Hisoblash usullari fanidan laboratoriya ishlari

Lagranj interpoliyatsion formulasi doir misollar.

	X		Variantlar																						X0
			1				2					3				4					5
			Y=				Y=					Y= EMBED Equation.3				Y= EMBED Equation.3					Y= EMBED Equation.3
	X0=0.41		Y0=1.5068				Y0=0.4346					Y0=0.0998				Y0=0.9171					Y0=0.6403				0.38
	X1=0.46		Y1=1.5841				Y1=0.4954					Y1=0.4439				Y1=0.8961					Y1=0.6782				0.43
	X2=0.52		Y2=1.6820				Y2=0.5725					Y2=0.4969				Y2=0.8678					Y2=0.7211				0.48
	X3=0.60		Y3=1.8221				Y3=0.6841					Y3=0.5646				Y3=0.8253					Y3=0.7746				0.74
	X4=0.65		Y4=1.9155				Y4=0.7602					Y4=0.6052				Y4=0.7961					Y4=0.8062
	X5=0.72		Y5=2.05444				Y5=0.9316					Y5=0.6593				Y5=0.7518					Y5=0.8485
	X		Variantlar																							X0
			6					7						8				9				10
			Y=					Y= EMBED Equation.3						Y= EMBED Equation.3				Y= EMBED Equation.3				Y= EMBED Equation.3
	X0=0,11		Y0=1,1163					Y0=0,1104						Y0=0,1098				Y0=0,9940				Y0=0,3317				0,08
	X1=0,16		Y1=1,1735					Y1=0,1514						Y1=0,1593				Y1=0,9872				Y1=0,4000				0,18
	X2=0,22		Y2=1,2461					Y2=0,2236						Y2=0,2182				Y2=0,9759				Y2=0,4690				0,33
	X3=0,30		Y3=1,3498					Y3=0,3093						Y3=0,2956				Y3=0,9553				Y3=0,5477				0,44
	X4=0,35		Y4=1,4191					Y4=0,3650						Y4=0,3429				Y4=0,9394				Y4=0,5916
	X5=0,42		Y5=1,5220					Y5=0,4466						Y5=0,4078				Y5=0,9131				Y5=0,6481
	X			Variantlar																							X0
				11					12				13						14				15
				Y=					Y= EMBED Equation.3				Y= EMBED Equation.3						Y= EMBED Equation.3				Y= EMBED Equation.3
	X0=0.21			Y0=1.2337					Y0=0.2131				Y0=0.2085						Y0=0.9780				Y0=0.4582				0.19
	X1=0.26			Y1=1.2969					Y1=0.2660				Y1=0.2571						Y1=0.9664				Y1=0.5099				0.28
	X2=0.32			Y2=1.3771					Y2=0.3314				Y2=0.3146						Y2=0.9492				Y2=0.5657				0.43
	X3=0.40			Y3=1.4918					Y3=0.4228				Y3=0.3894						Y3=0.9211				Y3=0.6324				0.54
	X4=0.45			Y4=1.5683					Y4=0.4830				Y4=0.4350						Y4=0.9004				Y4=0.6708
	X5=0.52			Y5=1.6820					Y5=0.5726				Y5=0.4969						Y5=0.8678				Y5=0.7211
	X		Variantlar																								X0
			16			17					18						19					20
			Y=			Y= EMBED Equation.3					Y= EMBED Equation.3						Y= EMBED Equation.3					Y= EMBED Equation.3
	X0=0.31		Y0=1.3634			Y0=0.3208					Y0=0.3051						Y0=0.9523					Y0=0.5568					0.28
	X1=0.36		Y1=1.4333			Y1=0.3776					Y1=0.3523						Y1=0.9359					Y1=0.6000					0.33
	X2=0.42		Y2=1.5220			Y2=0.4466					Y2=0.4078						Y2=0.9131					Y2=0.6481					0.53
	X3=0.50		Y3=1.6487			Y3=0.5463					Y3=0.4794						Y3=0.8776					Y3=0.7071					0.64
	X4=0.68		Y4=1.7332			Y4=0.6131					Y4=0.5227						Y4=0.8525					Y4=0.7416
	X5=0.62		Y5=1.8539			Y5=0.7139					Y5=0.5810						Y5=0.8139					Y5=0.7874
X		Variantlar																						X0
		21			22					23					24					25
		Y=			Y= EMBED Equation.3					Y= EMBED Equation.3					Y= EMBED Equation.3					Y= EMBED Equation.3
X0=051		Y0=1.6653			Y0=0.5593					Y0=0.4882					Y0=08722					Y0=0.7141				0.48
X1=0.56		Y1=1.7506			Y1=0.6269					Y1=0.5312					Y1=0.8472					Y1=0.7483				0.58
X2=0.62		Y2=1.8589			Y2=0.7139					Y2=0.5810					Y2=0.8139					Y2=0.7874				0.73
X3=0.70		Y3=2.0138			Y3=0.8423					Y3=0.6442					Y3=0.7648					Y3=0.8367				0.84
X4=0.75		Y4=2.1170			Y4=0.9316					Y4=0.6816					Y4=0.7317					Y4=0.8660
X5=0.82		Y5=2.2705			Y5=1.0717					Y5=0.7311					Y5=0.6822					Y5=0.9055

LABORATORIYA ISHI № 17

Tugunlar orasidagi masofa teng bo’lmaganda Nyuton
interpolyatsion formulasi.
Bo’lingan chekli ayirmalar. Biz oldin chekli ayirmalar tushunchasi bilan tanishgan edik va uning teng oraliqli interpolyatsion formulalarda ishlatilishini ko’rib o’tgan edik.
Tugunlar orasi teng bo’lmagan interpolyatsion formulalar va empirik formulalarda bo’lingan chekli ayirmalar ishlatiladi.
Bizga EMBED Equation.3 berilgan bo’lsin bu funktsiya jadval usulida berilgan bo’lsin argumentning EMBED Equation.3 qiymatlari mos keladi.
Birinchi tartibli bo’lingan chekli ayirma
EMBED Equation.3
va hokazo.
Shunga o’xshash ikkinchi tartibli bo’lingan chekli ayirma
EMBED Equation.3
Umuman tartibli bo’lingan chekli ayirmani EMBED Equation.3 tartibli bo’lingan chekli ayirma bilan ifodalash mumkin.

EMBED Equation.3 (1)

Bo’lingan chekli ayirmalarda elementlarning o’rnini almashtirgan bilan qiymati o’zgarmaydi (simmetrik funksiya).

EMBED Equation.3 (2)
Amaliy vazifalarni bajarishda bo’lingan chekli ayirmalar jadvalidan foydalangan ma’qul.
Bo’lingan chekli ayirmalar jadvali.

X	Y	BO’LINGAN chekli ayirmalar
		1-tartibli	2-tartibli	3-tartibli	4-tartibli
X0	y 0
		[x0, x1]
X1	y 1		[x0 ,x1 , x2]
		[x1, x2]		[x0 ,x1 , x2 , x3]
X2	y2		[x1 ,x2 , x3]		[x0 ,x1 , x2 , x3]
		[x2, x3]		[x1 ,x2 , x3 , x4]
X3	y3		[x2 ,x3 , x4]
		[x3, x4]
X4	y4

Bo’lingan chekli ayirmalar tushunchasi yordamida Lagranj interpolyatsion formulasini Nyutonning birinchi interpolyatsion formulasiga o’xshash interpolyatsion formula bilan yechish mumkin. Buning uchun bitta lemma isbot qilamiz.

n –tartibli ko’phadning n+1 tartibli bo’lingan chekli ayirmasi haqidagi lemma.
Lemma: n – tartibli EMBED Equation.3 ko’phadning EMBED Equation.3 tartibli bo’lingan chekli ayirmasi nolga teng.
Isbot qilish kerak EMBED Equation.3
Isbot:
EMBED Equation.3
EMBED Equation.3 ko’phad EMBED Equation.3 tartiblidir.
EMBED Equation.3
EMBED Equation.3 tartibli ko’phaddir.
Chunki EMBED Equation.3 ko’phadning ildizi. EMBED Equation.3 .
Demak Bezu teoremasiga asosan EMBED Equation.3 ga bo’linadi. Huddi shunday mulohaza qilsak
EMBED Equation.3
ko’phad nolinchi darajali ko’phaddir ya’ni
EMBED Equation.3
Bundan
EMBED Equation.3
Lemma isbot qilindi.
EMBED Equation.3 - n-darajali Lagranj interpolyatsion formulasi bo’lsin
EMBED Equation.3 (1)
Tubandagi belgilashlarni kiritamiz.
EMBED Equation.3 (2)
Lemmaga asosan
EMBED Equation.3 (3)
Ta’rifga asosan
EMBED Equation.3 (4)
Bundan
EMBED Equation.3 (5)
Bo’lingan chekli ayirmaning ta’rifiga asosan
EMBED Equation.3 (6)
(5) va (6) dan foydalanib
EMBED Equation.3
EMBED Equation.3 Lemmaga asosan.
EMBED Equation.3 (7)
(7) formulaga tugunlar orasidagi masofa teng bo’lganda Nyuton interpolyatsion formulasi deyiladi.