Hisoblash usullari fanidan laboratoriya ishlari

LABORATORIYA ISHI №19
Nuqtali kvadratik approksimatsiya.
Biz oldin funktsiyalarni almashtirishning interpolyatsiya usulini ko’rib o’tgan edik. Bunda n+1 ta tugunga ega bo’lsa, n - tartibli ko’phad yozar edik. n etarli darajada katta bo’lganda bu ko’phadni amaliy jihatdan hisoblash maqsadga muvofiq emas. Lekin, shu tugunlardan o’tmaydigan tugunlarga yaqin nuqtalardan o’tadigan funktsiyalarning analitik ko’rinishini yozish mumkin. Nuqtalar soni n+1 ta bo’lsin. Uni shunday biror ko’phad shaklida izlaymiz.
EMBED Equation.3 (1)
Bunda m<n . Bu ko’phadni berilgan funktsiyadan farqi
EMBED Equation.3 (2)
bo’lsin. Xatoning kvadratlari yig’indisi EMBED Equation.3 ni shunday tanlash kerakki, (2) minimumga ega bo’lishi kerak. Xosil qilingan ko’phad berilgan funktsiya uchun appraktsimatsiyalovchi deb aytiladi, bu ko’phadni ko’rish jarayoni nuqtali kvadratik appraktsimatsiya yoki funktsiyani nuqtali kvadratik appraktsimatsiyalash deyiladi. (2) da kvadratni olinishi EMBED Equation.3 ning yig’indida biri musbat, biri manfiy bo’lishi mumkin. EMBED Equation.3 ni minimumini topish uchun EMBED Equation.3 lar bo’yicha hosila olib, ularni 0 ga tenglashtirib kritik nuqtalarini topamiz.
EMBED Equation.3
Bu erda EMBED Equation.3 barcha EMBED Equation.3 bo’yicha
EMBED Equation.3 EMBED Equation.3 EMBED Equation.3
(3) tenglamalar sistemasi EMBED Equation.3 larga nisbatan m+1 noma’lumli m+1 tenglamalar sistemasidir. Sistema to’liq. x_i lar ustma ust tushmasa yagona yechimga ega.
(3) sistema uchun quyidagi belgilashlarni kiritamiz:
EMBED Equation.3
Bu belgilashlar yordamida (3) ni quyidagi shaklda ham yozish mumkin.

EMBED Equation.3

Bu erda EMBED Equation.3

(3) sistemani tuzish uchun kichik kvadratlar usulini sxemasi tavsiya qilinadi. Bu 1-jadvalda keltirilgan. U erda m=2 deb qabul qilingan.
1-jadval

x^0	x	x^2.	x^3	x^4	y	xy	x^2 y
1 1 1 1 1	x0 x1 x2 x3 x4	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3	EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3 EMBED Equation.3
S0	S1	S2	S3	S4	t0	t1	t2

Misol: Ushbu
EMBED Equation.3
2-darajali appraktsimaktsiyalovchi ko’phadini tanlang.
Jadval 2