كتاب التحليل العددي.pdf
TRANSCRIPT
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6.............................................................. 1-1
7 ........................... 4.0 OIDUTS REPOLEVED FOSORCIM2 -1
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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
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75................................. 1-4
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76............................................ 2-4
67......................................... : 3-4
68................................................. 1-5
89........................................................................
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www.enkt.ten (
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noitcudortni 1-1
laciremun) ( sesylana
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elcarO tfoSelpoeP naaB PAS sdrawdEDJ
stol
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pihsrenwO fo tsoC latoT
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. . ( )
) (.
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lecxe .
LECXE BALHTAM , DACHTAM
ECEFFO
NARTROF
4.0 OIDUTS REPOLEVED FOSORCIM2 -1 ( )SWODNIW
. EGAUGNAL NARTROF
C C
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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
:
(
)
/ ( )
( * )
+
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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)
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ELBAIRAV ENO NI NOITAUQE FO NOITULOS
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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
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(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
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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
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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
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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
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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
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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
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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
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) 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/
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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
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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
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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
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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
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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|
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#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); }
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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
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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
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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
-
( )
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-
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-
x .
xI =xA
321
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321
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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
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-
: .33 33 A .43 .53 T .63 (1) .73 T .83 A T A T .93 ( 1000.0 ) > .04
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-
. : 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