-mavzu: Chiziqli algebraik tenglamalar sistemasini taqribiy yechishning oddiy iteratsiya va Zeydel usullari. Iteratsion jarayon yaqinlashishining zaruriy shartlari

|
-mavzu: Chiziqli algebraik tenglamalar sistemasini taqribiy yechishning oddiy iteratsiya va Zeydel usullari. Iteratsion jarayon yaqinlashishining zaruriy shartlari
|
bet | 11/11 | Sana | 23.01.2024 | Hajmi | 1,34 Mb. | | #143898 |
Bog'liq 11-mavzu Chiziqli fazoning ta’rifi va misollar. Chiziqli fazoni15-mavzu: Chiziqli algebraik tenglamalar sistemasini taqribiy yechishning oddiy iteratsiya va Zeydel usullari. Iteratsion jarayon yaqinlashishining zaruriy shartlari.
- tartibli ta chiziqli algebraik tenglamalar sistemasining ko’rinishi quyidagi ifodadan iiborat:
(1)
Bu yerda lar ma’lum sonlardan iborat bo’lib, noma’lumlarning
k o e f f i ts i ye n t l a r i deyiladi, - noma’lumlar, - (1) sistema tenglamalarining ozod hadlari, ular ham ma’lum sonlardan iborat.
Tenglamalar sietemasining matritsa ko’rinishi:
(2)
Bu yerda
(3)
Zeydel usulining mohiyati. A matritsaning diagonal elementlari noldan farqli bo’lsin i=1,2,3,…,n. Sistemaning birinchi tenglamasini x1 ga, ikkinchi tenglamasini x2 ga va hokazo n – tenglamasini xn ga nisbatan yechib,
(4)
sistemaga ega bo’lamiz, bu yerda
Zeydel usuli ketma – ket yaqinlashish usulining boshqacha ko’rinishi bo’lib, unda (k+1) – yaqinlashishni hisoblashda yangi topilgan x1,x2,...., xk noma’lumning (k+1)-yaqinlashishi (iteratsiya usulida k– yaqinlashish) e’tiborga olinadi.
xi noma’lumning k – yaqinlashishi ma’lum bo’lsin, u holda (k+1) – yaqinlashish quyidagi formula bilan topiladi.
Argar keltirilgan (4) sistema uchun ; i,j=1,2,3,…n. shartlarning birortasi o’rinli bo’lsa Boshlang’ich yaqinlashish qanday tanlanishidan qat’iy nazar tenglamalar sistemasi yagona yechimga yaqinlashadi. (1)sistema uchun iteratsiya usulida yaqinlashish sharti
Masalan: Quyidagi tenglamalar sistemasining yechimini
aniqlikda topish talab etilsin.
Tenglamalar sistemasini yechish uchun quyidagi amallarni bajaramiz.
1. SHartni tekshiramiz,ya’ni
2. Shart bajarilayapti demak tenglamalar sistemasini Zeydel usulida yechimni topish mumkin.
3. Tenglamalar sistemasini mos ravishda x1, x2,x3 noma’lumlarga nisbatan yechamiz:
4.k=0 deb birinchi yaqinlashishni aniqlaymiz.
5.k=1 deb ikkinchi yaqinlashishni aniqlaymiz:
6. Yaqinlashish shartini tekshiramiz:
Topilgan yechim berilgan aniqlikni qanoatlantirmayapti shuning uchun jarayonni davom ettiramiz.
7.k=3 deb uchunchi yaqinlashishni aniqlaymiz:
8. Yaqinlashish shartini tekshiramiz:
Topilgan yechim berilgan aniqlikni qanoatlantiryapti. Demak tenglamalar sistemasining taqribiy yechimi: x1=0.3618; x2 =0.1481; x3=-0.4743 ekan.
Topshi- riq
Tartibi
|
A matiritsaning koeffitsienlari
|
Ozod
had
|
Topshi- riq
Tartibi
|
A matiritsaning koeffitsienlari
|
Ozod
had
|
1
|
2
|
3
|
|
1
|
2
|
3
|
|
1
|
1
|
13.47
|
-2.03
|
3.29
|
2.32
|
2
|
1
|
9.66
|
2.01
|
3.03
|
-2.29
|
2
|
2.75
|
11.11
|
2.28
|
4.75
|
2
|
3.22
|
12.41
|
1.65
|
2.64
|
3
|
0.28
|
6.25
|
-9.21
|
2.25
|
3
|
1.69
|
-2.17
|
13.65
|
-6.48
|
3
|
1
|
15.75
|
2.91
|
3.6
|
-2.84
|
4
|
1
|
12.88
|
0.28
|
0.99
|
-2.64
|
2
|
3.63
|
12.02
|
6.71
|
9.81
|
2
|
1.77
|
9.79
|
2.81
|
4.78
|
3
|
2.28
|
3.48
|
15.78
|
2.71
|
3
|
2.83
|
3.02
|
11.79
|
-2.71
|
5
|
1
|
12.85
|
0.75
|
3.21
|
-1.74
|
6
|
1
|
-6.75
|
0.24
|
1.21
|
0.08
|
2
|
-0.97
|
11.04
|
4.48
|
2.83
|
2
|
7.75
|
19.75
|
0.95
|
-1.75
|
3
|
0.77
|
1.43
|
9.71
|
0.92
|
3
|
2.81
|
2.63
|
13.45
|
4.86
|
7
|
1
|
17.28
|
3.48
|
2.64
|
-2.22
|
8
|
1
|
3.75
|
0.28
|
1.05
|
1.28
|
2
|
3.44
|
12.35
|
2.66
|
2.38
|
2
|
0.75
|
4.95
|
3.07
|
3.75
|
3
|
4.48
|
2.88
|
-14.37
|
-4.75
|
3
|
4.88
|
-0.88
|
6.75
|
0.08
|
9
|
1
|
18.88
|
0.29
|
1.75
|
-4.35
|
10
|
1
|
9.77
|
0.37
|
1.43
|
-2.33
|
2
|
0.78
|
19.99
|
8.78
|
2.35
|
2
|
3.23
|
18.91
|
8.71
|
0.78
|
3
|
4.75
|
0.75
|
10.37
|
-0.47
|
3
|
4.48
|
-9.77
|
15.75
|
3.78
|
11
|
1
|
7.71
|
2.83
|
1.08
|
2.39
|
12
|
1
|
17.79
|
3.21
|
6.71
|
0.73
|
2
|
0.77
|
16.61
|
-8.91
|
-0.33
|
2
|
2.22
|
-3.33
|
-0.7
|
2.81
|
3
|
0.48
|
-8.84
|
18.63
|
6.61
|
3
|
2.93
|
3.96
|
14.75
|
-0.78
|
13
|
1
|
13.75
|
2.69
|
0.71
|
3.33
|
14
|
1
|
3.78
|
-0.75
|
1.21
|
2.83
|
2
|
2.33
|
12.78
|
3.75
|
-6.36
|
2
|
0.48
|
3.73
|
0.75
|
-7.38
|
|
|
Bosh sahifa
Aloqalar
Bosh sahifa
-mavzu: Chiziqli algebraik tenglamalar sistemasini taqribiy yechishning oddiy iteratsiya va Zeydel usullari. Iteratsion jarayon yaqinlashishining zaruriy shartlari
|