********************

:

6.............................................................. 1-1

7 ........................... 4.0 OIDUTS REPOLEVED FOSORCIM2 -1

:

11......................................................................... 1-2

61......................................................... 2-2

91......................... 3-2

24................................................................................ 4-2

: 44......................... 1-3 54................................................................. 2-3 54................................................. 3-3 54................................................. 4-3 64......................................................... 5-3 45......................................................... 6-3

:

75................................. 1-4

76............................................ 2-4

67......................................... : 3-4

68................................................. 1-5

89........................................................................

..

.

) ..

www.enkt.ten (

.

.

/

afeeafer@liamtoh.moc

.. ,

.

.

,

.

.

noitcudortni 1-1

laciremun) ( sesylana

( )

.

.

elcarO tfoSelpoeP naaB PAS sdrawdEDJ

stol

...

...

.

pihsrenwO fo tsoC latoT

.

.

. . ( )

) (.

.

lecxe .

LECXE BALHTAM , DACHTAM

ECEFFO

NARTROF

4.0 OIDUTS REPOLEVED FOSORCIM2 -1 ( )SWODNIW

. EGAUGNAL NARTROF

C C

bal htam .

NARTROF .

NARTROF

SRETCARAHC 1

NARTROF :

SRETCARAHC SREBMUN 1.1

9 8 7 6 5 4 3 2 1 0

SRETCARAHC CITABAHPLA 2.1

:

Z Y X.. C B A

SRETCARAHC LEICEPS 3.1

:

(

)

/ ( )

( * )

+

NARTROF 1.2

: NARTROF

RAGETNI 2.2

LEAR

NOISECERP LEBUOD 3.2

XELPMOC 4.2

DNA LEAR EGAMI

LACIGOL 5.2

CIREMUNAHPLA 6.2

NOITANROFNI LRETIL

(ESLUF RO EURT)

ELBAIRAV ENO NI NOITAUQE FO NOITULOS

1-2 DOHTEM NOITCESIB

:

0 = 4 + 2^X

5^X ( ) X ( ^) .X

:

4=2^X

4=X

2-=X

. , 2-=X

:

0=01-2^X4+3^X

: DOHTEM NOITCESIB

} B , A { )X(F 0X )B(F , )A(F

)X(F 0.0=)0X(F }B,A{

B A

: )B(F , )A(F

(1-2)

( NOITULOS TNOIP )0X

10000.0

P )DOHTEM NOITCESIB(

:

TRATS

) B , A ( DEAR

2 / ) B+ A(=P OD

01 2^A*4 + 3^A =1Y

01- 2^B*4 + 3^B =2Y

01 2^P*4 + 3^P=0Y

001 PETS OTOG ) 10000.0 < 0Y ( FI

01 PETS OTOG ) 0.0 > 1Y*0Y( FI

A=A

B=B

02 PETS OTOG

P-=A OD

B=B

02 PETS OTOG

P , 0Y ETIRW

POTS

:

B A ( )P

10000.0

2. TEXT

3 . TAB C

4. :

C THIS PROGRAM TO CALCOLATE SOLUTION OF EQUATION BY USED BISECTION METHOD

RAED ( *,* ) A , B

P=(A +B ) / 2

Y1= A^3 + 4*A^2 10

Y2= B^3 + 4*B^2 -10

Y0=P^3 + 4*P^2 10

IF ( ASB( Y0) .LT. 0.00001 ) GOTO 100

IF (Y0*Y1 .GT. 0.0 ) GOTO 10

A=A

B=B

GOTO 20

A=-P

B=B

GOTO 20

WRITE(*,*) Y0 , P

STOP

END

ROF.NOITCESIB . 5 TXET

NARTROF

ELIPMOC DLIUB .6 ROF.NOITCESIB

. 7: )

0 )S(GNIRAW 0 S(RORRE

DLIUB DLIUB .8 EXE.NOITCESIB

ETUCEXE .9 EXE.NOITCESIB

2 RETNE 1 .01 RETNE

dohtem netuN 2-2

( )dohtem netwen

,

) ( dohtem noitcesib( )

.

0x )x(f : ( )

)x(' f/)x(f*1x=2X

:

(0x ) 2X

1X

1x )x(F

. )x( ' F

:

1x daeR .1 01-2^1x*4+3^1x=)1x(f .2

1x*8+ 2^1x*3 = )1x( ' f .3 )1x(f,2x etirw 10000.0 > )1x(f fi .4 otog 2x=1x, )1x( ' f /)1x(f (-1x=2x .5

)2( pets p, 2x etirw .6 pots .7

( 1x) ( )1x

)1x( f 2x 10000.0

)x ( ' f, )x ( f 1x 2x=1x

.

-: NARTROF

1x )*,*(daeR 01 01-2^1x*4+3^1x=)1x(f 1x*8+ 2^1x*3 = )1x( ' f 02 otog )10000.0 > )1x(f fi ))x( ' f/)x(f(-1x=2x 2x=1x 01 otog 02 p, 2x )*,*(etirW pots dne

OTOG OD :

1x )*,*(daeR N,1=I,005 OD 01-2^1x*4+3^1x=)1x(f

1x*8+ 2^1x*3 = )1x( ' f 02 OTOG )10000.0 > )1x(f fi ))x( ' f/)x(f(-1x=2x 2x=1x 005 OTOG 005 EUNITNOC 02 p, 2x )*,*(etirW pots dne

:

(2-2)

: 0X

) 2-3(

2-3

Newton-Raphson Method And

Modified Newton-Raphson Method

: ,

(1) y,x )1(0=)y,x(2F 0=)y,x(1F

:

x +)iy,ix(1F=)1+iy,1+ix(1F

F

y +)ix-1+ix( 1 F

.+)iy-1+iy( 1

x +)iy,ix(2F=)1+iy,1+ix(2FF

y +)ix-1+ix( 2 F

.+)iy-1+iy( 2

1+iy,1+ix 0=)1+iy,1+ix(2F=)1+iy,1+ix(1F

h = ix - 1+ix k = iy - 1+iy

x +)iy,ix(1F= 0

F

+h 1 y

F

k 1

x +)iy,ix(2F= 0F

+h 2 y

F

k 2

xF

+h 1 y

F

)iy,ix(1F - = k 1

xF

+h 2 y

F

)iy,ix(2F - = k 2

naibocaJ

=Jy

Fy

Fx

Fx

F

22

11 0

:

=Hy

FixiyF

yFixiyF

-

-

2(,)2

1(,)1

J/

=K22(,)

11(,)

xFixiyF

xFixiyF

-

-

J/

y,x

h+ix=1+iX k+iy=1+iY

.

: (n)

n ( 2n) x : ,

y x . y

y x

y 2F x 1F .

-ix=1+ixx

FixiyFixiy

(,)

(,)1

)1(.. 1

-iy=1+iyx

FixiyFixiy

(,)

(,)1

)2(. 1

0=y-)x(nis3 )x(nis+x=y:

: .

y-)x(nis+x=)y,x(1F y-)x(nis3=)y,x(2F

y x

xF

)x(soc+1= 1 y

F

1- = 1

xF

)x(soc3= 2 y

F

1- = 2

=J3(soc)1

1(soc)1

-

+-

x

x

)x(soc3+))x(soc+1(- =

)i( 1-)x(soc2 =

=H(3(nis))1

((nis))1

---

-+--

xy

xxy J/])y-)x(nis3(-)y-)x(nis+x([ = J/

)ii(.. J/))x(nis2-x( =

=K3(soc)(3(nis))

1(soc)((nis))

xxy

xxxy

--

+-+- = J/

J/))y-)x(nis+x()x(soc3+))x(soc+1()y-)x(nis3(-(=-)x(soc)x(nis3+)x(socx3+)x(socy+)x(soc)x(nis3-y+)x(nis3-(= J/))x(socy3 J/))x(socx3+)x(socy2-y+)x(nis3-(= )iii( J/)) y2-x3 ()x(soc+)x(nis3-y ( =

: I

: y,X : weny,wenX

(i) : caJ (ii) x : H (iii) y : k 10000.0 : E

( ): .1 . y , x .2 0=I .3 . y , x , I .4 1-)x(soc*2=caj .5 .... 0=caj .6

31 caj/))y*2-x*3()x(soc+)x(nis*3-y(=k caj/))x(nis*2-x(=h .7 k+y=weny h+x=wenx .8 i 1 .9

y ,x , I .01

E x .11 5 x .....

E y .21 5 y .....

.31 trahC wolF

orez yb ediviD 0=nibocaJ

tratS

, wenx etirW

1-)x(soc*2=caJ

daeR weny,wenx

0=I

h+x=wenX k+y=wenY

, wenx=X weny=Y

=caJ

caj/))x(nis*2-x(=H-)x(nis*3-y(=K

caj/))x(soc*x*3+)x(soc*y*2

1+I=I

-x| E

)2-4(

# include # include # include void main() { FILE *stream; int i; float x,y,xnew,ynew,jac,h,k; printf("Enter initial value of x, y :"); scanf("%f %f",&xnew,&ynew); i=0; stream = fopen("newraf.FIL", "w+"); fprintf(stream,"The initial value of x,y is (%2.1f ,%2.1f ) \n",xnew,ynew);

fprintf(stream,"\n i\t x\t\t y\t"); fprintf(stream,"\n_______________________________________");

fprintf(stream,"\n %d\t %f\t %f\t",i,xnew,ynew); do { x=xnew; y=ynew; jac=2*cos(x)-1 ; if( jac ==0) { fprintf(stream," divide by zero , Jacobin=0 "); break; } h=(x-2*sin(x))/jac; k= (y-3*sin(x)-2*y*cos(x)+3*x*cos(x))/jac; xnew=x+h; ynew=y+k; i++; fprintf(stream,"\n %d\t %f\t %f\t",i,xnew,ynew); }while(fabs(x-xnew)>.00001 & fabs(y-ynew)>.00001); fclose(stream); }

)0,0 ( x,y x,y

Enter initial value of x, y :2 1

The initial value of x,y is (2.0,1.0) i x y __________________________________ 0 2.000000 1.000000 1 1.900996 2.851493 2 1.895512 2.843267 3 1.895494 2.843241 4 1.895494 2.843241

Enter initial value of x, y :-2 -1 The initial value of x,y is (-2.0,-1.0) i x y __________________________________ 0 -2.000000 -1.000000 1 -1.900996 - 2.851493 2 -1.895512 - 2.843267 3 -1.895494 - 2.843241 4 -1.895494 - 2.843241

Enter initial value of x, y :0.5 0.5 The intial value of x,y is (0.5,0.5) i x y __________________________________ 0 0.500000 0.500000 1 -0.107617 -0.161425 2 0.000840 0.001259 3 -0.000000 - 0.000000 4 0.000000 0.000000

y-)x(nis3=)y,x( 1F y-)x(nis+x=)y,x( 2F

y x

xF

)x(soc3= 1 y

F

1- = 2

))ix(soc3( /)iy-)ix(nis3( ix = 1+iX )1-(/)iy-)ix(nis+ix( iy = 1+iY

: .1 . y , x .2 0=I .3 . y , x , I .4 )1-(/)y-)x(nis+x(-y=weny ))x(soc*3(/)y-)x(nis*3(-x=wenx .5 i 1 .6 y,x,I weny ,wenx , I .7 E x .8

5 )wenx( x ..... E y .9

5 )weny( y ..... .01

flow chart

) 2-5(

Start

Write xnew ,

Read xnew,ynew

I=0

X=xnew , Y=ynew

Xnew=x-(3*sin(x)-y)/(3*cos(x)) Ynew=y-(x+sin(x)-y)/(-1)

I=I+1

|x-xnew|

#include # include # include void main() { FILE *stream; int i; float x,y,xnew,ynew; printf("Enter initial value of x, y"); scanf("%f %f",&xnew,&ynew); i=0; stream = fopen("modnera.FIL", "w+"); fprintf(stream,"The initial value of x,y is (%2.1f ,%2.1f)\n",xnew,ynew);

fprintf(stream,"\n i\t x\t\t y\t"); fprintf(stream,"\n_______________________________________");

fprintf(stream,"\n %d %f %f",i,xnew,ynew); do { x=xnew; y=ynew; xnew=x-(3*sin(x)-y)/(3*cos(x)); ynew=y-(x+sin(x)-y)/(-1); i++; fprintf(stream,"\n %d %f %f",i,xnew,ynew); }while(fabs(x-xnew)>.00001 & fabs(y-ynew)>.00001); fclose(stream); }

Enter initial value of x, y :2 1 The initial value of x,y is (2.0,1.0) i x y

__________________________________

0 2.000000 1.000000 1 3.384041 2.909297 2 2.137745 3.143961 3 1.757074 2.981289 4 1.697342 2.739774 5 2.321275 2.689346

6 2.079209 3.052638 7 1.783338 2.952727

8 1.751365 2.760836 9 2.104744 2.735107 10 2.004736 2.965549 11 1.811609 2.912052 12 1.813539 2.782754 13 1.992840 2.784222 14 1.954219 2.905093 15 1.844689 2.881608 16 1.852782 2.807414 17 1.941547 2.813287 18 1.925796 2.873602 19 1.867622 2.863443 20 1.873738 2.823891 21 1.917870 2.828201 22 1.910927 2.858242 23 1.880843 2.853638 24 1.884470 2.833162

676538.2 075609.1 52 627058.2 323309.1 62 445848.2 149788.1 72 070838.2 419988.1 82 624938.2 630109.1 92 000748.2 164998.1 03 539548.2 536198.1 13 606048.2 966298.1 23 413148.2 382898.1 33 731548.2 305798.1 43 706448.2 235398.1 53 309148.2 460498.1 63 762248.2 109698.1 73 991448.2 115698.1 83 439348.2 994498.1 93 365248.2 077498.1 04 847248.2 602698.1 14 527348.2 900698.1 24 295348.2 099498.1 34 898248.2 821598.1 44 299248.2 458598.1 54 684348.2 557598.1 64 914348.2 932598.1 74 760348.2 903598.1 84 511348.2 676598.1 94 563348.2 626598.1 05 133348.2 563598.1 15 351348.2 004598.1 25 771348.2 685598.1 35 403348.2 165598.1 45 782348.2 924598.1 55 791348.2 744598.1 65 902348.2 145598.1 75 372348.2 825598.1 85 462348.2 164598.1 95

(1-2 )

Enter initial value of x, y :0.5 0.5

The initial value of x,y is (0.5,0.5) i x y __________________________________ 0 0.500000 0.500000 1 0.143613 0.979426 2 0.328876 0.286733 3 0.088597 0.651855 4 0.217908 0.177078 5 0.056940 0.434095 6 0.144872 0.113849 7 0.037329 0.289237 8 0.074650 0.264690 9 0.024699 0.192775 10 0.064273 0.049396 11 0.016411 0.128502 12 0.042838 0.032821 13 0.010924 0.085663 14 0.028556 0.021848 15 0.007278 0.057107 16 0.019036 0.014556 17 0.004850 0.038071 18 0.012691 0.009701 19 0.003233 0.025381 20 0.008460 0.006466 21 0.002155 0.016920 22 0.005640 0.004311

23

0.001437 0.011280

24 0.003760 0.002874 25 0.000958 0.007520 26 0.002507 0.001916 27 0.000639 0.005013 28 0.001671 0.001277 29 0.000426 0.003342

30 0.001114 0.000851 31 0.000284 0.002228 32 0.000743 0.000568 33 0.000189 0.001485 34 0.000495 0.000378 35 0.000126 0.000990 36 0.000330 0.000252 37 0.000084 0.000660 38 0.000220 0.000168 39 0.000056 0.000440 40 0.000147 0.000112 41 0.000037 0.000293 42 0.000098 0.000075 43 0.000025 0.000196 44 0.000065 0.000050 45 0.000017 0.000130 46 0.000043 0.000033 47 0.000011 0.000087 48 0.000029 0.000022 49 0.000007 0.000058 50 0.000019 0.000015 51 0.000005 0.000039 52 0.000013 0.000010

) 2-2(

Enter initial value of x, y :-2 -1 The initial value of x,y is (-2.0,-1.0) i x y __________________________________ 0 -2.000000 -1.000000 1 -3.384041 -2.909297 2 -2.137745 -3.143961 3 -1.757074 -2.981289 4 -1.697342 -2.739774 5 -2.321275 -2.689346 . . . . . . 57 -1.895541 -2.843209

58 -1.895528 -2.843273

462348.2- 164598.1- 95

x y

y ( iy,1+ix) y (iy,ix)

))ix(soc*3( /) iy-)ix(nis3( ix = 1+iX )1-(/)iy-)1+ix(nis+1+i x( iy = 1+iY

:

;))x(soc*3(/)y-)x(nis*3(-x=wenx ;)1-(/)y-)wenx(nis+wenx(-y=weny

: 1 2: y ,x fo eulav laitini retnE

)0.1,0.2( si y,x fo eulav laitini ehT y x i

__________________________________ 000000.1 000000.2 0 169341.3 140483.3 1 202149.2 761750.2 2 100218.2 200158.1 3 914078.2 375639.1 4 742728.2 083278.1 5 088558.2 003419.1 6 299438.2 084388.1 7 144948.2 756409.1 8 110938.2 903988.1 9 733648.2 550009.1 01 480148.2 333298.1 11 997448.2 587798.1 21 541248.2 688398.1 31

14 1.896649 2.844028 15 1.894678 2.842685 16 1.896078 2.843639 17 1.895080 2.842959 18 1.895790 2.843442 19 1.895285 2.843099 20 1.895644 2.843343 21 1.895388 2.843169 22 1.895570 2.843293 23 1.895441 2.843205 24 1.895532 2.843267 25 1.895467 2.843223 26 1.895514 2.843255 27 1.895481 2.843232 28 1.895504 2.843248 29 1.895487 2.843237 30 1.895499 2.843245

) 2-3( Enter initial value of x, y :0.5 0.5 The initial value of x,y is (0.5,0.5) I x y 0 0.500000 0.500000 1 0.143613 0.286733 2 0.095576 0.191007 3 0.063669 0.127295 4 0.042432 0.084850 5 0.028283 0.056563 6 0.018854 0.037708 7 0.012569 0.025138 8 0.008379 0.016759 9 0.005586 0.011172 10 0.003724 0.007448 11 0.002483 0.004966 12 0.001655 0.003310 13 0.001103 0.002207

174100.0 637000.0 41 189000.0 094000.0 51 456000.0 723000.0 61 634000.0 812000.0 71 192000.0 541000.0 81 491000.0 790000.0 91 921000.0 560000.0 02 680000.0 340000.0 12 750000.0 920000.0 22 830000.0 910000.0 32

(4-2) :( )

y-)x(nis3 =)y,x(2F , y-)x(nis+x =)y,x(1F y x

xF

, )x(soc+1= 1 y

F

1- = 2

))ix(soc+1( /)iy -)ix(nis+ix( ix = 1+iX )1-(/)iy-)ix(nis3( iy = 1+iY

000000.2 000000.3 0 063324.0 344620.111- 1 993029.1- 000000.0846553114301341- 17 117807.0- 000000.65242098739472194 27 NAN+ NAN+ 37

y x *

x . y

*

. . . )y,x(1F *

, *

y,x ( . )E .

y x

> > > 42 > )0,0( 16 > )8.2-,8.1-( 5>)8.2-,8.1-( 5-,3-

42 > )0,0( 45 > )0,0( 5 > )0,0( 6.0,6.0

33>)8.2-,8.1-( 57>)8.2-,8.1-( 5>)8.2,8.1( 57.0,5.1

72>)0,0( 5>)8.2,8.1( 5.2,5.1 92>)8.2,8.1( 75>)8.2,8.1( 4>)8.2,8.1( 2,2 42>)0,0( 16>)8.2,8.1( 5>)8.2,8.1( 5,3 64>)0,0( 411>)8.2,8.1( 4,7

03>)0,0( 7>)8.2-,8.1-( 21,01 13>)0,0( 541>)8.2,8.1( 02,02

(5-2 )

0

01

02

03

04

05

06

07

.. 2.. 1.

12

3

)0,0(

)1,2(

(6-2 )

: . : 1 .. x 2 ..

y

-,2-(

05.0

1

5.1

2

5.2

3

5.3

43210

(7-2 )

0

1

2

3

4

50.2259.19.158.1

(8-2 )

( 1,2)

( 1,2)

01

2

3

4

43210

(9-2 )

0

5.0

1

5.1

2

5.2

3

5.3

45.335.225.115.00

(01-2 )

( 1,2)

x y

( 1,2)

1,2(

.2,8.1(

( 1,2)

x y

( 1,2)

4-2 noitisop eslaF

)

( laimonylop noitalopretni egnargaL ( ) noitisop eslaf ( )

0

3. do y1=x1**3 +4*x1**2-10 y2=x2**3+4*x2**2-10

X3=x2 f(x2)*((x2- x1)/f(x2)-f(x2)) Y3=x3**2+4*x3**2-10

4. if(y30.0)goto (7) 6. x1=x1

x3=x2 goto (3)

7. x1=x3 x2=x2 goto 3

8. write x3 ,y3 9. stop

:

Read(*,*)x1 ,x2 y1=x1**3 +4*x1**2-10 y2=x2**3+4*x2**2-10

X3=x2 f(x2)*((x2- x1)/f(x2)-f(x2)) Y3=x3**2+4*x3**2-10

if(abs(y3).lt.0.000001)goto 8 if(y1*y3.gt.0.0)goto x1=x1 x3=x2 goto 3 x1=x3 x2=x2 goto 3 write(*,*) x3 ,y3 stop End

(11-2 )

1-3

)y,x(f = xd/yd

. :

dohteM sreluE 2-3 )x(F=y

n `x=x )0y,0x( ( ) w

n/)x-`x( = w

)iy,ix(f * w+ iy=1+iy

1-n..,1,0 = I :

w - w - . y -

dohteM reluE dednetxE ME

MEE )iy,ix(` f * !2/2w + )iy,ix(f * w+ iy=1+iy

1-n..,1,0 = I

reluE dednetxE eroM 3-3 dohteM

MEEM )iy,ix(`` f * !3/3w + )iy,ix(` f * !2/2w + )iy,ix(f * w+ iy=1+iy

dohteM reluE deifidoM 4-3

( rotciderP )1+i`y ME

2/) )1+i`y,1+ix(f + )iy,ix(f ( =gva (rotcerroC )1+iy

gva *w + iy = 1+iy

dohteM attuK-egnuR 5-3

6/)4k+3k2+2k2+1k( + iy =1+iY

)iy,ix(f *w = 1K )2/1k+iy,2/w+ix(f *w =2K )2/2k+iy,2/w+ix(f *w = 3K )3k+iy,w+ix(f *w = 4K

: 1= 0y , 0 = 0x ; yx = xd/yd

()2/2x-(^e )1.0=w 1= `x

yx-=)y,x(f

iy*ix * w - iy = 1+iY f

y-`yx- = )1-(t+`yx-=``y yx-=`Y `y

)1-2x(y =y-y2x = y-)yx-(x- = ``Y

2/)1-2ix(*iy*2w + iy*ix * w - iy = 1+iY f

`y)1-2x(+)x2(y=```y )1-2x(y =``y `y

)2x-3(yx = )yx-()1-2x(+yx2 = ``Y

6/)2ix-3(*ix*iy*3w + 2/)1-2ix(*iy*2w + iy*ix * w - iy = 1+iY

:

iy*ix*w iy=1+I`Y 2/)1+i`y*1+ix iy*ix- ( =gvA gva *w + iy = 1+iY

: )iy*ix-(*w =1K ))2/1k+iy(*)2/w+ix(-( *w =2K ))2/2k+iy(*)2/w+ix(-( *w = 3K ))3k+iy(*)w+ix(-(*w=3K 6/)4k+3k2+2k2+1k( + iy =1+iY

. n

: I : N

( ) y x : iy,iX x : wenx x y : wenY

: 4k,3k,2k,1K :W y,x ( ) : F

:

.41 0=I .51 y,x n .61 n/1=w .71 y*x-=F .81 F*w=1k .91 )2/1k+y(*)2/w+x(-=F .02 F*w=2k .12 )2/2k+y(*)2/w+x(-=F .22 F*w=3k .32 )3k+y(*)w+x(-=F .42 F*w=4k .52 6/)4k+3k*2+2k*2+1k(+y=weny .62 w+x=wenx .72 weny=y wenx=x .82 (1+I=I )i 1 .92 5 n=I .03 weny .13 .23

Flow Chart

Start

F=-x*y

K1=w*F

Read n,x,y

I=0

F=-(x+w/2)*(y+k2/2) K3=w*F

W=1/n

F=-(x+w/2)*(y+k1/2)

K2=w*F

I=I+1

I=n yes

no

F=-(x+w)*(y+k3) K4=w*F

y

Ynew=y+(k1+2*k2+2*k3+k4)/6 Xnew=x+w

X=xnew,y=ynew

(1-3 ) :

- 1.0=w

w - - - -

- - -

w -

dnE

406.0

5406.0

506.0

5506.0

606.0

5606.0

706.0

2.118.06.04.02.00

w

y

(2-3 ) y w

5.0

6.0

7.0

8.0

9.0

1

1.1

157.05.052.00

(3-3 )

w 1 0 x y

.

02.0

4.0

6.0

8.0

1

2.1

2.118.06.04.02.00

x

y

(4-3 )

6-3

,

( )

. ( ) :

- 1=x

( 3=`x) - .

- (9710.01=)3(y )

: 9710.01= )3(y , 2571.1= )1(y ; y= 2xd/y2D

y 1=x `y : 3=x 1=x `y :

. 3=x y : `y

Y Y `Y 1g-2g 1g 1R 2g 2R ]2[ox D

0yp+0y=Y h/)0x-px(=P

1r-2r=H 1g-2g=Y )1r-2r(/)1r-d(=P )1g-2g(*)1r-2r(/)1r-d( +1g=]2[0X=Y

y 3=x y . 3=x

02=1.0/)1-3(=n w/)0x-x(=n 02

: reti,I : N

( ) x : tratst y : tratsx

w x : ot 4k,3k,2k,1K : krwX

w : h ( ) : F

: loT `y : 2g,1G y : ]1[X `y : ]2[X

`y 3=x y : 2r,1R : )(tsyskR : )(sevireD

( )

,

. ,

1-4

x .

xI =xA

321

*003

020100

321

*132333122232112131

xxx

xxx

aaaaaaaaa

ll

l =

A :

X =xB

B

=B003302201100

bb

b

xT=X T

=1T001nissoc0soc-nis0

qqqq

=2Tqq

qq

nis0soc010

soc-nis

=3Tqqqq

0nissoc0soc-nis100

x=xA xT=xtA

T 1= x=xT1-T=xTA1-T

B TA1-T

)11a-22a( nis soc+) 2nis- 2soc(21a

)22a-11a(/21a*2=)2(nat )1(2/))22a-11a(/21a*2(1-nat=

.

nT...*2T*1T=V . n

=A

012121

210

---

-

-1 .1 3T 1T .2 ( 1) .3

2/-=)2-2(/2-(1-nat= A TA1-T .4 T T .5

. . 321 .6

.

: .33 33 A .43 .53 T .63 (1) .73 T .83 A T A T .93 ( 1000.0 ) > .04

... 3 : 33a,22a,11a :

.14

:

.

. :

. : )(regiB . : )(lauqE

:tam_etirW. T : )(srevnidnaT

. : tam_daer. : tam_lluM

. : T. : A. : V . : K

Flow Chart

) 4-1(

#include #include #include

Start

A12>=a

I=1, j=2

A=Read

I=0 j=0

I=1,j=3

Tetha=tan-

Aii=

Tetha=PI/

ttii=ttjj=tii=tjj=cos(tetha)

Aij>=

a12&a13&a23=aA13>=a

Tetha=PI/

yesno

no yes yes

T=tt=I(

#include #include #define max 3 int i,j,k,n; void equal(float s[][3],float c[][3]); void input_mat(float a[max][max]); void mull_mat(float aa[][max],float b[][max],float c[][max]); void write_mat(float a[][max]); void biger(float a[][max]); void tandinvers(float t[][3],float tt[][3],float a[][max]); FILE *in,*out; main() { int k; float d[max][max],a[max][max],c[max][max],tt[3][3],t[3][3],v[max][max]={1,0,0,0,1,0,0,0,1}; clrscr(); n=3; k=0; in=fopen("jacread.fil","r"); input_mat(a); fclose(in); out=fopen("jacwrite.fil","w+"); fprintf(out,"The Array of A =\n"); write_mat(a); fprintf(out,"\n"); while( (fabs(a[0][1])>0.0001) || (fabs(a[0][2])>0.0001) || (fabs(a[1][2])>0.0001) ) { k=k+1; biger(a ( ; // find biger element of A fprintf(out,"\nCycle no. %d & The biger element : %5.4f\n",k,a[i][j]); tandinvers(t,tt,a ( ; // find Angle , T & invers T fprintf(out,"T = \n"); write_mat(t); mull_mat(tt,a,d); // multiplicat invers T in A mull_mat(d,t,a); // multiplicat A in T fprintf(out,"A = \n"); write_mat(a); mull_mat(v,t,c); //clculat Eigenvectors

equal(v,c); } fprintf(out,"The lamda is "); for(i=0;i=fabs(a[1][2])) {i=0;j=2;} else {i=1;j=2;} } void mull_mat(float aa[][max],float b[][max],float c[][max]) {int k; for(i=0;i

} } void write_mat(float c[][max]) { for(i=0;i

3.3660 -0.2768 0.1704 -0.2768 0.6136 0.0000

0.1704 0.0000 2.0204 Cycle no. 4 & The biger element : -0.2768 T = 0.9951 0.0991 0.0000

-0.0991 0.9951 0.0000 0.0000 0.0000 1.0000 A = 3.3936 -0.0000 0.1695 0.0000 0.5860 0.0169 0.1695 0.0169 2.0204 Cycle no. 5 & The biger element : 0.1695 T = 0.9927 0.0000 -0.1207 0.0000 1.0000 0.0000 0.1207 0.0000 0.9927 A = 3.4142 0.0020 0.0000 0.0020 0.5860 0.0168 0.0000 0.0168 1.9998 Cycle no. 6 & The biger element : 0.0168 T = 1.0000 0.0000 0.0000 0.0000 0.9999 0.0119 0.0000 -0.0119 0.9999 A = 3.4142 0.0020 0.0000 0.0020 0.5858 -0.0000 0.0000 -0.0000 2.0000 Cycle no. 7 & The biger element : 0.0020 T = 1.0000 -0.0007 0.0000 0.0007 1.0000 0.0000 0.0000 0.0000 1.0000 A = 3.4142 0.0000 0.0000 0.0000 0.5858 -0.0000 0.0000 -0.0000 2.0000 The lamda is

2414.3=0J 0005.0 1707.0- 0005.0 =V 8585.0=1J 0005.0 1707.0 0005.0 =V 0000.2=2J 1707.0 0000.0- 1707.0- =V

-

-

7 1000.0 ) (

T - .

- T -

nis=lka , nis-=kla , soc=kka ,soc=lla k=j,l=I -

.

2-4 laimonyloP gnitalopretnI segnargaL

. . .

n 1-n= 3

L L :

=iL(()(......)()(.......))

(()(......)()(.......))1211

1211

iiiiiiin

iin

xxxxxxxxxxxxxxxxxxxx

----------

-+

I. -+

x =

: = )x(L=Y

=

n

iiLxiy

1 ()*

.

C . .

: j,I : N

y : pX : iy,iX

)I( px : C

)I( ix : D : L

y : muS :

.24 ( )n .34 y,x .44 y px .54 1 D C .64 )y,x( L .74

8 ... )ix-jx(*D=D )ix-px(*C=C .84 i .94 7 ... (... n=I) .05 D/C=L L .15 ( )mus Y L .25 j .35 5 ... (... n=j) .45 mus .55 .65

Flow Chart

) 4-2(

c # include

Start

I=

Read n

I=I+1

Read x,y

I=1

Read xp

J=1

I=1 , C=1 ,

I=j

D=D*(xp-xi) C=C*(xj-xi)

I=I+1

I=n

J=J+1

J=

L=C / D Sum=Sum+L*

Print

End

yes

yes

yes

no

no

no

yesno

# include void main() {int i,j,n; float L[9],sum,xp,x[9],y[9],C,D; sum=0; printf("Enter n value\n "); scanf("%d",&n); printf("\n Enter x and y \n "); for(i=0;i

}

1 dohtem egnargaL

00.4 00.3 00.2 00.1 00.0 x =px 00.23 00.81 00.8 00.2 00.0 y 00.5

000000.1 =)0(L 000000.5- =)1(L 000000.01 =)2(L 000000.01- =)3(L 000000.5 =)4(L ....+2y*2l+1y*1L=L 000000.05 =L

)x(nis a 2 ...................

dohtem egnargaL

00.09 00.06 00.03 00.0 x =px 00.1 8304520668.0 05.0 00.0 y 00.04

827160.0- =)0(L 147047.0 =)1(L 073073.0 =)2(L 383940.0- =)3(L

....+2y*2l+1y*1L=L 837146.0 =L

b 2 ..........................................

dohtem egnargaL

00.06 00.0 00.03 00.09 x =px 8304520668.0 00.0 05.0 00.1 y 00.04

383940.0- =)0(L 147047.0 =)1(L 827160.0- =)2(L 073073.0 =)3(L ....+2y*2l+1y*1L=L

837146.0 =L

c 2 ................................................

dohtem egnargaL 00.06 00.03 00.0 x =px 8304520668.0 05.0 00.0 y 00.04

111111.0- =)0(L 988888.0 =)1(L

222222.0 =)2(L ....+2y*2l+1y*1L=L 598636.0 =L

d 2 ....... .................................

dohtem egnargaL

00.09 00.06 00.03 x 00.04 =px 00.1 8304520668.0 05.0 y

655555.0 =)0(L 655555.0 =)1(L 111111.0- =)2(L ....+2y*2l+1y*1L=L

297746.0 =L

*

. *

2x2

)x(nis

.

.1 * d-c 2 *

-2 3x,2x,1x 2x,1x,0x L px c-2 . a

L d-2 px .

)059f,)05(x noisnemiD 'laimonylop fo eulav retne')*,*( etirW n )*,*( daeR 'noitulos dnif uoy taht x fo tniop retne')*,*( etirW 0x)*,*(daeR 1+n,1=i,01 oD

' = ) ',i,' (f', ',')',i,' ( x')*,*( etirW )i(f,)i(x)*,*(daeR

eunitnoC 01 0=pX 1+n,1=i,02 oD

1=t 1+n.1=j,03 oD

fI neht)j.en.i( ))j(x-)i(x(/)))j(x-0x(*t(=T eslE fidnE

03 eunitnoC )i(f(*t+xp=xP

02 eunitnoC px)*,*(etirW potS dnE

: 3-4

( . )

:

1c = nxna. + 3x3a + 2x2a + 1x1A

2c =nxnb ......... + 3x3b + 2x2b + 1x1B

nc=nxnd.. +3x3d+ 2x2d+ 1x1D

1x n nb 2b 1b , na 2a 1a nx 2x

nd. 3d 2d 1d

:

n1a .. 2,1a 1,1a

n2b 2,2b 1,2b

nnd . .. 2nd 1nd

: ,

1X

2X

.

.

nx

: ,

1C

2C

.

.

nc

) ,

( xirtam detnemgua

: ,

1c n1a .. 2,1a 1,1a

2c n2b 2,2b 1,2b

nnD . .. 2nd 1nd nc

1,1a ) 2,1a

(

:

1,1a/1,2b=1,2F

1,1a*1,2f 1,2b =1,2b

2,1a*12f - 2,2b= 2,2b

3,1a*2,1f 2,3b=3,2b

:

. 1 ( 1,2)

( 1,1)

12b =12b: ( 1,2) . 2 11a*12f

. 3

. 4

: ( ) n

x = nnd 0 0

. 1x

.1

)05( x )05(f )05 ,05( b )05,05(a .2

n daer .3

od .4

a xirtam fo stnemele daer n ot 1 = I morf

od .5

)1-n(*n=m

od .6

m ot 2=I morf

1-i ot 1=k morf

)k,k(a / )k,I(a=)k,I(f sey 0 )k,k( a( fi

od .7

1+ n ot 1 = j morf

)j,k(a*)k,I(f )j,I(a=)j,I(a

on

od .8

1 I ot 1 I= l morf

1+n ot 1=j morf

) j , l(a =) j, l(b

) j , 1 + l( a = ) j , l ( a

) j , l( b =) j , 1 + l( a

od .9

1 pets 1 ot n = I morf

n ot 1 + I = j morf

) 1+ n ,I(a =1s

) I , I (a /)2s 1s( = )i(x

0 = 2s

od .01

n ot 1 = I morf

)i(x etirw

pots .11

:

) . 1 ( 1 + n

. 2

( n )

3 .

4.

) (FORTRAN language

Dimension a(50,50) ,b (50, 50 ) , f(50) , x(50)

Write(*,*)' enter numbers of your elements (n)'

Read (*,*)n

Do 10,i=1,n

Do 20 , j=1 n+1

Write(*,*)' ','a ( ',I,j,' ) = '

Read(*,*)a(I,j)

20 continue

10 continue

m=n*(n-1)/2

do 30,i=2,m

do 40,k=1,n-1

if(a(k,k).ne.0)then

do 40,j=1.n+1

a(i.j)=a(I,j)-f(I,k)*a(k,j)

40 continue

else

do 50,L=i-1,i-1

do 60,j=1,n+1

b(l,j)=a(l,j)

a(l,j)=a(l+1,j)

a(l+1,j)=b(l,j)

60 continue

50 continue

do 70,i=n,1,-1

do 80,j=i+1,n

s2=s2 + a(I,j)*(j)

80 continue

s1=a(I,n+1)

x(i)=(s1 s2)/a(I,i)

s2=0.0

70 continue

do 90,i=1,n

write(*,*)' x ( ',I, ' ) = ',x(i)

90 continue

stop

end

n)*,*( daer

.

n

2/)1-n(*n=M

m (i) ,

k 1 I 1 k 1 I

:

) j , k(a*) k ,I( f ) j , I(a= )j,I(A

eunitnoc

:

) j , l(a =)j,l(b

)j , 1 +l(a=)j,l(a

) j ,l(b=)j ,1 + l(a

:

a=C

b=A

b=C

:

n

()n

j , I 2s " j

2/5=x 5=x2

2s 1s

( 1 + n,I) :

) I , I(a/)2s 1s(=)i(X

2s

DOHTEM GNITTIF

5-1 DOHTEM GNITTIF

.... .

... tiF tseB

=S=

n

0 i

d2

iy-)x(mP = d

)x(jgja = mxma+.+ x1a+0a = )x( mP

=0 S kaS

- 5791 -

7.944 0791 0691 0591 0491 0391 2.114 9.223 7.202 34.631 5.011

(1-5 ) ()

xba=Y

)xba ( nl = )y( nL )b(nl x + )a(nl = )y(nL x B + A = Z

)y(nl = Z )a(nl=A )b(nl=B

=S=

n

0 i

= d2=

+-n

i

AxBz0

()2

=

=+-=n

iAxBZ

AS

0 2(()1)0

=

=+-=n

i

AxBZxBS

0

2(())0

iz = ix B + An ixiz = 2ix B +ix A

)y(nl= z )iy(nl = ix B + An )iy(nlix = 2ix B +ix A

= D

ixix2

nix )1(.. 2)ix ( 2ix n =

=A

(nl)2

(nl)

ixiyix

iyix /])iy(nlix ix -)iy(nl 2ix [ = D /

)2(..D

=B

(nl)

(nl)

ixixiy

niy )3(. D /]ix )iy(nl -)iy(nlix n[ = D /

Be = b , Ae = a

b a )4(.. xb a=Y

.

0791 0691 0591 0491 0391 2.114 9.223 7.202 34.631 5.011

(2-5 )

:

)y(nl*2X )y(nl*X 2X )y(nL y x 15.4324 51.141 009 7.4 5.011 03 03.5687 36.691 0061 9.4 34.631 04 23.97231 95.562 0052 3.5 7.202 05 34.89702 46.643 0063 7.5 9.223 06

94.39492 43.124 0094 0.6 2.114 07 50.17657 53.1731 00531 6.62 37.3811 052

(3-5 )

0005 = 2)052(-00531*5 = D 52.3= 0005/)53.1731*052-00531*6.62( =A 140.0= 0005/)6.62*052-53.1731*5( = B 97.52 = 52.3e=a 40.1 = 140.0e=b xba=Y x)40.1(*97.52=Y 8.105= 57)40.1(*97.52=)57(y

.

07 06 05 04 03 x 64.124 13.792 27.902 49.741 63.401 y

(4-5 )

: I : N

( )y( )x : iy,iX : x x : Y

(1) ...... : atled x : xmus

y : ymus 2x x : xxmus y*x : yxmuS

(2) a : a (3) b : b

: .75 n .85 0=I .95 iy ix .06 (ix+xmus=xmus )ix .16 ()iy(nl+ymus=ymus ))iy(nl .26 (2ix+xxmus=xxmus )2ix .36 ()iy(nl*ix+yxmus=yxmus ))iy(nl*ix .46 1+I=I .56 4 n

Flow Chart

Start

read x , y

sumy=sumy+y

Read n

I=0

sumxy=sumxy+x*ln(y)

sumx=sumx+x

sumxx=sumxx+x*x

I=I+1

I

) 5-1(

#include #include void main() { int i,n,sumx,x,sumxx,xi[5]; double a,b,sumy,y,sumxy,delta,yi[5]; printf("Enter Points Number: "); scanf("%d",&n); printf("Enter Values of x and y "); sumx=0;sumy=0;sumxy=0;sumxx=0; for(i=0;i

yi[i]=log(yi[i]); sumx=sumx+xi[i]; sumy=sumy+yi[i]; sumxx=sumxx+xi[i]*xi[i]; sumxy=sumxy+xi[i]*yi[i]; } delta = n*sumxx-sumx*sumx ; a=exp( (sumy*sumxx-sumx*sumxy)/delta); b=exp( (n*sumxy-sumx*sumy)/delta ); printf("Enter value of x"); scanf("%d",&x); y=a*pow(b,x); // writing printf("a= %lf , b= %lf\n\n",a,b); printf(" y = a * b ^ x\n\n"); printf("x :");for(i=0;i

sumxx=sumxx+xi[i]*xi[i]; sumxy=sumxy+xi[i]*yi[i]; }

delta = n*sumxx-sumx*sumx ; a=exp( (sumy*sumxx-sumx*sumxy)/delta); b=exp( (n*sumxy-sumx*sumy)/delta ); printf("Enter value of x "); scanf("%d",&x); y=a*pow(b,x); // writing stream = fopen("fitwrite.FIL", "w+"); fprintf(stream,"a= %lf , b= %lf\n\n",a,b); fprintf(stream," y = a * b ^ x\n\n"); fprintf(stream,"x :"); for(i=0;i

5791 ( 7.944) ( 8.105)

.

005

001

051

002

052

003

053

004

054

005

055

006

06030

(2-5 )

5791

0

05

001

051

002

052

003

053

004

054

005

055

006

06030

(3-5)

.

:

ten.enkt@enkt