Chiziqli algebraik tenglamalar

Ketma-ket yaqinlashish usuli

Soddalik uchun ketma-ket yaqinlashish usuli algoritmini quyidagi uch noma’lumli CHATSda ko’rib chiqamiz:
_ a₁₁x₁  a₁₂ x₂  a₁₃ x₃  b₁,

a

^ 21^x1
 a₂₂ x₂
 a₂₃ x₃
 b₂ ,
(5.4.2)

^ a x  a x  a
x  b

 ^{31 1 32 2}
33 3 3

Bu sistemani matritsa ko’rinishida ifodalaymiz:

^{ a}11
^a12
^a13 ^
 ^x₁ 
 ^b₁ 

^{ a}21 ^a22
a₂₃ ^  ^ x₂ ^  ^b₂ ^,

^ a a
_a   _x   _b 

yoki
 31 32

33   ³   ³ 

^{ a}11
^a12


^a13 ^
Ax=B,
 ^x₁ 


 ^b₁ 

  ^{ }  

bu erda
A  _ a₂₁
^a22
^{a23 } ,
x  _ x_{2  ,}
B  _ b_{2  .}

^ a a a ^
 _x 
 _b 

 ^{31 32 33} 
 ³ 
 ³ 

(6.4.2) sistemani unga teng kuchli sistema bilan almashtiramiz

(5.4.3)

yoki


x = (E-A)x + B.
(5.4.3) sistemani quyidagi ko’rinishda yozib olamiz:

 ^x₁ 
_1  a₁₁
 a₁₂
 a_{13 }
 ^x₁ 
 ^b₁ 

^ x₂ ^  ^  a₂₁
1  a₂₂  a₂₃
^  ^ x₂ ^  ^b₂ ^ ,

 _x   _{ a}

• 
  a 1  a
  

  _x   _b 

 ³   31 32
33   ³ 
 3 

va iteratsiya jarayonini quramiz:

_{ x1}(k ) <sub>


</sub>1  a  a

 a _

_{ x} (k 1) _

^{ } 11 12 13 ^


^ x₂ ^{(k ) }  ^  a₂₁ 1  a
(5.4.4)

yoki
 (k ) 

x
 ³ 
^  a

_x (k )
 E  Ax ^{(k 1)}

• 
  B .
  

Bu usulda, iteratsiya jarayonining yaqinlashishi uchun etarli shart quyidagicha ifodalanadi:

 ⁿ   ⁿ 

max_1  a _jj 
_ a_{ij }  1
yoki
max_1  a_ii 
^{ a}ij ^{  1 .}

^j _ i1,i j _ ⁱ _ j1, ji _
Agar CHATSda tenglamalar soni n ta bo’lsa, ketma-ket yaqinlashish usulining umumiy formulasi

_x (k) _{ x} (k 1 ) _
n
_{a x} (k 1 ) _{ b}

yoki

1. 
   i ^ ij j i j 1
   

x ^(k)  (1 a
_)x (k1 ) _{ a
x} (k1 ) _{ b}

i
ko’rinishga ega bo’ladi.
Misol. Quyidagi

2. 
   i
   

ij j i

n
j1,ji


_1,1x₁  0,2x₂  0,3x₃  1,
_0,1x₁  0,9x₂  0,2x₃  3,
^ 0,2x  0,1x 1,2x  2
 ^{1 2 3}

CHATSni ketma-ket yaqinlashish usuli yordamida eching. Bu sistemani matritsa ko’rinishida:

^{1.1  0.2 0.3}  ^x1   ¹
     
_ 0.1 0.9 0.2__ x_{2 }  _ 3_

^ 0.2
 0.1
1.2 ^
 _x 
 ₂

   3   
yozib olamiz.
(6.4.4) formulaga asosan iteraцion jarayonni quyidagicha yozishimiz mumkin:




_{ x1}(k ) <sub>


</sub>  0.1

0.2

 0.3_{
 x1}(k 1) _


^ x₂ ^{(k ) }  ^  0.1
0.1 
(5.4.5)





^ _x3^{(k ) }  0.2

Dastlabki yaqinlashish sifatida nolь vektorni olamiz va (6.5) formula yordamida birinchi yaqinlashishni aniqlaymiz:




_{ x1}(1) _
  0.1
0.2
 0.3_
 ⁰
 ¹ 
 ¹

^ x₂ ^{(1) }  ^  0.1
0.1
 0.2^  ^ 0^  ^ 3^  ^ 3^ .

^{ (1)  } 0.2
0.1
 0.2^{ } 0^
 ₂  ₂




 _x2  ^
  
   

YAna bir marta iteratsiya jarayonini bajaramiz:

_{ x} (2) _

_  0.1

 0.2

 0.3_

 ¹

 ¹ 

_ 0.9_

 ¹  _

      


^ x ^{(2) }  _  0.1

0.1

 0.2__ 3_  _ 3_  _ 2.8_ .

^ (2) _

      

_ x₃

_{ } 0.2

0.1

 0.2_

 ²

 ²

_1.7 _

Ikkita ketma-ket yaqinlashish bir-biridan etarlicha kam farq qilguncha iterцiya jarayonini davom ettiramiz. Hosil qilingan


vektorlar ketma-ketligi sistemaning aniq echimiga intiladi.

Umumiy holda iteratsiya jarayoni
_x(k ) _{ x}(k 1) _{ }
shart bajarilganda

tugallanadi. Bu erda

^ berilgan aniqlik.

3. Oddiy iteratsiya usuli

CHATSda noma’lumlar soni ko’p bo’lganda, Kramer, Gauss, teskari matritsa usullarining aniq echimlar beruvchi chiziqli sхema juda murakkab bo’lib qoladi. Bunday hollarda sistema ildizlarini topish uchun taqribiy sonli echish usullaridan foydalanish qulay bo’ladi. SHunday usullardan biri oddiy iteratsiya usulidir. Bu usulni (6.1) sistemasi uchun ko’rib o’tamiz.
Buning uchun bu sistemani

_a_i x
j 1

ko’rinishda yozib olamiz.

, i =1,2,...,n (5.4.6)

Bu sistemani matritsa ko’rinishida quyidagicha yozish mumkin:
^{АX  B} ,

bu erda
^a11 _{A } ^a21
...
^an1


^a12 ^a22
...
^an 2
..
...
...
...
^a1n ^a2 n _;
...
^ann
b₁ B  ^b2 ;
...
b_n
x₁ X  ^x2 .
...
x_n

(6.4.6) da
a_ii  0
( i  1,2, n )
deb faraz qilamiz. (5.4.6) dagi birinchi

tenglamani
^x₁ ga nisbatan, ikkinchi tenglamani
^x₂ ga nisbatan va nihoyat oхirgisini

^x_n ga nisbatan echib, quyidagi sistemaga ega bo’lamiz:

_x₁  ₁  0  ₁₂ x₂  ₁₃ x₃  ...  _1n x_n

x

 

2 n

n

^ 2 2

  ₂₁ x₁
 0  
23 ^x3
 ...   x

(5.4.7)

(6.4.7) ni ushbu

_.............................................................
^_x_n  _n  _n1 x₁  _{n 2} x₂  _n3 x₃  ...  _nn1 x_n1  0

0
_{α } ^α21
...
^α12
0


... α
, ^{β }
β₁ β₂
...
β_n
^, X 
x₁ x₂
...
x

n

matritsalar yordamida quyidagicha yozishimiz mumkin

X    X
(5.4.8)

(6.4.8) sistemani ketma-ket yaqinlashishlar usuli bilan echamiz:

Х⁽⁰⁾=,
^{X }1^{    X }0 ^ ,
^{X }2 ^{    X }1^ ,....

Ushbu jarayonni umumiy holda quyidagicha yozish mumkin:

^{X }k ^{    X }k 1^ , Х ⁽⁰⁾= , k=1,2,3, … (5.4.9)

Agar bu
_X ( k
ketma-ketlikning
k  
dagi limiti mavjud bo’lsa, bu limit

(6.4.1) sistemaning echimi bo’ladi.
Quyidagi
belgilashni kiritamiz.
_X k  _
k 

x
1

x
k 
2
...

x
k 
n

Agar iхtiyoriy >0 uchun
_xk 1 _{ x}k  _{ }
tengsizlik barcha i =1,2,...,n lar

i i

uchun bajarilsa,
_X k 1 _{ x}k 1 _{, x}k 1 _{,..., x}k 1 _
vektor (1.7.1) sistemaning 

1 2 n


aniqlikdagi echimi deb ataladi.