Toshkent axborot texnologiyalari universiteti huzuridagi dasturiy mahsulotlar va apparat dasturiy majmualar yaratish

Aniq integralni taqribiy hisoblash usullari

Nyuton-Kotes formulalari J ^NK ( f ) .
h


J ( f )  int( f , a,b)) integralni hisoblash uchun Lagranj interpolyatsion ko’phadi

formulasidan foydalanamiz:

J ^NK ( f )  J (L ( f ; x)) 
h n

b

_ L_n ( f ; x)dx  _a
b

n n
_ f (x_i )l_i (x)dx  _ f (x_i ) p_i
(1)

bu yerda

a
p  l (x)dx 
b


i0

b _ x  x_{j dx}

i0
(2)

i _a i

^{a ji x}i ^{ x}j

1. formula
   

x_i1 - x_i  h , hol uchun Nyuton - Kotes formulasi deyiladi, (2) Nyuton -

Kotes koeffitsientlari deyiladi. (2) da x  x  th almashtirishni bajarsak

dx  hdt, x  t, a  0,b  n, h  (b - a)/ n va

_{p } b  a ⁿ _(1)ni t(t 1)...(t  n) _dt

(3)

i _{n 0}

i!(n  i)!(t  i)

ko’rinishni hosil qilamiz. (3) ni hosil qilishda

x - x_j  (t - j)h, x_i - x_j
 (i - j)h

tengliklardan foydalandik.

To’g’ri to’rtburchaklar formulasi

J ^TT ( f ) .
h

Kvadratura formulasi (integral yig’indi)

_b n

J ( f )  _a

f (x)dx
 _p_if(_i )
i=0
(4)

da _i  x_i  h / 2,

p_i  h,

i  0, 1, ..., n 1
deb ushbu markaziy to’g’ri to’rtburchaklar

formulasi

J ^TT ( f )
ga kelamiz:

n1 n1
h


J ^TT ( f )  h_ f (x  h / 2) h_ f _ .
h i i 0.5


i0 i0
Markaziy to’g’ri to’rtburchaklar formulasida egri chiziqli trapetsiya yuzi chizmada

ko’rsatilgan asoslari h va

f (x_i  h / 2)
ga teng to’g’ri to’rtburchak yuzalarining

yig’indisi J_h^TT(f) ga almashtirilmoqda.

Trapetsiya formulasi J^T ( f ) .
h
Kvadratura formulasida_i  x_i , p₀  p_n  h / 2, p_i  h,i  1,..., n 1deb olamiz

n1



J^T ( f ) 

^{fi  fi1} h  ^h {f +2(f +...+f )+f }
(5)

h

i0

_{2 2} 0 1 n-1 n

5. formula trapetsiya formulasi deyiladi. Trapetsiya formulasida egri chiziqli trapetsiya yuzi chizmada ko’rsatilgan asoslari f_i, f_i+1, h balandlikka ega trapetsiyalar yuzalarining yig’indisi J_h^T(f) bilan almashtirilmoqda.

   
   

Simpson formulasi J^C ( f ) .
h

J ( f )
integralni taqribiy hisoblash uchun {(x_i , f (x_i )), i  0,1,..., 2n} jadval olib

xar bir [x_2i , x_2i2 ] {i  0,1,..., 2n - 2 } kesmada Nyutonning ikkinchi darajali ko’pxadini

quramiz. Bu funktsiyalar

[x₀ ; x_2n ]
kesmada uzluksiz ikkinchi darajali (parabolik)

interpolyatsiya splayni S( f , x) ni tashqil qiladi.

 ^{f (x2i )  (x - x2i ) f [x2i , x2i1]}

S ( f , x)  ^(x - x )(x - x ) f [x , x , x ]
(6)

^ 2i
2i1 2i 2i1 2i2

^x  x  x
, i  0.1,..., n -1

 2i

h
2i 2

so’ng

J ( f )  J (S)  J^C ( f )
deb qabul qilamiz va


J^C ( f )
ni Simpson formulasi deb

ataymiz. Ravshanki,

h


C


n1


^x2i2

_h n1

J_h ( f )  _

i0
^x2 i

L_2,i ( f ; x)dx 

^{[ f}2i ^{ 4 f}2i1 ^{ f}2i2 ^{] }

i0
3

^h { f  4( f ...  f )  2( f ...  f )  f }

₃ 0 1 2m1 2 2m2 2m

Oraliq natija quyidagicha yaratiladi. interpolyatsiya ko’phadini integrallaymiz.

[x₀ , x₂ ]
kesmada Nyutonning 2-darajali

Lemma 1. Ushbu sodda Simpson formulasi o’rinli:

x<sub>2

</sub> 2 0 1 2 h 2

N (x)dx h( f  4 f  f ) / 3  J ^C (N ).

x₀

Isbot.

a₀  f₀ , a₁  f [x₀ , x₁], a₂  f [x₀ , x₁, x₂ ] deb quyidagilarni olamiz:

x₂ x₂
 ² 
0 1 0 2 0 1 0 1 2

N (x)dx  (a

x₀ x₀
0 1 2 h 2

• 
  
  a (x  x )  a (x  x )(x  x )dx  2ha
  

 2a h²  2a h³ / 3

 2hf₀  2h

2
3
( f₁  f₀ ) / h  2 ₃ ( f₀  2 f₁  f₂ ) / 2h

h

2
 h( f  4 f  f ) / 3  J ^C (N ).

Lemma 2.

r^C ( f )  f (x)  J^C ( f )
desak

r^C (x^ )  0,  0,1, 2,3 .

Isbot.   0,1, 2 hollar ravshan,   3 hol elementar ko’rsatiladi:

h h

h

1 (x

• 
  
  x )
  

x  x 1
(x²  x² ) 3

r^C (x³) 

(x⁴  x⁴)  ^{2 0} [x³  4( ^{0 2} )³  x³] 

(x⁴  x⁴)  ^{2 0} [x²  x²]  0

h ₄ 2 0 ₆ 0 ₂
2 ₄ 2 0


_{6 2} 0 2

Integralni taqribiy hisoblashga doir algoritmlar va dasturlar.

Misol.
¹ dx

^{I  }1 x
0
integralning qiymatini trapetsiyalar va Simpson formulalari yordamida

taqribiy hisoblang.

Yechish.

^0,1 kesmani

n  10
ta ^{x0, x1,x1, x2 ,. ,x9, x10}kesmalarga ajratamiz. Har bir

^xi nuqtada

y_i 

f x_i  i  0,1,2,...,10
qiymatlarni hisoblaymiz va quyidagi jadvalga

joylashtiramiz.

i	xi	yi
0	0	1,000
1	0,1	0,909
2	0,2	0,833
3	0,3	0,769
4	0,4	0,715
5	0,5	0,667
6	0,6	0,625
7	0,7	0,588
8	0,8	0,556
9	0,9	0,526
10	1,0	0,500

Trapetsiyalar formulasiga ko’ra
¹ dx _ y y _

^{T }1  x ^ 2
  h

⁰  y

• 
  
  y
  

 ......  y



10

I



^{1 2 9} ₂ 

₀  
 0,1 (0,5  0,909  0,833  0,769  0,715  0,667  0,625  0,588 
0,556  0,526  0, 25)  0,1 6,938  0,694
Simpson formulasiga ko’ra

¹ dx

^h    
 

^{IS  }1 x ^ 3
0


y₀  y₁₀
 4 y₁  y₃  y₅  y₇  y₉
 2 y₂  y₄  y₆  y₈ 

^{ 0,1  }^{0,5  0, 25} ^{ 4 } ^{0,909  0, 769  0, 667  0,588  0,526} ^

3

^{ 2 } ^{0,833  0, 715  0, 625  0,556}^_ <sup>


 0,1 </sup> ^{0, 75  4  3, 459  2  2, 729} ^
3

^{ 0,1 } ^{0, 75  13,836  5, 458} ^{ 0, 693}
3

1.