Algoritmlarni loyhalash Mustaqil ish Mavzu: Chiziqli masalalri uchun egizak masala, uni tuzish va iqtsodiy manosini tahlil qilish Gruruh: 431-21 /cal015 Bajardi: Madraximov Abdulhamid Tekshirdi

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 x_n 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,...., x_k noma’lumning (k+1)-yaqinlashishi (iteratsiya usulida k– yaqinlashish) e’tiborga olinadi.
x_i 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