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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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