CHATSni echish usullari
CHATSni echishda aniq (Gauss, Kramer,
teskari matritsa, Jardon) va taqribiy
(ketma-ket yaqinlashish, oddiy iteratsiya,
Zeydel) usullaridan foydalanish mumkin.
n
ta noma’lumli
n
chiziqli algebraik tenglamalar
sistemasi
berilgan bo’lsin. Bu yerda
a
ij
, b
i
(i,j=1,2,…n)
lar
berilgan sonli koeffitsientlar lar noma’lumlar
bo’lib, ularni aniqlash kerak
.
Masalaning qo’yilishi
n
n
nn
n
n
n
n
b
x
a
...
x
a
x
a
.
..........
..........
..........
..........
b
a
...
x
a
x
a
b
a
...
x
a
x
a
2
2
1
1
2
2
2
22
1
21
1
1
2
12
1
11
i
x
Agar berilgan sistemaga mos keluvchi asosiy
diterminant 0 dan farqli, ya’ni
bo’lsa, u yagona echimga ega bo’ladi.
Masalaning matematik echimi mavjudmi?
0
2
1
2
2 2
2 1
1
1 2
1 1
n n
n
n
n
n
a
...
a
a
.......
..........
a
...
a
a
a
...
a
a
3 ta noma’lumli 3 chiziqli algebraik tenglamalar sistemasi
berilgan bo’lsin. Bu yerda
a
ij
, b
j
(i,j=1,2,3)
lar berilgan
sonli koeffitsientlar
x, y, z
lar noma’lumlar bo’lib, ularni
aniqlash kerak
.
Masalaning qo’yilishi
3
3 3
3 2
3 1
2
2 3
2 2
2 1
1
1 3
1 2
1 1
b
z
a
y
a
x
a
b
z
a
y
a
x
a
b
z
a
y
a
x
a
Agar berilgan sistemaga mos keluvchi asosiy
diterminant 0 dan farqli, ya’ni
bo’lsa, u yagona echimga ega bo’ladi.
Masalaning matematik echimi mavjudmi?
0
3 3
3 2
3 1
2 3
2 2
2 1
1 3
1 2
1 1
a
a
a
a
a
a
a
a
a
Kramer usuli algoritmi
;
3 3
3 2
3 1
2 3
2 2
2 1
1 3
1 2
1 1
a
a
a
a
a
a
a
a
a
;
3 3
3 2
3
2 3
2 2
2
1 3
1 2
1
a
a
b
a
a
b
a
a
b
x
.
;
;
z
z
y
y
x
x
;
3 3
3
3 1
2 3
2
2 1
1 3
1
1 1
a
b
a
a
b
a
a
b
a
y
;
3
3 2
3 1
2
2 2
2 1
1
1 2
1 1
b
a
a
b
a
a
b
a
a
z
Quyidagi
ChATSni Kramer usulida eching.
Echish.
javob: (5; -4; 7)
Miso
l
;
12
8
7
5
-
7
-
8
-
3
3
9
2
;
60
8
7
3
7
-
8
-
2
-
3
9
5
-
x
3
8
7
5
2
7
8
3
5
3
9
2
z
y
x
z
y
x
z
y
x
;
48
8
3
5
-
7
-
2
-
3
3
5
-
2
y
;
84
3
7
5
-
2
-
8
-
3
5
-
9
2
z
;
5
12
60
x
x
;
4
12
48
y
y
.
7
12
84
z
z
Gauss usuli algoritmi
3
3 3
3 2
2
2 3
2 2
1
1 3
1 2
d
z
c
y
c
x
d
z
с
y
с
x
d
z
c
y
c
x
1
3
1
3 3
1
3 2
1
2
1
2 3
1
2 2
1
1 3
1 2
d
z
c
y
c
d
z
с
y
с
d
z
c
y
c
x
3
3 3
3 2
3 1
2
2 3
2 2
2 1
1
1 3
1 2
1 1
b
z
a
y
a
x
a
b
z
a
y
a
x
a
b
z
a
y
a
x
a
3
3
3
3 3
2
2
2
2 3
1
1 3
1 2
d
z
c
d
z
с
y
d
z
c
y
c
x
Gauss usuli, noma’lumlarni ketma-ket yo’qatishga asoslangan:
2
3
2
33
2
2
2
23
1
13
12
d
z
c
y
d
z
с
y
d
z
c
y
c
x
Quyidagi
ChATSni Gauss usulida eching.
Miso
l
13
2
4
13
4
3
2
1
2
z
y
x
z
y
x
z
y
x
Misol
Echish
.
13
2
4
13
4
3
2
1
2
z
y
x
z
y
x
z
y
x
25
,
3
5
,
0
25
,
0
5
,
6
2
5
,
1
1
2
z
y
x
z
y
x
z
y
x
25
,
2
5
,
0
75
,
1
5
,
7
3
5
,
3
1
2
z
y
z
y
z
y
x
7
9
7
2
7
15
7
6
1
2
z
y
z
y
z
y
x
7
24
7
4
-
7
15
7
6
1
2
z
z
y
z
y
x
6
3
1
z
y
x
3-o’lchovli
kvadrat matritsa berilgan bo’lsin.
Ta’rif.
A
matritsaga teskari matritsa deb shunday
A
-1
matritsaga
aytiladiki,
A
-1
∙A=E
bo’ladi. Bu erda
E
– birlik matritsa, ya’ni
Teskari matritsa
3 3
3 2
3 1
2 3
2 2
2 1
1 3
1 2
1 1
a
a
a
a
a
a
a
a
a
A
1
0
0
0
1
0
0
0
1
E
Agar
A
matritsa elementlaridan tuzilgan determinant holdan
farqli, ya’ni
detA ≠ 0
bo’lsa, bu matritsaga teskari matritsa
mavjud va u quyidagi formula yordamida hisoblanadi.
Bu yerda
Δ=detA ; A
ij
- a
ij
- elementlarning algebraik
to’ldiruvchilari.
Teskari matritsa
33
23
13
32
22
12
31
21
11
1
1
A
A
A
A
A
A
A
A
A
Δ
A
Teskari matritsa
;
3 3
3 2
2 3
2 2
1 1
a
a
a
a
A
;
3 3
3 1
2 3
2 1
1 2
a
a
a
a
A
;
3 2
3 1
2 2
2 1
1 3
a
a
a
a
A
;
3 3
3 2
1 3
1 2
2 1
a
a
a
a
A
;
3 3
3 1
1 3
1 1
2 2
a
a
a
a
A
;
3 2
3 1
1 2
1 1
2 3
a
a
a
a
A
;
23
22
13
12
31
a
a
a
a
A
;
2 3
2 1
1 3
1 1
3 2
a
a
a
a
A
;
2 2
2 1
1 2
1 2
3 3
a
a
a
a
A
matritsaga teskari matritsa toping.
Echish.
A
11
=-2 ; A
12
=-4; A
13
=8; A
21
=3; A
22
=6;
A
23
=-7; A
31
=-10; A
32
=-10; A
33
=20;
u holda
Misol
-
A
0
1
2
2
4
0
3
1
5
10
10
24
4
0
1
2
2
4
0
3
1
5
det
-
A
Δ
-
-
-
-
-
A
2
7
,
0
8
,
0
1
6
,
0
4
,
0
1
3
,
0
2
,
0
1
ChATSni echishda bu usuldan foydalanish uchun, uni
AX=B
( 1 )
ko’rinishda yozib olamiz. Bu yerda
(1) ni
A
-1
ga
ko’paytirib, sistemaning echimini matritsa ko’rinishida
hosil qilamiz
X=A
-1
B
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