aritmitiki-analisi1 - ΑΡΙΘΜΗΤΙΚΗ...

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Unformatted text preview: ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ 1 1. Δίνεται η εξίσωση f ( x ) = x − cos x = 0 που είναι συνεχής ως διαφορά συνεχών 2 (g(x)=x πολυωνυμική και h(x)=1/2cosx τριγωνομετρική) και για την οποία ισχύει: 1 ⎫ f (0) = − ⎪ ⎪ 2 ⎬ ⇒ f (0) ⋅ f (1) < 0 ⇒ 1 f (1) = 1− cos1 = 0.500076152 ≈ 0.5⎪ ⎪ 2 ⎭ ⇒ϑ . Bolzano ∃x ∈ (0,1) : f ( x) = 0. Άρα δείξαμε ότι η f(x) έχει στο (0,1) μία τουλάχιστον ρίζα. Αν δείξουμε ότι η f(x) είναι και (γνησίως) μονότονη στο [0,1] θα έχουμε δείξει ότι η δοθείσα εξίσωση έχει στο [0,1] μοναδική ρίζα. 1 Έχουμε: f '( x) = 1 + sin x 2 Φτιάχνουμε σταδιακά την ανισότητα ως εξής: 0 ≤ x≤1 sin 0 ≤ sin x ≤ sin1 1 ≤ sin x ≤ 0.017... 1 1 1 1 + ≤ 1 + sin x ≤ 1 + ?0.017 2 2 2 Άρα συμπεραίνουμε ότι f '( x) > 0 ∀x ? [0,1] ,δηλαδή η f(x) είναι γνησίως 1 αύξουσα στο [0,1], με αποτέλεσμα η εξίσωση f ( x ) = x − cos x = 0 να έχει 2 μοναδική ρίζα στο [0,1]. Ακολουθεί ο κώδικας του ζητούμενου προγράμματος και αμέσως μετά το πρόγραμμα εκτελεσμένο. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------C C C C C C PROGRAM ASK1 SYNERGATES DIMOS NIKOLAOS, KAMPOLIS GEORGIOS BISEC, QUANEW, SECAN, NEWTO DOUBLE PRECISION F,DF,EPS,A,B,X,FX,E,Y EXTERNAL F,DF WRITE(6,*) 'GIVE A,B,NIT,EPS' READ(5,*) A,B,NIT,EPS WRITE(6,*) 'INPUT:' WRITE(6,*) 'A=',A,' B=',B WRITE(6,*) 'NIT=',NIT,' EPS=',EPS WRITE(6,*) ' ' WRITE(6,*) 'OUTPUT:' CALL BISECT(F,A,B,NIT,EPS,X,FX,E,KIT,ITEST) 1 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ WRITE(6,*) 'ITEST=',ITEST,' KIT=',KIT WRITE(6,*) 'X=',X,' FX=',FX WRITE(6,*) 'E=',E WRITE(6,*) 'BISEC HAS FINISHED!!!' C WRITE(6,*) 'GIVE M,NIT,EPS AND X5 OF BISEC FOR X' READ(5,*) M,NIT,EPS,X WRITE(6,*) 'INPUT:' WRITE(6,*) 'M=',M,' NIT=',NIT,' EPS=',EPS WRITE(6,*) 'X=',X WRITE(6,*) ' ' WRITE(6,*) 'OUTPUT:' CALL NEWTON(F,DF,M,NIT,EPS,X,FX,KIT,ITEST) WRITE(6,*) 'ITEST=',ITEST,' KIT=',KIT IF(ITEST.EQ.0) STOP WRITE(6,*) 'X=',X,' FX=',FX WRITE(6,*) 'QUANEW HAS FINISHED!!!' C WRITE(6,*) 'GIVE NIT, EPS,Y=0.5,X=0.25' READ(5,*) NIT,EPS,Y,X WRITE(6,*) 'INPUT:' WRITE(6,*) 'NIT=',NIT,' EPS=',EPS WRITE(6,*) 'Y=',Y WRITE(6,*) 'X=',X WRITE(6,*) ' ' WRITE(6,*) 'OUTPUT:' CALL SECANT(F,NIT,EPS,Y,X,FX,KIT,ITEST) WRITE(6,*) 'ITEST=',ITEST,' KIT=',KIT IF(ITEST.EQ.0) STOP WRITE(6,*) 'X=',X,' FX=',FX WRITE(6,*) 'SECAN HAS FINISHED' C WRITE(6,*) 'GIVE M,NIT,EPS,X=0.5' READ(5,*) M,NIT,EPS,X WRITE(6,*) 'INPUT:' WRITE(6,*) 'M=',M,' NIT=',NIT,' EPS=',EPS WRITE(6,*) 'X=',X WRITE(6,*) ' ' WRITE(6,*) 'OUTPUT:' CALL NEWTON(F,DF,M,NIT,EPS,X,FX,KIT,ITEST) WRITE(6,*) 'ITEST=',ITEST,' KIT=',KIT IF(ITEST.EQ.0) STOP WRITE(6,*) 'X=',X,' FX=',FX WRITE(6,*) 'NEWTO HAS FINISHED.THANK YOU!!!' STOP END C---------------------------------------------------------------SUBROUTINE BISECT(F,A,B,NIT,EPS,X,FX,E,KIT,ITEST) C C SOLUTION OF THE EQUATION F(X)=0 BY THE BISECTION METHOD C C USES D.P. FUNCTION F C C INPUT: 2 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C C C C C C C C C C C C C C F: FUNCTION [A,B]: SEARCH INTERVAL NIT: MAXIMUM NUMBER OF ITERATIONS EPS: ERROR TEST NUMBER OUTPUT: X: APPROXIMATE ROOT FX:=F(X) E: ERROR BOUND KIT: NUMBER OF EXECUTED ITERATIONS ITEST = 0, F(A)F(B)>0 = 1, ERROR TEST SATISFIED = 2, ERROR TEST NOT SATISFIED DOUBLE PRECISION F,A,B,EPS,X,FX,FA,FB,E,P EXTERNAL F ITEST=1 E=B-A FA=F(A) FB=F(B) P=FA*FB IF(P.GT.0.D0) GO TO 40 C DO 10 K=0,NIT KIT=K X=A+E/2.D0 FX=F(X) WRITE(6,*) 'K=',K,' X=',X,' FX=',FX P=FA*FX IF(P.LE.0.D0) GO TO 20 A=X FA=FX GO TO 30 20 B=X FB=FX 30 E=B-A IF(1.D0/2.D0**(K+1).LE.EPS) RETURN 10 CONTINUE ITEST=2 RETURN 40 ITEST=0 RETURN END C----------------------------------------------------------------SUBROUTINE QUANEW(F,DF,M,NIT,EPS,X,FX,KIT,ITEST) C C SOLUTION OF THE EQUATION F(X)=0 BY THE QUASI-NEWTON's METHOD C C USES D.P. FUNCTIONS F AND DF C C INPUT: C F: FUNCTION C DF: DERIVATIVE OF F C M: ROOT MULTIPLICITY PARAMETER (NORMALLY M=1) C NIT: MAXIMUM NUMBER OF ITERATIONS C EPS: CONVERGENCE TEST NUMBER C X: INITIAL VALUE 3 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C C C C C C C C C OUTPUT: X: APPROXIMATE ROOT FX:=F(X) KIT: NUMBER OF EXECUTED ITERATIONS ITEST = 0, ZERO DERIVATIVE (DIVISION BY ZERO) = 1, CONVERGENCE TEST SATISFIED = 2, CONVERGENCE TEST NOT SATISFIED DOUBLE PRECISION F,DF,EPS,X,FX,DFX,Y,EE EXTERNAL F,DF EE=1.D-99 ITEST=1 FX=F(X) C DO 10 K=1,NIT KIT=K Y=X IF ((K-((K/3)*3)).EQ.1) DFX=DF(X) IF(DABS(DFX).LT.EE) GO TO 20 X=X-DFLOAT(M)*FX/DFX FX=F(X) WRITE(6,*) 'K=',K,' X=',X,' FX=',FX IF(DABS(X-Y).LE.EPS) RETURN 10 CONTINUE ITEST=2 RETURN 20 ITEST=0 RETURN END C----------------------------------------------------------------SUBROUTINE SECANT(F,NIT,EPS,Y,X,FX,KIT,ITEST) C C SOLUTION OF THE EQUATION F(X)=0 BY THE SECANT METHOD C C USES D.P. FUNCTION F C C INPUT: C F: FUNCTION C Y,X: INITIAL VALUES C NIT: MAXIMUM NUMBER OF ITERATIONS C EPS: CONVERGENCE TEST NUMBER C C OUTPUT: C X: APPROXIMATE ROOT C FX:=F(X) C KIT: NUMBER OF EXECUTED ITERATIONS C ITEST = 0, DIVISION BY ZERO C = 1, CONVERGENCE TEST SATISFIED C = 2, CONVERGENCE TEST NOT SATISFIED C DOUBLE PRECISION F,EPS,X,Y,Z,FX,FY,D,EE EXTERNAL F EE=1.D-99 ITEST=1 FY=F(Y) FX=F(X) 4 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C DO 10 K=2,NIT KIT=K Z=X D=FX-FY IF(DABS(D).LE.EE) GO TO 20 X=X-FX*(X-Y)/D Y=Z FY=FX FX=F(X) WRITE(6,*) 'K=',K,' X=',X,' FX=',FX IF(DABS(X-Y).LE.EPS) RETURN 10 CONTINUE ITEST=2 RETURN 20 ITEST=0 RETURN END C----------------------------------------------------------------SUBROUTINE NEWTON(F,DF,M,NIT,EPS,X,FX,KIT,ITEST) C C SOLUTION OF THE EQUATION F(X)=0 BY NEWTON'S METHOD C C USES D.P. FUNCTIONS F AND DF C C INPUT: C F: FUNCTION C DF: DERIVATIVE OF F C M: ROOT MULTIPLICITY PARAMETER (NORMALLY M=1) C NIT: MAXIMUM NUMBER OF ITERATIONS C EPS: CONVERGENCE TEST NUMBER C X: INITIAL VALUE C C OUTPUT: C X: APPROXIMATE ROOT C FX:=F(X) C KIT: NUMBER OF EXECUTED ITERATIONS C ITEST = 0, ZERO DERIVATIVE (DIVISION BY ZERO) C = 1, CONVERGENCE TEST SATISFIED C = 2, CONVERGENCE TEST NOT SATISFIED C DOUBLE PRECISION F,DF,EPS,X,FX,DFX,Y,EE EXTERNAL F,DF EE=1.D-99 ITEST=1 FX=F(X) C DO 10 K=1,NIT KIT=K Y=X DFX=DF(X) IF(DABS(DFX).LT.EE) GO TO 20 X=X-DFLOAT(M)*FX/DFX FX=F(X) WRITE(6,*) 'K=',K,' X=',X,' FX=',FX IF(DABS(X-Y).LE.EPS) RETURN 10 CONTINUE 5 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ ITEST=2 RETURN 20 ITEST=0 RETURN END C----------------------------------------------------------------DOUBLE PRECISION FUNCTION F(X) DOUBLE PRECISION X F=X-5.D-1*DCOS(X) RETURN END C----------------------------------------------------------------DOUBLE PRECISION FUNCTION DF(X) DOUBLE PRECISION X DF=1.D0+5.D-1*DSIN(X) RETURN END ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------GIVE A,B,NIT,EPS 0 1 50 10E-10 INPUT: A= 0.000000000000000E+000 B= 1.00000000000000 NIT= 50 EPS= 1.000000000000000E-009 OUTPUT: K= 0 X= 0.500000000000000 K= 1 X= 0.250000000000000 K= 2 X= 0.375000000000000 K= 3 X= 0.437500000000000 K= 4 X= 0.468750000000000 K= 5 X= 0.453125000000000 K= 6 X= 0.445312500000000 K= 7 X= 0.449218750000000 K= 8 X= 0.451171875000000 K= 9 X= 0.450195312500000 K= 10 X= 0.449707031250000 K= 11 X= 0.449951171875000 K= 12 X= 0.450073242187500 K= 13 X= 0.450134277343750 K= 14 X= 0.450164794921875 K= 15 X= 0.450180053710938 K= 16 X= 0.450187683105469 K= 17 X= 0.450183868408203 K= 18 X= 0.450181961059570 K= 19 X= 0.450182914733887 K= 20 X= 0.450183391571045 K= 21 X= 0.450183629989624 K= 22 X= 0.450183510780334 K= 23 X= 0.450183570384979 K= 24 X= 0.450183600187302 K= 25 X= 0.450183615088463 K= 26 X= 0.450183607637882 K= 27 X= 0.450183611363173 FX= 6.120871905481362E-002 FX= -0.234456210855322 FX= -9.025381095615714E-002 FX= -1.540684171296819E-002 FX= 2.268315031650281E-002 FX= 3.583279719930776E-003 FX= -5.925551618970737E-003 FX= -1.174572173499699E-003 FX= 1.203495526914955E-003 FX= 1.424701366475878E-005 FX= -5.802162583392612E-004 FX= -2.829980403610821E-004 FX= -1.343788676563262E-004 FX= -6.006676554809465E-005 FX= -2.291008557664398E-005 FX= -4.331588364325789E-006 FX= 4.957699548169270E-006 FX= 3.130523164029952E-007 FX= -2.009268842806389E-006 FX= -8.481084679545781E-007 FX= -2.675281269293173E-007 FX= 2.276208194151863E-008 FX= -1.223830257135461E-007 FX= -4.981047269092542E-008 FX= -1.352419554123685E-008 FX= 4.618943116874163E-009 FX= -4.452626212181343E-009 FX= 8.315842459083456E-011 6 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= 28 X= 0.450183609500527 FX= -2.184733893795254E-009 K= 29 X= 0.450183610431850 FX= -1.050787734602210E-009 K= 30 X= 0.450183610897511 FX= -4.838146550056877E-010 K= 31 X= 0.450183611130342 FX= -2.003280874518509E-010 K= 32 X= 0.450183611246757 FX= -5.858480367493257E-011 ITEST= 1 KIT= 32 X= 0.450183611246757 FX= -5.858480367493257E-011 E= 1.164153218269348E-010 BISEC HAS FINISHED!!! GIVE M,NIT,EPS AND X5 OF BISEC FOR X 1 40 10E-10 0.453125 INPUT: M= 1 NIT= 40 EPS= 1.000000000000000E-009 X= 0.453125000000000 OUTPUT: K= 1 X= 0.450185207496634 FX= 1.943480655297591E-006 K= 2 X= 0.450183611295345 FX= 5.735412145213559E-013 K= 3 X= 0.450183611294874 FX= 0.000000000000000E+000 ITEST= 1 KIT= 3 X= 0.450183611294874 FX= 0.000000000000000E+000 QUANEW HAS FINISHED!!! GIVE NIT, EPS,Y=0.5,X=0.25 40 10E-10 0.5 0.25 INPUT: NIT= 40 EPS= 1.000000000000000E-009 Y= 0.500000000000000 X= 0.250000000000000 OUTPUT: K= 1 X= 0.448244860260044 FX= -2.359709907393370E-003 K= 2 X= 0.450260402756096 FX= 9.349995579999160E-005 K= 3 X= 0.450183583753726 FX= -3.353314903531412E-008 K= 4 X= 0.450183611294483 FX= -4.760636329592671E-013 K= 5 X= 0.450183611294874 FX= 5.551115123125783E-017 ITEST= 1 KIT= 5 X= 0.450183611294874 FX= 5.551115123125783E-017 SECAN HAS FINISHED GIVE M,NIT,EPS,X=0.5 1 40 10E-10 0.5 INPUT: M= 1 NIT= 40 EPS= 1.000000000000000E-009 X= 0.500000000000000 OUTPUT: K= 1 X= 0.450626693077243 K= 2 X= 0.450183647577775 K= 3 X= 0.450183611294874 FX= 5.395252478607726E-004 FX= 4.417680643520328E-008 FX= 3.330669073875470E-016 7 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= 4 X= 0.450183611294874 FX= 0.000000000000000E+000 ITEST= 1 KIT= 4 X= 0.450183611294874 FX= 0.000000000000000E+000 NEWTO HAS FINISHED.THANK YOU!!! ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Καταρχήν, πρέπει να κάνουμε ορισμένες επισημάνσεις για τον κώδικα. Το πρώτο, το τρίτο και το τέταρτο υποπρόγραμμα είναι ακριβώς τα ίδια με τα αντίστοιχα υποπρογράμματα της δισκέτας του βιβλίου, με τη διαφορά ότι στο πρώτο χρησιμοποιήσαμε διαφορετικό κριτήριο σταματήματος. Πιο συγκεκριμένα, επειδή εζητείτο η μέθοδος της διχοτόμησης να σταματά μέχρι να ισχύει η xk − x ≤ 10−10 χρησιμοποιήσαμε τον τύπο της θεωρίας: ανισότητα b − ak b −a xk − x ≤ k = ... = 0 k +1 0 . Επίσης μια επεξήγηση που πρέπει να δοθεί στο 2 2 δετερο υποπρόγραμμα είναι ο τύπος IF ((K-((K/3)*3)).EQ.1) DFX=DF(X). Με αυτή τη συνθήκη ελέγχω αν το ακέραιο υπόλοιπο (K-((K/3)*3) της διαίρεσης του Κ που είναι ο αριθμός των επαναλήψεων με το τρία είναι ίσο με τη μονάδα και μόνο τότε αλλάζω την παράγωγο f ΄(x). Οι παρατηρήσεις που πρέπει να γίνουν επί των αποτελεσμάτων είναι ότι η πρώτη μέθοδος συγκλίνει πολύ πιο αργά από τις υπόλοιπες. Όπως φαίνεται και από το OUTPUT η πρώτη συγκλίνει στη ρίζα για Κ=32, γεγονός που την καθιστά πιο αργή συγκρινόμενη με τις άλλες. Ώστοσο, όπως έχει αναφερθεί και στις διαλέξεις μπορεί να είναι ‘κουτή’ αλλά είναι σίγουρη. Η δεύτερη μέθοδος συκλίνει στη ρίζα για Κ=3 έχοντας όμως πάρει σαν αρχική τιμή την x5 της διχοτόμησης. Η τρίτη συγκλίνει στη ρίζα για Κ=5 γεγονός που την κατατάσει στην τρίτη πιο γρήγορη μέθοδο. Τέλος, η τέταρτη τερματίζεται για Κ=4 και είναι μαζί με την δεύτερη μέθοδο που αποτελεί παραλλαγή της η πιο γρήγορη από τις υπόλοιπες. 2. Ακολουθεί ο κώδικας του δεύτερου προγράμματος, το πρόγραμμα εκτελεσμένο και ορισμένες επεξηγήσεις επί του κώδικα καθώς και ορισμένα σχόλια. C C C C PROGRAM NEWTC COMPLEX NEWTON METHOD DIMENSION MATR(-10:10,-10:10),A(10) COMPLEX*16 F,DF,Z,FZ,A INTEGER K,L,S 8 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ CHARACTER MATR REAL H DOUBLE PRECISION EPS,EP1 EXTERNAL F,DF WRITE(6,*) 'GIVE M,NIT,EPS,EP1. I USE C=4,N=10' READ(5,*) M,NIT,EPS,EP1 WRITE(6,*) 'INPUT:' WRITE(6,*) 'M=',M,' NIT=',NIT,' EPS=',EPS WRITE(6,*)'EP1=',EP1 WRITE(6,*) 'OUTPUT:' H=0.4 L=0 DO 10 J=-10,10 DO 30 I=-10,10 Z =(DCMPLX(H*I,H*J)) CALL NEWTCOM(F,DF,M,NIT,EPS,Z,FZ,KIT,ITEST) IF (ITEST.EQ.1) THEN A(10)=Z K =0 DO 20 S=1,9 IF (CDABS(A(S)-A(10)).LE.EP1) K=K+1 20 CONTINUE IF(K.EQ.0)THEN L=L+1 A(L) =Z END IF IF (CDABS(Z-A(1)).LE.EP1) THEN MATR(I,J) ='1' ELSE IF (CDABS(Z-A(2)).LE.EP1) THEN MATR(I,J)='2' ELSE IF (CDABS(Z-A(3)).LE.EP1) THEN MATR(I,J)='3' ELSE IF (CDABS(Z-A(4)).LE.EP1) THEN MATR(I,J)='4' END IF ELSE IF(ITEST.EQ.0.OR.ITEST.EQ.2)THEN MATR(I,J)='*' END IF 30 CONTINUE 10 CONTINUE DO 50 J=-10,10 50 WRITE(6,60)(MATR(I,-J),I=-10,10) 60 FORMAT(21(A2)) DO 80 S=1,4 80 WRITE(6,*)'ROOT',S,A(S) STOP END C----------------------------------------------------------------SUBROUTINE NEWTCOM(F,DF,M,NIT,EPS,Z,FZ,KIT,ITEST) C C SOLUTION OF THE COMPLEX EQUATION F(Z)=0 BY NEWTON'S METHOD C C USES COMPLEX*16 FUNCTIONS F AND DF C C INPUT: C F: COMPLEX FUNCTION C DF: COMPLEX DERIVATIVE OF F 9 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C C C C C C C C C C C C C M: ROOT MULTIPLICITY PARAMETER NIT: MAXIMUM NUMBER OF ITERATIONS EPS: CONVERGENCE TEST NUMBER Z: INITIAL COMPLEX NUMBER OUTPUT: Z: APPROXIMATE ROOT FZ:=F(Z) KIT: NUMBER OF EXECUTED ITERATIONS ITEST = 0, ZERO DERIVATIVE (DIVISION BY ZERO) = 1, CONVERGENCE TEST SATISFIED = 2, CONVERGENCE TEST NOT SATISFIED COMPLEX*16 F,DF,Z,Y,FZ,DFZ DOUBLE PRECISION EPS,EE,EP1 EXTERNAL F,DF EE=1.D-99 ITEST=1 C DO 10 K=1,NIT KIT=K FZ=F(Z) DFZ=DF(Z) IF(CDABS(DFZ).LE.EE) GO TO 20 Y=Z Z=Z-DFLOAT(M)*FZ/DFZ IF(CDABS(Z-Y).LE.EPS)RETURN 10 CONTINUE ITEST=2 RETURN 20 ITEST=0 RETURN END C----------------------------------------------------------------COMPLEX*16 FUNCTION F(Z) COMPLEX*16 Z F=Z*(Z*(Z**2+4.D0*Z+4.D0)+5.D0)+18.D0 RETURN END C----------------------------------------------------------------COMPLEX*16 FUNCTION DF(Z) COMPLEX*16 Z DF=4.D0*Z*(Z**2+3.D0*Z+2.D0)+5.D0 RETURN END ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------GIVE M,NIT,EPS,EP1. I USE C=4,N=10 1 30 10E-10 10E-6 INPUT: M= 1 NIT= 30 EPS= 1.000000000000000E-009 10 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ EP1= 1.000000000000000E-005 OUTPUT: 222222124333333333333 222222111333333333333 222222221333333333333 222222114333333333333 222222212333333333333 222222214333333333333 222222211333333333333 222222212133333333333 2222223114333333333*3 222222211121124112212 ********************* 111111122212213221121 1111114223444444444*4 111111121244444444444 111111122444444444444 111111123444444444444 111111121444444444444 111111223444444444444 111111112444444444444 111111222444444444444 111111213444444444444 ROOT 1 (-2.55357472917640,-0.780383525820651) ROOT 2 (-2.55357472917640,0.780383525820650) ROOT 3 (0.553574729176397,1.48935906714894) ROOT 4 (0.553574729176397,-1.48935906714894) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Όσον αφορά τον κώδικα χρησιμοποίησα δυο πίνακες, ένα πίνακα χαρακτήρων 21x21 που τον χρειάστηκα για το τύπωμα του πίνακα μου, και ένα πίνακα μιγαδικών στον οποίο τύπωνα κάθε νέα ρίζα που έβρισκα σύμφωνα με τα ζητούμενα από την άσκηση κριτήρια για την εύρεση της νέας ρίζας. Επίσης μια παρατήρηση επί των αποτελεσμάτων είναι ότι εφόσον δεν υπάρχουν πραγματικές ρίζες ο άξονας χ΄χ (άξονας των πραγματικών) θα πρέπει να είναι γεμάτος αστεράκια μιας και κανένα σημείο του δεν συγκλίνει σε κάποια ρίζα, πράγμα το οποίο επαληθεύεται από τον πίνακα που πήραμε. Για επαλήθευση των ριζών που βρέθηκαν χρησιμοποίησα το Mathematica τα αποτελέσματα του οποίου ακολουθούν. N*1 Sz o ^ l 2 v + e 5 z* z ^z 4+ + 48 * z 0 ^ , 3 + 4 z- 0 - 34 ®0 z + δ - 3 4. 0 2 8 ®0 . 3 28 5 8 53 5 δ 5 7 ,5 , . 7 58 7 . 7 z- 90 4 ®1 z + δ 0 4 ®1 .8 . 8 5 3 59 5 6 53 3 δ 5 ,7 7 5 . 36 5 5 . P^ 0 l2 o+, t5 zz ^+ 41 +*4 , * z z ^, 3 1 + 40 * 8 z1 11 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ 3000 2500 2000 1500 1000 500 -10 -5 …i G c r s a … p h 5 10 Στο τέλος της εργασίας βρίσκονται και δύο γραφικές παραστάσεις του πίνακα που έγιναν χρησιμοποιώντας το ArrayVisualizer. 3. Ακολουθεί ο κώδικας της τρίτης άσκησης, το προγραμμα εκτελεσμένο και στη συνέχεια κάποια σχόλια επί των αποτελεσμάτων. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------C C C C PROGRAM ASKHSH3 REGULA FALSI METHOD DOUBLE PRECISION F,A,B,EPS,X,FX,E1,E2 EXTERNAL F OPEN (UNIT=6,FILE='ASKHSH3.doc') READ(5,*) NIT EPS=1D-10 DO I=1,20 A=0.5*(I-1) B=0.5*I WRITE(6,*) 'INPUT:' WRITE(6,*) 'A=',A,' B=',B WRITE(6,*) 'NIT=',NIT,' EPS=',EPS WRITE(6,*) ' ' WRITE(6,*) 'OUTPUT:' CALL RFALSI(F,A,B,NIT,EPS,X,FX,E1,E2,KIT,ITEST) WRITE(6,*) 'ITEST=',ITEST IF(ITEST.EQ.1) WRITE(6,*) 'X=',X, ‘KIT=’,KIT END DO STOP END C----------------------------------------------------------------SUBROUTINE RFALSI(F,A,B,NIT,EPS,X,FX,E1,E2,KIT,ITEST) C C SOLUTION OF THE EQUATION F(X)=0 BY THE REGULA FALSI METHOD C USES D.P. FUNCTION F C C INPUT: C F: FUNCTION C [A,B]: SEARCH INTERVAL C NIT: MAXIMUM NUMBER OF ITERATIONS C EPS: ERROR TEST NUMBER 12 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C C C C C C C C C C OUTPUT: X: APPROXIMATE ROOT FX:=F(X) E1,E2: ERROR BOUND KIT: NUMBER OF EXECUTED ITERATIONS ITEST = 0, F(A)F(B)>0 = 1, ERROR TEST SATISFIED = 2, ERROR TEST NOT SATISFIED DOUBLE PRECISION F,A,B,EPS,X,FX,FA,FB,E1,E2,P,FE1,FE2 EXTERNAL F ITEST=1 FA=F(A) FB=F(B) P=FA*FB IF(P.GT.0.D0) GO TO 40 DO 10 K=0,NIT KIT=K X=(((A*FB)-(B*FA))/(FB-FA)) FX=F(X) WRITE(6,*) 'K=',K,' X=',X,' FX=',FX P=FA*FX IF(P.LE.0.D0) GO TO 20 A=X FA=FX GO TO 30 20 B=X FB=FX 30 E1=X-EPS E2=X+EPS FE1=F(E1) FE2=F(E2) IF((FE1*FE2).LE.0) RETURN 10 CONTINUE ITEST=2 RETURN 40 ITEST=0 RETURN END C----------------------------------------------------------------DOUBLE PRECISION FUNCTION F(X) DOUBLE PRECISION X F=DSIN(X)+(X/5.D0)-1 RETURN END ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- INPUT: A= 0.000000000000000E+000 B= 0.500000000000000 13 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ NIT= 60 EPS= OUTPUT: ITEST= 0 INPUT: A= 0.500000000000000 NIT= 60 EPS= 1.000000000000000E-010 B= 1.00000000000000 1.000000000000000E-010 OUTPUT: K= 0 X= 0.955122396347984 FX= 7.408910803086366E-003 K= 1 X= 0.947243676149803 FX= 1.257845189267526E-003 K= 2 X= 0.945910057795558 FX= 2.116695177227790E-004 K= 3 X= 0.945685750101109 FX= 3.556644567925815E-005 K= 4 X= 0.945648063272522 FX= 5.974664097463389E-006 K= 5 X= 0.945641732503767 FX= 1.003617415396008E-006 K= 6 X= 0.945640669070826 FX= 1.685853394661763E-007 K= 7 X= 0.945640490437884 FX= 2.831854284401913E-008 K= 8 X= 0.945640460431572 FX= 4.756876892386686E-009 K= 9 X= 0.945640455391188 FX= 7.990481609709832E-010 K= 10 X= 0.945640454544517 FX= 1.342219668742928E-010 K 11 X= 0.945640454402295 FX= 2.254640918408768E-011 ITEST= 1 X= 0.945640454402295 KIT= 11 INPUT: A= 1.00000000000000 B= 1.50000000000000 NIT= 60 EPS= 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 1.50000000000000 NIT= 60 EPS= B= 2.00000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 2.00000000000000 NIT= 60 EPS= B= 2.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 2.50000000000000 NIT= 60 EPS= B= 3.00000000000000 1.000000000000000E-010 OUTPUT: K= 0 X= 2.63778026513854 FX= 1.032378509690268E-002 K= 1 X= 2.65167115010949 FX= 8.908566845837740E-004 K= 2 X= 2.65286570710463 FX= 7.539158511771227E-005 K= 3 X= 2.65296677086409 FX= 6.369585407117739E-006 K= 4 X= 2.65297530919642 FX= 5.380690015766021E-007 K= 5 X= 2.65297603046823 FX= 4.545268961209104E-008 K= 6 X= 2.65297609139673 FX= 3.839554008067125E-009 K= 7 X= 2.65297609654359 FX= 3.243407764585982E-010 K= 8 X= 2.65297609697836 FX= 2.739808380169961E-011 ITEST= 1 X= 2.65297609697836 KIT= 8 INPUT: A= 3.00000000000000 B= 3.50000000000000 NIT= 60 EPS= 1.000000000000000E-010 14 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ OUTPUT: ITEST= 0 INPUT: A= 3.50000000000000 NIT= 60 EPS= B= 4.00000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 4.00000000000000 NIT= 60 EPS= B= 4.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 4.50000000000000 NIT= 60 EPS= B= 5.00000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 5.00000000000000 NIT= 60 EPS= B= 5.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 5.50000000000000 NIT= 60 EPS= B= 6.00000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 6.00000000000000 NIT= 60 EPS= B= 6.50000000000000 1.000000000000000E-010 OUTPUT: K= 0 X= 6.06678785373681 FX= -1.354927814128493E-003 K= 1 X= 6.06792434878878 FX= -1.750146341250591E-005 K= 2 X= 6.06793902827990 FX= -2.248436422602751E-007 K= 3 X= 6.06793921686920 FX= -2.888393857425342E-009 K= 4 X= 6.06793921929187 FX= -3.710509677290474E-011 ITEST= 1 X= 6.06793921929187 KIT= 4 INPUT: A= 6.50000000000000 B= 7.00000000000000 NIT= 60 EPS= 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 7.00000000000000 NIT= 60 EPS= B= 7.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 7.50000000000000 NIT= 60 EPS= B= 8.00000000000000 1.000000000000000E-010 OUTPUT: 15 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ ITEST= 0 INPUT: A= 8.00000000000000 NIT= 60 EPS= B= 8.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 8.50000000000000 NIT= 60 EPS= B= 9.00000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 9.00000000000000 NIT= 60 EPS= B= 9.50000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 INPUT: A= 9.50000000000000 NIT= 60 EPS= B= 10.0000000000000 1.000000000000000E-010 OUTPUT: ITEST= 0 Παρατηρούμε λοιπόν ότι η συνάρτηση έχει τρεις ρίζες στα διαστήματα [0.5,1], [2.5,3], [6,6.5]. 4. Ακολουθεί ο κώδικας της τέταρτης άσκησης, το πρόγραμμα εκτελεσμένο και τέλος οι παρατηρήσεις μας και τα σχόλια μας επί των αποτελεσμάτων. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- C C PROGRAM ASKHSH4 DOUBLE PRECISION G,A,X,GX,NIT,C,EPS EXTERNAL G OPEN (UNIT=6,FILE='ASKHSH4.DOC') READ(5,*) A,NIT,EPS WRITE(6,*) 'Xo=',A,'','EPS=',EPS DO C=0.5,4,0.5 WRITE(6,*) ' ' WRITE(6,*) 'C=',C WRITE(6,*) ' ' CALL EPMETH(G,A,X,GX,NIT,C,ITEST,EPS) WRITE(6,*)'ITEST=',ITEST 16 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ ENDDO STOP END C----------------------------------------------------------------SUBROUTINE EPMETH(G,A,X,GX,NIT,C,ITEST,EPS) C DOUBLE PRECISION G,A,X,GX,NIT,C,Y,EPS EXTERNAL G X=A ITEST=1 DO 10 K=0,NIT Y=X GX=G(X,C) WRITE(6,*) 'K=',K,' ','X=',X,' ','GX=',GX X=GX IF(DABS(X-Y).LE.EPS) RETURN 10 CONTINUE ITEST=0 RETURN END C----------------------------------------------------------------DOUBLE PRECISION FUNCTION G(X,C) DOUBLE PRECISION X,C G=C*X*(1.D0-X) RETURN END ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Xo= 0.250000000000000 C= 0.500000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 EPS= 1.000000000000000E-009 0.250000000000000 9.375000000000000E-002 4.248046875000000E-002 2.033793926239014E-002 9.962153744474733E-003 4.931454618623090E-003 2.453567686983776E-003 1.223773846294582E-003 6.111381119338539E-004 3.053823110709979E-004 1.526445263575414E-004 7.631061300305726E-005 3.815239484670018E-005 1.907546962073382E-005 9.537552873596284E-006 4.768730954340733E-006 2.384354106772909E-006 1.192174210814201E-006 5.960863947674262E-007 2.980430197242181E-007 1.490214654472882E-007 7.451072161994552E-008 3.725535803404894E-008 1.862767832304362E-008 9.313838988026612E-009 4.656919450639507E-009 2.328459714476304E-009 1.164229854527290E-009 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 9.375000000000000E-002 4.248046875000000E-002 2.033793926239014E-002 9.962153744474733E-003 4.931454618623090E-003 2.453567686983776E-003 1.223773846294582E-003 6.111381119338539E-004 3.053823110709979E-004 1.526445263575414E-004 7.631061300305726E-005 3.815239484670018E-005 1.907546962073382E-005 9.537552873596284E-006 4.768730954340733E-006 2.384354106772909E-006 1.192174210814201E-006 5.960863947674262E-007 2.980430197242181E-007 1.490214654472882E-007 7.451072161994552E-008 3.725535803404894E-008 1.862767832304362E-008 9.313838988026612E-009 4.656919450639507E-009 2.328459714476304E-009 1.164229854527290E-009 5.821149265859293E-010 1.00000000000000 17 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 X= X= X= X= X= X= X= X= X= 0.250000000000000 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 7.508929631879595E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 7.508929631879595E-002 6.945089389714401E-002 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 6.945089389714401E-002 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 5.357062532685648E-002 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 2.244503406282446E-002 2.194125450874311E-002 2.145983585932567E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 5.357062532685648E-002 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 2.244503406282446E-002 2.194125450874311E-002 2.145983585932567E-002 2.099931130421647E-002 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0 2.099931130421647E-002 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 1.493325968842430E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 1.493325968842430E-002 1.471025744350238E-002 18 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ C= 1.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 0.250000000000000 0.281250000000000 0.303222656250000 0.316918015480042 0.324721480416351 0.328916160858846 0.331095479977086 0.332206894673744 0.332768210707458 0.333050292975022 0.333191692986411 0.333262483066890 0.333297900670471 0.333315615118692 0.333324473755110 0.333328903426483 0.333331118350472 0.333332225834544 0.333332779582099 0.333333056457256 0.333333194895180 0.333333264114228 0.333333298723773 0.333333316028552 0.333333324680942 0.333333329007138 0.333333331170235 0.333333332251784 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.281250000000000 0.303222656250000 0.316918015480042 0.324721480416351 0.328916160858846 0.331095479977086 0.332206894673744 0.332768210707458 0.333050292975022 0.333191692986411 0.333262483066890 0.333297900670471 0.333315615118692 0.333324473755110 0.333328903426483 0.333331118350472 0.333332225834544 0.333332779582099 0.333333056457256 0.333333194895180 0.333333264114228 0.333333298723773 0.333333316028552 0.333333324680942 0.333333329007138 0.333333331170235 0.333333332251784 0.333333332792559 GX= GX= GX= GX= GX= GX= 0.375000000000000 0.468750000000000 0.498046875000000 0.499992370605469 0.499999999883585 0.500000000000000 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.468750000000000 0.622558593750000 0.587448477745056 0.605881909350181 0.596972553180900 0.601490809823944 0.599249038803200 0.600374070741602 0.599812614806900 0.600093604813524 0.599953175688585 0.600023406674417 0.599988295293110 0.600005852010944 0.599997073908913 0.600001463024139 0.599999268482580 0.600000365757372 0.599999817120979 0.600000091439427 0.599999954280266 0.600000022859862 0.599999988570068 2.00000000000000 K= K= K= K= K= K= ITEST= 0 1 2 3 4 5 X= X= X= X= X= X= 1 0.250000000000000 0.375000000000000 0.468750000000000 0.498046875000000 0.499992370605469 0.499999999883585 C= 2.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.250000000000000 0.468750000000000 0.622558593750000 0.587448477745056 0.605881909350181 0.596972553180900 0.601490809823944 0.599249038803200 0.600374070741602 0.599812614806900 0.600093604813524 0.599953175688585 0.600023406674417 0.599988295293110 0.600005852010944 0.599997073908913 0.600001463024139 0.599999268482580 0.600000365757372 0.599999817120979 0.600000091439427 0.599999954280266 0.600000022859862 19 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= ITEST= 23 24 25 26 27 28 X= X= X= X= X= X= 1 0.599999988570068 0.600000005714966 0.599999997142517 0.600000001428741 0.599999999285629 0.600000000357185 GX= GX= GX= GX= GX= GX= 0.600000005714966 0.599999997142517 0.600000001428741 0.599999999285629 0.600000000357185 0.599999999821407 C= 3.00000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.250000000000000 0.562500000000000 0.738281250000000 0.579666137695313 0.730959919514135 0.589972546734074 0.725714822502555 0.597158456707920 0.721680702870406 0.602572997924649 0.718436340290250 0.606856695721807 0.715744939738252 0.610362362932014 0.713460446544187 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.562500000000000 0.738281250000000 0.579666137695313 0.730959919514135 0.589972546734074 0.725714822502555 0.597158456707920 0.721680702870406 0.602572997924649 0.718436340290250 0.606856695721807 0.715744939738252 0.610362362932014 0.713460446544187 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 0.705612995493528 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.705612995493528 0.623169888252535 0.704487535883574 0.624554543004792 0.703458497450602 0.625813919445430 0.702512573021337 0.626965973304834 0.701638924868269 0.628025231933906 0.700828619964809 0.629003596209091 0.700074216495365 0.629910923681297 0.699369455724817 0.630755460371971 0.698709028748742 0.631544165681195 0.698088397425715 0.632282960415936 0.697503655150788 0.632976918606236 0.696951417353971 0.633630417606786 0.696428734470708 0.634247256822708 0.695933022106734 0.634830752544366 0.695462004504959 0.635383814384710 0.695013668407940 0.635909007402235 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.623169888252535 0.704487535883574 0.624554543004792 0.703458497450602 0.625813919445430 0.702512573021337 0.626965973304834 0.701638924868269 0.628025231933906 0.700828619964809 0.629003596209091 0.700074216495365 0.629910923681297 0.699369455724817 0.630755460371971 0.698709028748742 0.631544165681195 0.698088397425715 0.632282960415936 0.697503655150788 0.632976918606236 0.696951417353971 0.633630417606786 0.696428734470708 0.634247256822708 0.695933022106734 0.634830752544366 0.695462004504959 0.635383814384710 0.695013668407940 0.635909007402235 0.694586225120818 20 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= ITEST= 56 57 58 59 60 X= X= X= X= X= 0 0.694586225120818 0.636408602979691 0.694178079099387 0.636884620791817 0.693787801772042 C= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 0.636408602979691 0.694178079099387 0.636884620791817 0.693787801772042 0.637338863653079 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.656250000000000 0.789550781250000 0.581561207771301 0.851717192854103 0.442032556877904 0.863239214382603 0.413200455971483 0.848630437047546 0.449598864274130 0.866109039311399 0.405874599670697 0.843991431544969 0.460844632582750 0.869634000208470 0.396797470614596 0.837722332749592 0.475802690867609 0.872950715807630 0.388177172525017 0.831234793394251 0.490992290757739 0.874716014109724 0.383557680694024 0.827544151961258 0.499501899805932 0.874999131636688 0.382814779451054 0.826936684297132 0.500893415612791 0.874997206329900 0.382819833356697 0.826940829909067 0.500883928084298 0.874997265348996 0.382819678432711 0.826940702830853 0.500884218913637 0.874997263549195 0.382819683157156 0.826940706706137 0.500884210044721 0.874997263604089 0.382819683013059 0.826940706587940 0.500884210315224 0.874997263602415 0.382819683017454 0.826940706591545 0.500884210306974 0.874997263602466 0.382819683017320 0.826940706591435 0.500884210307225 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 3.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= GX= GX= GX= GX= GX= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.250000000000000 0.656250000000000 0.789550781250000 0.581561207771301 0.851717192854103 0.442032556877904 0.863239214382603 0.413200455971483 0.848630437047546 0.449598864274130 0.866109039311399 0.405874599670697 0.843991431544969 0.460844632582750 0.869634000208470 0.396797470614596 0.837722332749592 0.475802690867609 0.872950715807630 0.388177172525017 0.831234793394251 0.490992290757739 0.874716014109724 0.383557680694024 0.827544151961258 0.499501899805932 0.874999131636688 0.382814779451054 0.826936684297132 0.500893415612791 0.874997206329900 0.382819833356697 0.826940829909067 0.500883928084298 0.874997265348996 0.382819678432711 0.826940702830853 0.500884218913637 0.874997263549195 0.382819683157156 0.826940706706137 0.500884210044721 0.874997263604089 0.382819683013059 0.826940706587940 0.500884210315224 0.874997263602415 0.382819683017454 0.826940706591545 0.500884210306974 0.874997263602466 0.382819683017320 0.826940706591435 0.500884210307225 0.874997263602464 0.382819683017324 0.826940706591439 21 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= ITEST= C= 57 58 59 60 X= X= X= X= 0 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 GX= GX= GX= GX= 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 GX= GX= 0.750000000000000 0.750000000000000 4.00000000000000 K= K= ITEST= 0 X= 0.250000000000000 1 X= 0.750000000000000 1 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Xo= 0.500000000000000 C= 0.500000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 EPS= 1.000000000000000E-009 0.500000000000000 0.125000000000000 5.468750000000000E-002 2.584838867187500E-002 1.259012473747134E-002 6.215806748283127E-003 3.088585247375562E-003 1.539522944272628E-003 7.685764066883431E-004 3.839928484977126E-004 1.919226989950076E-004 9.594293233630902E-005 4.796686364502187E-005 2.398228141250696E-005 1.199085313134261E-005 5.995354675391894E-006 2.997659365557105E-006 1.498825189797717E-006 7.494114716603835E-007 3.747054550214148E-007 1.873526573086184E-007 9.367631110380010E-008 4.683815116427442E-008 2.341907448523101E-008 1.170953696838898E-008 5.854768415637862E-009 2.927384190679774E-009 1.463692091055098E-009 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.125000000000000 5.468750000000000E-002 2.584838867187500E-002 1.259012473747134E-002 6.215806748283127E-003 3.088585247375562E-003 1.539522944272628E-003 7.685764066883431E-004 3.839928484977126E-004 1.919226989950076E-004 9.594293233630902E-005 4.796686364502187E-005 2.398228141250696E-005 1.199085313134261E-005 5.995354675391894E-006 2.997659365557105E-006 1.498825189797717E-006 7.494114716603835E-007 3.747054550214148E-007 1.873526573086184E-007 9.367631110380010E-008 4.683815116427442E-008 2.341907448523101E-008 1.170953696838898E-008 5.854768415637862E-009 2.927384190679774E-009 1.463692091055098E-009 7.318460444563517E-010 C= 1.00000000000000 K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 X= X= X= X= X= X= X= X= X= 0.500000000000000 0.250000000000000 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.250000000000000 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 7.508929631879595E-002 K= K= K= K= K= 9 10 11 12 13 X= X= X= X= X= 7.508929631879595E-002 6.945089389714401E-002 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 GX= GX= GX= GX= GX= 6.945089389714401E-002 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 5.357062532685648E-002 22 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 5.357062532685648E-002 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 2.244503406282446E-002 2.194125450874311E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 2.244503406282446E-002 2.194125450874311E-002 2.145983585932567E-002 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0 2.145983585932567E-002 2.099931130421647E-002 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 2.099931130421647E-002 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 1.493325968842430E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.375000000000000 0.351562500000000 0.341949462890625 0.337530041579157 0.335405268916094 0.334362861749125 0.333846507648091 0.333589525468896 0.333461330949499 0.333397307566332 0.333365314310779 0.333349322287882 0.333341327427137 0.333337330284377 C= 1.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.500000000000000 0.375000000000000 0.351562500000000 0.341949462890625 0.337530041579157 0.335405268916094 0.334362861749125 0.333846507648091 0.333589525468896 0.333461330949499 0.333397307566332 0.333365314310779 0.333349322287882 0.333341327427137 23 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.333335331784892 0.333334332553122 0.333333832941730 0.333333583137157 0.333333458235152 0.333333395784219 0.333333364558770 0.333333348946050 0.333333341139691 0.333333337236512 0.333333335284923 0.333333334309128 GX= 0.500000000000000 0.333337330284377 0.333335331784892 0.333334332553122 0.333333832941730 0.333333583137157 0.333333458235152 0.333333395784219 0.333333364558770 0.333333348946050 0.333333341139691 0.333333337236512 0.333333335284923 0.500000000000000 0.625000000000000 0.585937500000000 0.606536865234375 0.596624740865082 0.601659148631890 0.599163543748598 0.600416478978050 0.599791326874127 0.600104227701753 0.599947858990589 0.600026063707993 0.599986966447711 0.600006516351461 0.599996741718113 0.600001629114403 0.599999185436164 0.600000407280259 0.599999796359456 0.600000101820168 0.599999949089890 0.600000025455049 0.599999987272474 0.600000006363762 0.599999996818119 0.600000001590941 0.599999999204530 0.600000000397735 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.625000000000000 0.585937500000000 0.606536865234375 0.596624740865082 0.601659148631890 0.599163543748598 0.600416478978050 0.599791326874127 0.600104227701753 0.599947858990589 0.600026063707993 0.599986966447711 0.600006516351461 0.599996741718113 0.600001629114403 0.599999185436164 0.600000407280259 0.599999796359456 0.600000101820168 0.599999949089890 0.600000025455049 0.599999987272474 0.600000006363762 0.599999996818119 0.600000001590941 0.599999999204530 0.600000000397735 0.599999999801132 3.00000000000000 0 X= 0.500000000000000 1 X= 0.750000000000000 2 X= 0.562500000000000 3 X= 0.738281250000000 4 X= 0.579666137695313 5 X= 0.730959919514135 6 X= 0.589972546734074 7 X= 0.725714822502555 8 X= 0.597158456707920 9 X= 0.721680702870406 10 X= 0.602572997924649 11 X= 0.718436340290250 12 X= 0.606856695721807 13 X= 0.715744939738252 14 X= 0.610362362932014 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.750000000000000 0.562500000000000 0.738281250000000 0.579666137695313 0.730959919514135 0.589972546734074 0.725714822502555 0.597158456707920 0.721680702870406 0.602572997924649 0.718436340290250 0.606856695721807 0.715744939738252 0.610362362932014 0.713460446544187 0 X= 0.500000000000000 1 2.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= X= X= X= X= X= X= X= X= X= X= X= X= 1 2.00000000000000 K= ITEST= C= 14 15 16 17 18 19 20 21 22 23 24 25 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 24 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.713460446544187 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 0.705612995493528 0.623169888252535 0.704487535883574 0.624554543004792 0.703458497450602 0.625813919445430 0.702512573021337 32 X= 0.626965973304834 33 X= 0.701638924868269 34 X= 0.628025231933906 35 X= 0.700828619964809 36 X= 0.629003596209091 37 X= 0.700074216495365 38 X= 0.629910923681297 39 X= 0.699369455724817 40 X= 0.630755460371971 41 X= 0.698709028748742 42 X= 0.631544165681195 43 X= 0.698088397425715 44 X= 0.632282960415936 45 X= 0.697503655150788 46 X= 0.632976918606236 47 X= 0.696951417353971 48 X= 0.633630417606786 49 X= 0.696428734470708 50 X= 0.634247256822708 51 X= 0.695933022106734 52 X= 0.634830752544366 53 X= 0.695462004504959 54 X= 0.635383814384710 55 X= 0.695013668407940 56 X= 0.635909007402235 57 X= 0.694586225120818 58 X= 0.636408602979691 59 X= 0.694178079099387 60 X= 0.636884620791817 0 C= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 0.705612995493528 0.623169888252535 0.704487535883574 0.624554543004792 0.703458497450602 0.625813919445430 0.702512573021337 0.626965973304834 GX= 0.701638924868269 GX= 0.628025231933906 GX= 0.700828619964809 GX= 0.629003596209091 GX= 0.700074216495365 GX= 0.629910923681297 GX= 0.699369455724817 GX= 0.630755460371971 GX= 0.698709028748742 GX= 0.631544165681195 GX= 0.698088397425715 GX= 0.632282960415936 GX= 0.697503655150788 GX= 0.632976918606236 GX= 0.696951417353971 GX= 0.633630417606786 GX= 0.696428734470708 GX= 0.634247256822708 GX= 0.695933022106734 GX= 0.634830752544366 GX= 0.695462004504959 GX= 0.635383814384710 GX= 0.695013668407940 GX= 0.635909007402235 GX= 0.694586225120818 GX= 0.636408602979691 GX= 0.694178079099387 GX= 0.636884620791817 GX= 0.693787801772042 3.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.500000000000000 0.875000000000000 0.382812500000000 0.826934814453125 0.500897694844753 0.874997179503880 0.382819903774472 0.826940887670016 0.500883795893397 0.874997266166866 0.382819676285819 0.826940701069839 0.500884222943868 0.874997263524249 0.382819683222636 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.875000000000000 0.382812500000000 0.826934814453125 0.500897694844753 0.874997179503880 0.382819903774472 0.826940887670016 0.500883795893397 0.874997266166866 0.382819676285819 0.826940701069839 0.500884222943868 0.874997263524249 0.382819683222636 0.826940706759849 25 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0 0.826940706759849 0.500884209921797 0.874997263604850 0.382819683011062 0.826940706586302 0.500884210318973 0.874997263602391 0.382819683017515 0.826940706591595 0.500884210306859 0.874997263602466 0.382819683017318 0.826940706591434 0.500884210307229 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.500884209921797 0.874997263604850 0.382819683011062 0.826940706586302 0.500884210318973 0.874997263602391 0.382819683017515 0.826940706591595 0.500884210306859 0.874997263602466 0.382819683017318 0.826940706591434 0.500884210307229 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 4.00000000000000 K= K= K= ITEST= 0 X= 0.500000000000000 GX= 1 X= 1.00000000000000 GX= 2 X= 0.000000000000000E+000 GX= 1 1.00000000000000 0.000000000000000E+000 0.000000000000000E+000 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Xo= 0.750000000000000 C= 0.500000000000000 K= K= K= K= K= 0 1 2 3 4 X= X= X= X= X= EPS= 1.000000000000000E-009 0.750000000000000 9.375000000000000E-002 4.248046875000000E-002 2.033793926239014E-002 9.962153744474733E-003 GX= GX= GX= GX= GX= 9.375000000000000E-002 4.248046875000000E-002 2.033793926239014E-002 9.962153744474733E-003 4.931454618623090E-003 26 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 4.931454618623090E-003 2.453567686983776E-003 1.223773846294582E-003 6.111381119338539E-004 3.053823110709979E-004 1.526445263575414E-004 7.631061300305726E-005 3.815239484670018E-005 1.907546962073382E-005 9.537552873596284E-006 4.768730954340733E-006 2.384354106772909E-006 1.192174210814201E-006 5.960863947674262E-007 2.980430197242181E-007 1.490214654472882E-007 7.451072161994552E-008 3.725535803404894E-008 1.862767832304362E-008 9.313838988026612E-009 4.656919450639507E-009 2.328459714476304E-009 1.164229854527290E-009 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 2.453567686983776E-003 1.223773846294582E-003 6.111381119338539E-004 3.053823110709979E-004 1.526445263575414E-004 7.631061300305726E-005 3.815239484670018E-005 1.907546962073382E-005 9.537552873596284E-006 4.768730954340733E-006 2.384354106772909E-006 1.192174210814201E-006 5.960863947674262E-007 2.980430197242181E-007 1.490214654472882E-007 7.451072161994552E-008 3.725535803404894E-008 1.862767832304362E-008 9.313838988026612E-009 4.656919450639507E-009 2.328459714476304E-009 1.164229854527290E-009 5.821149265859293E-010 C= 1.00000000000000 K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 X= X= X= X= X= X= X= X= X= 0.750000000000000 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 7.508929631879595E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.187500000000000 0.152343750000000 0.129135131835938 0.112459249561653 9.981216674968249E-002 8.984969811841606E-002 8.177672986644556E-002 7.508929631879595E-002 6.945089389714401E-002 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 6.945089389714401E-002 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 5.357062532685648E-002 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 6.462746723403166E-002 6.045075771294583E-002 5.679646360487655E-002 5.357062532685648E-002 5.070081342894604E-002 4.813024094658925E-002 4.581372085301251E-002 4.371482383461476E-002 4.180383801172335E-002 4.005627713921295E-002 3.845177180095951E-002 3.697323304632644E-002 3.560621308442848E-002 3.433841067421475E-002 3.315928422658373E-002 3.205974609616436E-002 3.103191877641383E-002 3.006893879346789E-002 2.916479771330257E-002 2.831421228764471E-002 2.751251767017490E-002 2.675557904162321E-002 2.603971803177066E-002 2.536165111659654E-002 2.471843776923658E-002 2.410743660348496E-002 2.352626810389391E-002 2.297278281299762E-002 2.244503406282446E-002 27 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= 38 X= 39 X= 40 X= 2.244503406282446E-002 GX= 2.194125450874311E-002 GX= 2.145983585932567E-002 GX= 2.194125450874311E-002 2.145983585932567E-002 2.099931130421647E-002 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 2.099931130421647E-002 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 1.493325968842430E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 2.055834022896508E-002 2.013569487599520E-002 1.973024866785602E-002 1.934096595536058E-002 1.896689299127416E-002 1.860714996153172E-002 1.826092393184079E-002 1.792746258899632E-002 1.760606867411645E-002 1.729609501995875E-002 1.699694011701931E-002 1.670804414367777E-002 1.642888540457068E-002 1.615897712893416E-002 1.589786458708075E-002 1.564512248865159E-002 1.540035263096668E-002 1.516318176980855E-002 1.493325968842430E-002 1.471025744350238E-002 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.281250000000000 0.303222656250000 0.316918015480042 0.324721480416351 0.328916160858846 0.331095479977086 0.332206894673744 0.332768210707458 0.333050292975022 0.333191692986411 0.333262483066890 0.333297900670471 0.333315615118692 0.333324473755110 0.333328903426483 0.333331118350472 0.333332225834544 0.333332779582099 0.333333056457256 0.333333194895180 0.333333264114228 0.333333298723773 0.333333316028552 0.333333324680942 0.333333329007138 0.333333331170235 0.333333332251784 0.333333332792559 GX= GX= GX= GX= GX= GX= 0.375000000000000 0.468750000000000 0.498046875000000 0.499992370605469 0.499999999883585 0.500000000000000 C= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0 1.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 0.750000000000000 0.281250000000000 0.303222656250000 0.316918015480042 0.324721480416351 0.328916160858846 0.331095479977086 0.332206894673744 0.332768210707458 0.333050292975022 0.333191692986411 0.333262483066890 0.333297900670471 0.333315615118692 0.333324473755110 0.333328903426483 0.333331118350472 0.333332225834544 0.333332779582099 0.333333056457256 0.333333194895180 0.333333264114228 0.333333298723773 0.333333316028552 0.333333324680942 0.333333329007138 0.333333331170235 0.333333332251784 C= 2.00000000000000 K= K= K= K= K= K= 0 1 2 3 4 5 X= X= X= X= X= X= 0.750000000000000 0.375000000000000 0.468750000000000 0.498046875000000 0.499992370605469 0.499999999883585 28 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ ITEST= C= 1 2.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 1 0.750000000000000 0.468750000000000 0.622558593750000 0.587448477745056 0.605881909350181 0.596972553180900 0.601490809823944 0.599249038803200 0.600374070741602 0.599812614806900 0.600093604813524 0.599953175688585 0.600023406674417 0.599988295293110 0.600005852010944 0.599997073908913 0.600001463024139 0.599999268482580 0.600000365757372 0.599999817120979 0.600000091439427 0.599999954280266 0.600000022859862 0.599999988570068 0.600000005714966 0.599999997142517 0.600000001428741 0.599999999285629 0.600000000357185 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.468750000000000 0.622558593750000 0.587448477745056 0.605881909350181 0.596972553180900 0.601490809823944 0.599249038803200 0.600374070741602 0.599812614806900 0.600093604813524 0.599953175688585 0.600023406674417 0.599988295293110 0.600005852010944 0.599997073908913 0.600001463024139 0.599999268482580 0.600000365757372 0.599999817120979 0.600000091439427 0.599999954280266 0.600000022859862 0.599999988570068 0.600000005714966 0.599999997142517 0.600000001428741 0.599999999285629 0.600000000357185 0.599999999821407 C= 3.00000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.750000000000000 0.562500000000000 0.738281250000000 0.579666137695313 0.730959919514135 0.589972546734074 0.725714822502555 0.597158456707920 0.721680702870406 0.602572997924649 0.718436340290250 0.606856695721807 0.715744939738252 0.610362362932014 0.713460446544187 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.562500000000000 0.738281250000000 0.579666137695313 0.730959919514135 0.589972546734074 0.725714822502555 0.597158456707920 0.721680702870406 0.602572997924649 0.718436340290250 0.606856695721807 0.715744939738252 0.610362362932014 0.713460446544187 0.613303913283469 0.711486669703957 0.615820165612589 0.709757067712418 0.618005917634065 0.708223810210027 0.619928534584856 0.706851439776987 0.621637445586563 0.705612995493528 K= K= K= K= 24 25 26 27 X= X= X= X= 0.705612995493528 0.623169888252535 0.704487535883574 0.624554543004792 GX= GX= GX= GX= 0.623169888252535 0.704487535883574 0.624554543004792 0.703458497450602 29 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.703458497450602 0.625813919445430 0.702512573021337 0.626965973304834 0.701638924868269 0.628025231933906 0.700828619964809 0.629003596209091 0.700074216495365 0.629910923681297 0.699369455724817 0.630755460371971 0.698709028748742 0.631544165681195 0.698088397425715 0.632282960415936 0.697503655150788 0.632976918606236 0.696951417353971 0.633630417606786 0.696428734470708 0.634247256822708 0.695933022106734 0.634830752544366 0.695462004504959 0.635383814384710 0.695013668407940 0.635909007402235 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.625813919445430 0.702512573021337 0.626965973304834 0.701638924868269 0.628025231933906 0.700828619964809 0.629003596209091 0.700074216495365 0.629910923681297 0.699369455724817 0.630755460371971 0.698709028748742 0.631544165681195 0.698088397425715 0.632282960415936 0.697503655150788 0.632976918606236 0.696951417353971 0.633630417606786 0.696428734470708 0.634247256822708 0.695933022106734 0.634830752544366 0.695462004504959 0.635383814384710 0.695013668407940 0.635909007402235 0.694586225120818 K= K= K= K= K= ITEST= 56 57 58 59 60 X= X= X= X= X= 0 0.694586225120818 0.636408602979691 0.694178079099387 0.636884620791817 0.693787801772042 GX= GX= GX= GX= GX= 0.636408602979691 0.694178079099387 0.636884620791817 0.693787801772042 0.637338863653079 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.656250000000000 0.789550781250000 0.581561207771301 0.851717192854103 0.442032556877904 0.863239214382603 0.413200455971483 0.848630437047546 0.449598864274130 0.866109039311399 0.405874599670697 0.843991431544969 0.460844632582750 0.869634000208470 0.396797470614596 0.837722332749592 0.475802690867609 0.872950715807630 0.388177172525017 0.831234793394251 0.490992290757739 0.874716014109724 0.383557680694024 0.827544151961258 0.499501899805932 0.874999131636688 0.382814779451054 0.826936684297132 C= 3.50000000000000 K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0.750000000000000 0.656250000000000 0.789550781250000 0.581561207771301 0.851717192854103 0.442032556877904 0.863239214382603 0.413200455971483 0.848630437047546 0.449598864274130 0.866109039311399 0.405874599670697 0.843991431544969 0.460844632582750 0.869634000208470 0.396797470614596 0.837722332749592 0.475802690867609 0.872950715807630 0.388177172525017 0.831234793394251 0.490992290757739 0.874716014109724 0.383557680694024 0.827544151961258 0.499501899805932 0.874999131636688 0.382814779451054 30 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= K= ITEST= C= 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= X= 0 0.826936684297132 0.500893415612791 0.874997206329900 0.382819833356697 0.826940829909067 0.500883928084298 0.874997265348996 0.382819678432711 0.826940702830853 0.500884218913637 0.874997263549195 0.382819683157156 0.826940706706137 0.500884210044721 0.874997263604089 0.382819683013059 0.826940706587940 0.500884210315224 0.874997263602415 0.382819683017454 0.826940706591545 0.500884210306974 0.874997263602466 0.382819683017320 0.826940706591435 0.500884210307225 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= GX= 0.500893415612791 0.874997206329900 0.382819833356697 0.826940829909067 0.500883928084298 0.874997265348996 0.382819678432711 0.826940702830853 0.500884218913637 0.874997263549195 0.382819683157156 0.826940706706137 0.500884210044721 0.874997263604089 0.382819683013059 0.826940706587940 0.500884210315224 0.874997263602415 0.382819683017454 0.826940706591545 0.500884210306974 0.874997263602466 0.382819683017320 0.826940706591435 0.500884210307225 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 0.874997263602464 0.382819683017324 0.826940706591439 0.500884210307218 GX= 0.750000000000000 4.00000000000000 K= ITEST= 0 X= 0.750000000000000 1 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Για να καταλάβουμε τι συμβαίνει χρησιμοποιήσαμε τρία διαφορετικά x0 , το x0 =0.25, το x0 =0.5 και το x0 =0.75. Και στις τρεις αυτές περιπτώσεις βλέπουμε ότι η μέθοδος συγκλίνει σε ρίζα για C=0.5,1.5,2,2.5 και 4. Για C=1 δοκιμάζοντας ακόμα και για Κ=1000 βλέπουμε ότι GX= 9.894270975829328E-004. Αποφαινόμαστε λοιπόν ότι η μέθοδος δεν συγκλίνει για αυτό το C παρόλο που φθίνει συνεχώς. Για C=3 βλέπουμε ότι η μέθοδος πάλι δεν συγκλίνει αλλά παλινδρομεί μεταξύ των αριθμών 0.63...-0.69...Για C=3.5 η μέθοδος πέφτει σε πηγάδι (0.50..-0.87..-0.38..-0.82..) και άρα δεν συγκλίνει. Όταν χρησιμοποίησα x0 =0.2 παρατήρησα ότι για C=4 η μέθοδος δεν συνέκλινε ακόμα και για Κ=600. Το ίδιο παρατήρησα όταν χρησιμοποίησα x0 =0.7. Βάζοντας όμως 0.25 και 0.75 η μέθοδος συνέκλινε ακόμα και με μία επανάληψη. Βλέπουμε λοιπόν πόσο σημαντικός είναι ο ρόλος του αρχικού σημείου. Όταν το x0 είναι κοντά στη ρίζα βλέπουμε ότι έχουμε σύγκλιση και μάλιστα τετραγωνική, ενώ όταν το x0 δεν είναι κοντά στη ρίζα μπορεί να ταλαιπωρηθούμε ή και να μη συγκλίνει, όπως στην περίπτωσή μας. 31 ΑΡΙΘΜΗΤΙΚΗ ΑΝΑΛΥΣΗ-ΠΡΩΤΗ ΕΡΓΑΣΤΗΡΙΑΚΗ ΑΣΚΗΣΗ Μια ιδέα αντιμετώπισης αυτού του προβλήματος που παρουσιάζει η μέθοδος αυτή είναι να κάνουμε λίγα βήματα της διχοτόμησης (‘κουτή’ αλλά σίγουρη) και να ξεκινάμε με το π.χ. x5 αυτής της μεθόδου ως x0 της επαναληπτικής (όπως έγινε και με το δεύτερο υποερώτημα της πρώτης άσκησης). Τέλος, υπάρχει σε δισκέτα ο κώδικας των τεσσάρων προγραμμάτων για περισσότερη διευκόλυνση.__ 32 ...
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This note was uploaded on 10/02/2009 for the course G 001 taught by Professor Shmmygr during the Spring '07 term at National Technical University of Athens, Athens.

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