Question

ALGUÉM ME AJUDA????????

O objetivo do meÌ?todo de Lagrange eÌ? determinar o valor do polinoÌ‚mio interpolador p na abcissa upsilon, sem a necessidade de determinar previamente os coeficientes de p. Dados as abscissas dos nodos x subscript 1 less than x subscript 2 less than... less than x subscript n, constrói-se um conjunto de n polinoÌ‚mios auxiliares Li dados por:



L subscript 1 left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus x subscript 2 right parenthesis left parenthesis x minus x subscript 3 right parenthesis left parenthesis x minus x subscript 4 right parenthesis... left parenthesis x minus x subscript n right parenthesis over denominator left parenthesis x subscript 1 minus x subscript 2 right parenthesis left parenthesis x subscript 1 minus x subscript 3 right parenthesis left parenthesis x subscript 1 minus x subscript 4 right parenthesis... left parenthesis x subscript 1 minus x subscript n right parenthesis end fraction
L subscript 2 left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus x subscript 1 right parenthesis left parenthesis x minus x subscript 3 right parenthesis left parenthesis x minus x subscript 4 right parenthesis... left parenthesis x minus x subscript n right parenthesis over denominator left parenthesis x subscript 2 minus x subscript 1 right parenthesis left parenthesis x subscript 2 minus x subscript 3 right parenthesis left parenthesis x subscript 2 minus x subscript 4 right parenthesis... left parenthesis x subscript 2 minus x subscript n right parenthesis end fraction
L subscript 3 left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus x subscript 1 right parenthesis left parenthesis x minus x subscript 2 right parenthesis left parenthesis x minus x subscript 4 right parenthesis... left parenthesis x minus x subscript n right parenthesis over denominator left parenthesis x subscript 3 minus x subscript 1 right parenthesis left parenthesis x subscript 3 minus x subscript 2 right parenthesis left parenthesis x subscript 3 minus x subscript 4 right parenthesis... left parenthesis x subscript 3 minus x subscript n right parenthesis end fraction
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L subscript n left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus x subscript 1 right parenthesis left parenthesis x minus x subscript 2 right parenthesis left parenthesis x minus x subscript 3 right parenthesis... left parenthesis x minus x subscript n minus 1 end subscript right parenthesis over denominator left parenthesis x subscript n minus x subscript 1 right parenthesis left parenthesis x subscript n minus x subscript 2 right parenthesis left parenthesis x subscript n minus x subscript 3 right parenthesis... left parenthesis x subscript n minus x subscript n minus 1 end subscript right parenthesis end fraction



ou seja,



L subscript 1 left parenthesis x right parenthesis equals product from 1 equals 1 comma j not equal to i to n of fraction numerator x minus x j over denominator x i minus x j end fraction comma space space i equals 1 comma... comma n.



Correlacionando o texto a notação de Lagrange, Dada a Tabela a seguir, determine o valor de f (8), fazendo a interpolação na forma de Lagrange:



x

3

7

10

F(x)

5

9

11

Answer 1

Os polinômios interpoladores são

$L_{1}(x) = \frac{(x-7)(x-10)}{(3-7)(3-10)} = \frac{(x-7)(x-10)}{28}$

$L_{2}(x) = \frac{(x-3)(x-10)}{(7-3)(7-10)} =\frac{(x-3)(x-10)}{-12}$

$L_{3}(x) = \frac{(x-3)(x-7)}{(10-3)(10-7)} = \frac{(x-7)(x-10)}{21}$

Combine os polinômios dessa forma para obter a função de interpolação:
$f(x) = F(x_1)L_1(x) + F(x_2)L_2(x) + F(x_3)L_3(x)$

$f(x) = 5 L_1(x) + 9 L_2(x) + 11 L_3(x)$

Finalmente, substitua x por 8. Como temos alguns termos repetidos, temos
x - 3 => 5
x - 7 => 1
x - 10 => -2

O resultado será
$f(8) = \frac{5.1.(-2)}{28} + \frac{9.5.(-2)}{-12} + \frac{11.5.1}{21}$

$f(8) = \frac{(-12).21.5.1.(-2)+28.21.9.5.(-2)+(-12).21.11.5.1}{28.(-12).21}$

$f(8) = \frac{2520-52920-13860}{-7056}$

$f(8) = \frac{-64260}{-7056}$

$f(8) = \frac{255}{28} = 9.10714285714285...$

Espero não ter errado na conta. Alguém me avisa se tiver algum erro!

Answer 2

Resposta corrigida de acordo com anexo espero que ajude