Hisoblash usullari fanidan laboratoriya ishlari

LABORATORIYA ISHI № 14

Gaussning birinchi interpolyatsion formulasi.
Gaussning birinchi interpolyatsion formulasida interpolyasiya tugunlari bir-biriga teng bo’lganda interpolyasiya tugunlarini quyidagicha belgilaymiz.
EMBED Equation.3
Bu erda
EMBED Equation.3
EMBED Equation.3
funktsiyani qiymatlari bu tugunlarda ma’lum EMBED Equation.3 EMBED Equation.3 daraja ko’rsatkichidan oshmaydigan shunday
EMBED Equation.3 (1)
shartni qanoatlantiruvchi polinom tuzish talab qilinadi.
Polinom ko’rinish tubandagicha axtariladi
EMBED Equation.3 (2)
(2) formulaning koeffitsienti EMBED Equation.3 huddi Nyutonning interpolyatsion formulalariga o’xshash topiladi. Buning uchun (1) foydalanamiz. Gauss interpolyatsion formulasini yozish uchun markaziy chekli ayirmalar jadvalidan foydalanamiz.
EMBED Equation.3 bular markaziy chekli ayirmalar deb aytiladi. Bunda
EMBED Equation.3
va hokazo.
Markaziy chekli ayirmalar jadvali.

x	y	y	^2y	^3y	^4y	^5y	^6y
x-4	y-4
		y-4
x-3	y-3		^2y-4
		y-3		^3y-4
x-2	y-2		^2y-3		^4y-4
		y-2		^3y-3		^5y-4
x-1	y-1		^2y-2		^4y-3		^6y-4
		y-1		^3y-2		^5y-3
x0	y0		^2y-1		^4y-2		^6y-3
		y0		^3y-1		^5y-2
x1	y1		^2y0		^4y-1		^6y-2
		y1		^3y0		^5y-1
x2	y2		^2y1		^4y0
		y2		^3y1
x3	y3		^2y2
		y3
x4	y4

EMBED Equation.3 ;

EMBED Equation.3 ;
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3 ;
Huddi shunday hisoblashlarni amalga oshirish natijasida quyidagilarga ega bo’lamiz.

EMBED Equation.3

EMBED Equation.3
Bu topilgan koeffitsientlarni (2) ga qo’yib quyidagiga ega bo’lamiz
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3 (3)
(3) ga Gaussning birinchi interpolyatsion formulasi deyiladi.
Bunga yangi EMBED Equation.3 o’zgaruvchi kiritib,
EMBED Equation.3

EMBED Equation.3 (4)

natijaga erishish mumkin.

LABORATORIYA ISHI № 15

Gaussning ikkinchi interpolyatsion formulasi.

Gaussning ikkinchi interpolyatsion formulasi EMBED Equation.3 ta interpolyatsiya tugunlari, ya’ni

EMBED Equation.3
uchun tubandagicha axtariladi.
EMBED Equation.3
EMBED Equation.3
EMBED Equation.3 (1)
(1) ning koeffitsentlarini topish huddi oldingidek.
EMBED Equation.3
EMBED Equation.3 (2)
(2) ga Gaussning ikkinchi interpolyatsion formulasi deyiladi.
Bunda ham yangi EMBED Equation.3 o’zgaruvchi kiritib, quyidagi natijani olamiz.
EMBED Equation.3
EMBED Equation.3 (3)
Gaussning birinchi va ikkinchi interpolyatsion formulalari bilan ishlash jarayonida markaziy chekli ayirmalar jadvalidan foydalanish maqsadga muvofiqir.

LABORATORIYA ISHI № 16

Langranj interpolyatsion formulasi.

B iz oldin bir qancha interpolyatsion formulalarni ko’rib o’tdik. Bularning hammasida interpolyatsiya tugunlari orasidagi masofa tengdir. Interpolyatsiya tugunlari teng bo’lmagan hol uchun Lagranj interpolyatsion formulasidan foydalaniladi. Hususiy holda interpolyatsiya qadamlari teng bo’lishi ham mumkin. EMBED Equation.3 oraliqda ixtiyoriy joylashgan EMBED Equation.3 funktsiya argumentlari berilgan va ularga mos EMBED Equation.3 funktsiya qiymatlari

EMBED Equation.3
berilgan bo’lsin (7­-chizma).
Daraja ko’rsatkichidan oshmaydigan shunday ko’phad qurish talab qilinadiki u
EMBED Equation.3 (1)
shartni qanoatlantirsin.
Buning uchun oldin EMBED Equation.3 ko’phadni topamiz bu ko’phad tubandagi shartni qanoatlantirsin
EMBED Equation.3 va EMBED Equation.3 , (2)
ya’ni Kroneker belgilashi bo’yicha
EMBED Equation.3 (3)
EMBED Equation.3 ko’phad EMBED Equation.3 nuqtada nolga aylanadi uning ko’rinishi
EMBED Equation.3 (4)
EMBED Equation.3 desak (3) ga asosan (4)ni tubandagicha yozish mumkin.
EMBED Equation.3
Bundan
EMBED Equation.3 (5)
(3) ni (2) ga olib qo’ysak
EMBED Equation.3 (6)
Bu ko’phad yordamida EMBED Equation.3 shartni EMBED Equation.3 qanoatlantiruvchi ko’phadni yozamiz.
EMBED Equation.3 (7)
ya’ni
EMBED Equation.3 (8)
(6) Lagranj interpolyatsion formulasini qisqaroq yozish uchun tubandagicha belgilashni kiritamiz
EMBED Equation.3 (9)
EMBED Equation.3 (10)
(7) va (8) belgilashga asosan (6) tubandagi

EMBED Equation.3 (11)

formulani hosil qilamiz.

Lagranj koeffitsientlarining ko’rinishiga nisbatan chiziqli x=at+b almashtirish invariantdir. Haqiqatan ham, (11) formulada
EMBED Equation.3
almashtirishlarni bajarib, surat va maxrajdagi EMBED Equation.3 ni qisqartirsak
EMBED Equation.3
formulaga ega bo’lamiz.

Lagranj koefitsentlarini hisoblash.

EMBED Equation.3

(8) Lagranj interpolyatsion formulasining birinchi koeffitsentini topish uchun diagonal elementlar ko’paytmasini birinchi satr elementlar ko’paytmasiga bo’lamiz. Ikkinchi koeffitsentni topish uchun diagonal elementlar ko’paytmasini ikkinchi satr elementlar ko’paytmasiga bo’lamiz va hokazo.

Misol. n=1 hol uchun Lagranj interpolyatsion formulasi ikki nuqtadan o’tuvchi to’g’ri chiziq tenglamasini beradi.
Ushbu
EMBED Equation.3
nuqtalardan o’tuvchi Lagranj interpolyatsion fomulasi.