Question

16. The polynomial f(x) by using lagrange's interpolation formula for the following data
X
0 1
2 5
f(x) 2
3
12
147
​

Answer 1

Given :  

x      f(x)

0       2

1        3

2       12

5       147

To find : The polynomial f(3) by using lagrange's interpolation formula

Solution:

f(0) =  2    => x₀ = 0     y₀ = 2

f(1)  = 3   => x₁ = 1     y₁ = 3

f(2) =12   => x₂ = 2     y₂  = 12

f(5) =147   => x₃ = 5     y₃  = 147

Using lagrange's interpolation formula

f(x) = y₀ (x - x₁)(x - x₂)(x-x₃) /  (x₀ - x₁)(x₀ - x₂)(x₀-x₃)   +  y₁ (x - x₀)(x - x₂)(x-x₃) / (x₁ - x₀)(x₁ - x₂)(x₁-x₃)    + y₂ (x - x₀)(x - x₁)(x-x₃)/(x₂ - x₀) (x₂ - x₁)(x₂-x₃)  +  y₃ (x - x₀)(x - x₁)(x-x₂)/(x₃ - x₀) (x₃ - x₁)(x₃-x₂)

f(3) = 2(3 - 1)(3 - 2)(3-5 )/(0 - 1)(0-2)(0 - 5)  +  3(3 - 0)(3- 2)(3-5 )/(1 - 0)(1-2)(1 - 5) +12(3 - 0)(3 - 1)(3-5 )/(2 - 0)(2-1)(2 - 5) + +147(3 - 0)(3 - 1)(3-2 )/(5 - 0)(5-1)(5 - 2)

= -0.8  - 4.5 +  24 + 14.7

= 35

f(3) = 35

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