Hisoblash usullari fanidan laboratoriya ishlari

11. A= EMBED Equation.3 ,12. A= EMBED Equation.3 ,

13. A= EMBED Equation.3 , 14. A= EMBED Equation.3 ,

15. A= EMBED Equation.3 , 16. A= EMBED Equation.3 ,

17. A= EMBED Equation.3 , 18. A= EMBED Equation.3 ,

19. A= EMBED Equation.3 , 20. A= EMBED Equation.3 ,

21. A= EMBED Equation.3 , 22. A= EMBED Equation.3 ,

23. A= EMBED Equation.3 ,24. A= EMBED Equation.3 ,

25. A= EMBED Equation.3 ,

LABORATORIYA ISHI № 10

Tenglamalar sistemasi uchun iteratsiya

(ketma-ket yakinlashish) usuli.

Aniq yechimni beruvchi Gauss usuli noma’lumlar soni ko’p bo’lgani bu usul bilan yechish murakkablashadi. Bunday hollarda tenglamalar sistemasini iteratsiya (ketma-ket yaqinlashish) usuli bilan yechish amaliy jihatdan foydali. Biz bu usulni chiziqli algebraik tenglamalar sistemasiga tadbiq qilishni ko’rsatamiz. Bizga chiziqli tenglamalar sistemasi berilgan bo’lsin.

EMBED Equation.3 (1)
(1) sistemani matritsa ko’rinishida yozamiz:
EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 ,
bular yordamida (1) tenglamalar sistemasini quyidagi matritsali tenglamalar ko’rinishida yozish mumkin:
EMBED Equation.3 (1*)
A matritsaning dioganal koeffitsientlari EMBED Equation.3 . (1) sistemaning birinchi tenglamasini EMBED Equation.3 ga nisbatan yechsak, ikkinchi tenglamasini EMBED Equation.3 nisbatan yechsak va x.k. U holda (1) sistemaga teng kuchli tenglamalar sistemasini hosil qilamiz.
EMBED Equation.3 (2)
bu erda EMBED Equation.3
(2) sistema o’rniga matritsali tenglama hosil bo’ladi.
EMBED Equation.3 (2*)
Bu usul odatda Yakobi metodi deyiladi. Albatta bu usulni qo’llash uchun barcha EMBED Equation.3 lar noldan farqli bo’lishi kerak. Bundan tashqari diagonal elementlarning moduli boshqa elementlar modullari yig’indisidan katta bo’lishi kerak. Yani quyidagi tengsizliklarning birortasi bajarilishi kerak.
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3 .
(2) sistemani ketma-ket yaqinlashish usuli bilan yechamiz. Nolinchi yaqinlashish sifatida EMBED Equation.3 vektor ustunni olamiz. (2*) foydalanib
EMBED Equation.3
birinchi yaiqnlashishni hosil qilamiz. Bundan foydalanib
EMBED Equation.3
hosil qilamiz. Shu tartibda EMBED Equation.3 - yaqinlashishni hisoblash uchun quyidagi formuladan foydalanamiz:
EMBED Equation.3 (3)
Agar yaqinlashish EMBED Equation.3 qatori limiti, ya’ni
EMBED Equation.3 ,
bu limit (2) sistemaning yechimidir. Haqiqatan ham (3) dan limit olsak,

EMBED Equation.3

yoki
EMBED Equation.3

limitik vektor X (2*) – sistemaning yechimidir. Huddi shunday (1) ning ham yechimidir. (3) quyidagicha yozamiz.

EMBED Equation.3 (3*)
Ayrim hollarda (1) sistemani (2) sistemaga keltirishda EMBED Equation.3 bo’lishi mumkin.
Masalan: EMBED Equation.3 tenglamaga iteratsiya usulini qo’llashda EMBED Equation.3 shaklda yozish mumkin. Bundan keyin EMBED Equation.3 bo’lishi shart emas. (3) va (3*) formulalar ildizni aniqlovchi usul, oddiy iteratsiya usuli deb aytiladi.