Reja: Aniq integralni taqribiy hisoblash. Umumiy mulohazalar

{ 7.1.1 - DASTUR }

{ * Aniq integralni to‘gri to‘rtburchak usulida”

hisoblash dasturi * }

uses crt;

label 40,120,130;

var

a,n,i:integer;

z,EPS,h,s,x,b:real;

function f(x:real):real;

begin

f:=1/(sqrt(5+4*x-x*x));

end;

begin

clrscr;

n:=10;a:=2;b:=3.5;EPS:=0.1;

40: h:=(b-a)/n;

s:=0;

for i:=1 to n do

begin x:=a+h*i; s:=s+f(x); end;

s:=s*h;

if n<> 10 then goto 130;

120: n:=n+10;

z:=s;goto 40;

130: if abs(s-z)>EPS then goto 120;

n:=n-10;h:=(b-a)/n;

for i:=0 to n do

begin

x:=a+h*i;

writeln('x=',x:4:2,' f(',x:4:2,')=',f(x):4:2);

end;

writeln('integralning qiymati : s=',s:6:3);

readln;

end.
2. Aniq integralni trapetsiya usulida hisoblash dasturlari:

1 '------------- 7.2 - DASTUR --------------------

2 REM SAVE"a:inttr’1",a

5 DEF FNF(X)=1/sqr(5+4*X- X^2)

6 DIM X(80)

8 PRINT"Berilganlarni kiriting:"

10 INPUT "Integral chegaralari a,b=";A,B

12 INPUT "Bolinishlar soni N=";N

14 INPUT "Aniqlik qiymati E=";E

20 H=(B-A)/N

25 X(0)=A : S=(FNF(A)+FNF(B))/2

30 FOR I=1 TO N-1

40 X(I)=X(I-1)+H

80 S=S+FNF(X(I))

90 NEXT I

92 S=S*H

94 REM Aniq integral qiymatini baholash

95 IF ABS(S-S1)

100 N=2*N:S1=S:GOTO 20

120 PRINT "S=";S

130 END

Berilganlarni kiriting:

Integral chegaralari a,b=? 2,3.5

Bolinishlar soni N=? 10

Aniqlik qiymati E=? 0.001

S= .523639

Ok 

7.2.1 - DASTUR

10 DEF FNF(X)=1/SQR (5+4*X-X*X)

12 READ A, B, EPS ‘Aniq integral chegaralari a,b va aniqlikni kiritish

14 DATA 2, 3.5, 0.01 ‘Aniq integral chegaralari a,b va aniqlik qiymatlari

16 PRINT “Aniq integralni trapetsiya usulida”

18 PRINT “ taqribiy hisoblash”

19 PRINT

32 N=10

34 H=(B-A)/N

36 S=(FNF(A)+FNF(3))/2

37 FOR I=1 TO N

38 X=A+H* I

58 S=S+FNF (X)

62 NEXT I

66 S=S*H

68 REM Aniq integral qiymatini baholash

70 IF N><10 THEN 74

72 N=N+10:Z=S:GOTO 34

74 IF ABS (S-Z)>EPS THEN 72

76 N=N-10:H=(B-A)/N

78 FOR I=0 TO N

80 X=A+H*I

82 PRINT “X (“;USING “###.####”;I:

84 PRINT “)=”;USING “###.####”;X;

86 PRINT “ F(“;USING “###.####”;I;

88 PRINT “)=”;USING “###.####”;FNF(X)

89 NEXT I

90 PRINT

92 PRINT “Integralning qiymati=”;USING “###.####”;S

98 END

Aniq integralni trapetsiya usulida

taqribiy hisoblash.

X(0)=2,00 F(0)=0,3333

X(1)= 2,15 F(1)= 0,3338

X(2)= 2,30 F(2)= 0,3350

X(3)= 2,45 F(3)= 0,3371

X(4)= 2,60 F(4)= 0,3402

X(5)= 2,75 F(5)= 0,3443

X(6)= 2,90 F(6)= 0,3494

X(7)= 3,05 F(7)= 0,3558

X(1)= 3,20 F(1)= 0,3637

X(9)= 3,35 F(9)= 0,3733

X(10)= 3,50 F(10)=0,3849

Integralning kiymati=0.5336
{ 7.2.1 - DASTUR }

{ Aniq integralni trapetsiya usulida hisoblash }

uses crt;

label 40,120,130;

var

a,n,i:integer;

EPS,h,s,b,x,z:real;

function f(x:real):real;

begin

f:=1/(sqrt(5+4*x-x*x));

end;

begin

clrscr;

a:=2;b:=3.5;EPS:=0.1;n:=10;

40: h:=(b-a)/n;

s:=(f(a)+f(b))/2;

for i:=1 to n-1 do

begin

x:=a+i*h;s:=s+f(x);

end;

s:=s*h;

if n<>10 then goto 130;

120: n:=n+10;z:=s;goto 40;

130: if abs(s-z)>EPS then goto 120;

n:=n-10;

h:=(b-a)/n;

for i:=0 to n do

begin

x:=a+h*i;

writeln('x=',x:4:2,' f(',x:4:2,')=',f(x):4:2);

end;

writeln(' integralning qiymati:s=',s:6:3);

readln;

end.
3. Aniq integralni Simpson usulida hisoblash dasturlari:

2 '------------- 7.3 - DASTUR --------------------

3 PRINT "Aniq integralni Simpon usulida yechish"

5 REM SAVE"SIM51",A

30 READ A,B,N,E

40 DEF FNF(X)=1/SQR(5+4*X-X^2)

70 X=A : H=(B-A)/N : C=1

90 FOR I=1 TO N-1

110 X=X+H : S=S+(C+3)*FNF(X)

112 C=-C

120 NEXT I

122 S=S*H/3

180 IF ABS(S-S1)

190 N=2*N : S1=S : GOTO 70

210 PRINT "Integralni qiymati S=";S

220 DATA 2,3.5,10,0.01

230 END

RUN

Aniq integralni Sim’on usulida yechish

Integralni qiymati S= .5210937

Ok 

7.3.1- DASTUR

10 DEF FNF(X)=1/SQR (5+4*X-X*X)

12 READ A, B, EPS ‘Aniq integral chegaralari a,b va aniqlikni kiritish

14 DATA 2, 3.5, 0.01 ‘Aniq integral chegaralari a,b va aniqlik qiymatlari

16 PRINT “Aniq integralni Cim’son usulida”

18 PRINT “ taqribiy hisoblash”

19 PRINT

32 N=10

34 H=(B-A)/N

36 S=FNF(A)+FNF(B)

38 C=1:X=A

54 FOR I=1 TO N-1

56 X=A+H*I

58 S=S+(C+3)*FNF (X)

60 C= -1

62 NEXT I

66 S=S*H/3

68 REM Aniq integral qiymatini baholash

70 IF N><10 THEN 74

72 N=N+10:Z=S:GOTO 34

74 IF ABS (S-Z)>EPS THEN 72

76 N=N-10:H=(B-A)/N

78 FOR I=0 TO N

80 X=A+H*I

82 PRINT “X (“;USING “###.####”;I:

84 PRINT “)=”;USING “###.####”;X;

86 PRINT “ F(“;USING “###.####”;I;

88 PRINT “)=”;USING “###.####”;FNF(X)

89 NEXT I

90 PRINT

92 PRINT “Integralning qiymati=”;USING “###.####”;S

98 END

Aniq integralni Simpson usulida

taqribiy hisoblash.

X(0)=2.00 F(0)=0.3333

X(1)= 2.15 F(1)= 0.3338

X(2)= 2.30 F(2)= 0.3350

X(3)= 2.45 F(3)= 0.3371

X(4)= 2.60 F(4)= 0.3402

X(5)= 2.75 F(5)= 0.3443

X(6)= 2.90 F(6)= 0.3494

X(7)= 3.05 F(7)= 0.3558

X(1)= 3.20 F(1)= 0.3637

X(9)= 3.35 F(9)= 0.3733

X(10)= 3.50 F(10)=0.3849

Integralning qiymati= 0. 5236
{ 7.3.1 - DASTUR }

{ Aniq integralni Simpson usulida hisoblash }

uses crt;

label 40,140,150;

var

n,i:integer;

a,b,EPS,h,s,c,x,z:real;

function f(x:real):real;

begin f:=1/(sqrt(5+4*x-x*x));

end;

begin

clrscr;

n:=10 ;a:=2;b:=3.5;EPS:=0.01;

40: h:=(b-a)/n;

s:=f(a)+f(b);

c:=1;x:=a;

for i:=1 to n-1 do

begin

x:=x+h;

s:=s+(c+3)*f(x);

c:=-c;

end;

s:=s*h/3;

if n<>10 then goto 150;

140: n:=n+10; z:=s; goto 40;

150: if abs(s-z)>EPS then goto 140;

n:=n-10;h:=(b-a)/n;

for i:=0 to n do

begin

x:=a+h*i;

writeln('x=',x:4:2,' f(',x:4:2,')=',f(x):4:2);

end;

writeln(' integralning qiymati :s=',s:6:3);

readln;

end.

.

5. Sonli integrallash. Gauss formulasi
The methods described above use the fixed points of a piece (the ends and the middle) and have a low order of accuracy (0 – methods of the right and left rectangles, 1 – methods of average rectangles and trapezes, 3 – a method of parabolas (Simpson)). If we can choose points in which we calculate values of function f (x) it is possible to receive methods of higher order of accuracy at the same quantity of calculations of subintegral function. So for two (as in a method of trapezes) calculations of values of subintegral function, it is possible to receive a method any more 1st, and 3rd order of accuracy:
.
Generally, using n points, it is possible to receive a method with accuracy order 2n-1. Values of knots of a method of Gaussa on nточкам are roots of a polynom of Lezhandra of degree n.

Values of knots of a method of Gaussa and their scales are resulted in directories of special functions. The method of Gaussa on five points is most known.

1-misol.

Gauss usuli yordamida hisoblang.

Yechimi.

.

.



.
The answer: 3.584.
2-misol

Gauss usuli yordamida hisoblang.

Yechimi.
.

.



.

Javobi: - 0.588.

O‘z-o‘zini tekshirish uchun savollar

1. 
   Qanday hollarda aniq integralni taqribiy hisoblanadi?
   
2. 
   Bo‘linish qadamini toping.
   
3. 
   Oraliqning bo‘linish nuqtalari qanday topiladi?
   
4. 
   To‘g’ri to‘rtburchaklar usuli va formulasini tushuntiring.
   
5. 
   Trapetsiyalar usuli va formulasini tushuntiring.
   
6. 
   Simpson usuli va formulasini tushuntiring.
   
7. 
   Aniq integralni taqribiy hisoblashlardagi xatoliklarini qanday baholaymiz?
   
8. 
   Simpson usulini boshqa usullardan farqi.
   
9. 
   Simpson usulida bo‘linish qadamini aniqlash
   
10. 
    Gauss kvadratur formulasi
    
11. 
    Tenglamani yechish
    

Nazorat savollari