III TEORIJA APROKSIMACIJE

III TEORIJA APROKSIMACIJEZADATAK 22.

Odrediti polinom minimalnih kvadrata reda 1 i 2 za podatke koji su prikazani u slijedeoj tabeli.

ixiyi

001,0

10,151,004

20,311,031

30,51,117

40,61,223

50,751,422

Koji stepen je bolja aproksimacija, odnosno koja ima manju greku?

a)

Polinom prvog reda oblika: y=ax + b ili y=a1x1 + a0 n=1,

broj nepoznatih = n+1 =2 (a=? i b=?)

Potrebne dvije normalne jednaine:

(1)

(2)

Formirat emo tabelu:

ixiyixi2xi yi

001,000

10,151,0040,02250,1506

20,311,0310,09610,3196

30,51,1170,250,5585

40,61,2230,360,7338

50,751,4220,56251,0665

Suma2,316,7971,29112,829

(1) a1, 2911 + b2,31=2,829 / 2,31

(2) a2,31 + b6 =6,797 /1,2911

a2,9 82 + b5,3361=6,535

a2,982 + b7,7466=8,7756

2,41 b=2,24 ( b=0,9295

(2) ( a=(6,797 -6b)/2,31 =0,5281

POLINOM PRVOG REDA: Y=0,5281x + 0,9295

GREKA:

yixiY=0,5281x + 0,9295(yi Y)2

1,000,92950,004970

1,0040,151,0090,000025

1,0310,311,0930,003844

1,1170,51,1930,005776

1,2230,61,2460,000529

1,4220,751,3260,009216

Suma0,024360

b)

Polinom drugog reda oblika: y=a2x2+ a1x1 + a0 n=2,

broj nepoznatih = n+1 =3 (a0 =? ; a1 =? i a2 =?)

Potrebne tri normalne jednaine:

(1)

(2)

(3)

Zatim emo formirati tabelu:

ixiyixi2xi3xi4xi yixi2 yi

001,000000

10,151,0040,02250.0033750,0005060,15060,02259

20,311,0310,09610,0297910,009230,31960,09908

30,51,1170,250,1250,06250,55850,27925

40,61,2230,360,2160,12960,73380,44028

50,751,4220,56250,421870,31641,06650,79987

Suma2,316,7971,29110,7960410,518242,8291,641

6a0 + 2,31a1 + 1,2911a2 = 6,797 /(-2,31/6) /(-1,2911/6)

2,31a0 + 1,2911a1 + 0,796041a2 = 2,829 +1,2911a0 + 0,796041a1 + 0,51824a2 = 1,641+

6a0 + 2,31a1 + 1,2911a2 = 6,797

0,40175a1 + 0,299a2 = 0,2122 /(-0,29897/0,40175)

0,29897a1 + 0,2404a2 = 0,1784 +

6a0 + 2,31a1 + 1,2911a2 = 6,797

0,40175a1 + 0,299a2 = 0,2122

0,0179a2 = 0,0205

a2=1,1473a1 =(0,2122-0,2991,1473)/0,40175= -0,3257a0=(6,797-1,29111,1473+2,310,3257)/6=1,01134

POLINOM DRUGOG REDA: Y=1,01134 - 0,3257X +1,1473X2GREKA:

xixi2Y=1,01134 - 0,3257X+1,1473X2yi(yi Y)2

001,011341,00,0001285

0,150,02250,98831,0040,000246

0,310,09611,020631,0310,0001075

0,50,251,1353151,1170,0003354

0,60,361,228951,2230,0000354

0,750,56251,41241,4220,0000921

Suma 0,0009449

Bolju aproksimaciju daje polinom drugog reda jer ima manju greku.

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