diag ( a ) ; D=diag (d, ) ; L=t r i l ( a) D; U=triu ( a) D; B= inv (D) (L+U) ; y=eig (B) ; p=max( abs ( y ) ) ; w=2/(1+ sqrt(1 p ^ 2 ) ) ; Created with LATEX Algorithms programmed in MATLAB 1