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Задать вопрос экспертам

Консультация № 184460

15.11.2011, 21:34

54.66 руб.

0 1 1

Образование

Здравствуйте! Прошу помощи в следующем вопросе:
Вопрос по вычислительной математике:

Построить интерполяционный полином Лагранжа. И выполнить проверку.
Дана табица

| i | x | y |
| 0 | -1 | 2 |
| 1 | -1/2 | 1 |
| 2 | 0 | -1 |
| 3 | 1/2 | 2 |

Обсуждение

давно

Орловский Дмитрий

Мастер-Эксперт
319965
1463

15.11.2011, 22:21

общий

это ответ

Здравствуйте, Евгений!
Составляем элементарные многочлены, отвечающие заданным узлам:
x=-1 ---> L₀(x)=(x+1/2)x(x-1/2)/(-1+1/2)(-1)(-1-1/2)=-(4/3)(x+1/2)x(x-1/2)
x=-1/2 ---> L₁(x)=(x+1)x(x-1/2)/(-1/2+1)(-1/2)(-1/2-1/2)=4(x+1)x(x-1/2)
x=0 ---> L₂(x)=(x+1)(x+1/2)(x-1/2)/(0+1)(0+1/2)(0-1/2)=-4(x+1)(x+1/2)(x-1/2)
x=1/2 ---> L₃(x)=(x+1)(x+1/2)x/(1/2+1)(1/2+1/2)(1/2+0)=(4/3)(x+1)(x+1/2)x
Интерполяционный многочлен:
L(x)=2*L₀(x)+1*L₁(x)+(-1)*L₂(x)+2*L₃(x)=
=(-8/3)(x+1/2)x(x-1/2)+4(x+1)x(x-1/2)+4(x+1)(x+1/2)(x-1/2)+(8/3)(x+1)(x+1/2)x
Проверка:
L(-1)=2+0+0+0=2
L(-1/2)=0+1+0+0=1
L(0)=0+0-1+0=-1
L(1/2)=0+0+0+2=2

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