ВУЗ:
Составители:
y(x, α) = g
0
(x) + a
1
g
1
(x) + a
2
g
2
(x) .
g
0
(x) = 0 , g
1
(x) = x(x −1) , g
2
(x) = x
2
(x − 1) .
x
i
x
1
= 1 , x
2
= 2 .
a
1
a
2
g
00
0
(x) = 0 , g
00
1
(x) = 2 , g
00
2
(x) = 6x −2 ,
Lg
0
(x) = 0 , Lg
1
(x) = −x
2
+x+ 2 , Lg
2
(x) = −x
3
+x
2
+6x−2 ,
Lg
1
(1) = 2 , Lg
2
(1) = 4 , f(1) = 1 ,
Lg
1
(2) = 0 , Lg
2
(2) = 6 , f(1) = 1 ,
½
2a
1
+ 4a
2
= 1 ,
6a
2
= 1 .
a
∗
1
= a
∗
2
=
1
6
.
˜y(x) =
1
6
x(x − 1) +
1
6
x
2
(x − 1) =
x
3
− x
6
. ¤
Ñîãëàñíî óñëîâèþ, ðåøåíèå áóäåì èñêàòü â âèäå y(x, α) = g0 (x) + a1 g1 (x) + a2 g2 (x) . Âûáåðåì áàçèñíûå ôóíêöèè g0 (x) = 0 , g1 (x) = x(x − 1) , g2 (x) = x2 (x − 1) . Çàäàäèì íàáîð òî÷åê xi : x1 = 1 , x 2 = 2 . Ïåðåéäåì ê ñîñòàâëåíèþ ñèñòåìû (4) ïîèñêà ïàðàìåò- ðîâ a1 è a2 . Ïðåäâàðèòåëüíî âûïîëíèì ïðîìåæóòî÷íûå âû÷èñëåíèÿ: g000 (x) = 0 , g100 (x) = 2 , g200 (x) = 6x − 2 , Lg0 (x) = 0 , Lg1 (x) = −x2 +x+2 , Lg2 (x) = −x3 +x2 +6x−2 , Lg1 (1) = 2 , Lg2 (1) = 4 , f (1) = 1 , Lg1 (2) = 0 , Lg2 (2) = 6 , f (1) = 1 , Òàêèì îáðàçîì, ïîëó÷àåì ñèñòåìó ½ 2a1 + 4a2 = 1 , 6a2 = 1 . Îòñþäà 1 a∗1 = a∗2 = . 6  ðåçóëüòàòå ïîëó÷àåì ïðèáëèæåííîå ðåøåíèå 1 1 2 x3 − x ỹ(x) = x(x − 1) + x (x − 1) = . ¤ 6 6 6 61