Question

Compute the value of f(x) for x = 2.5 from the following
table
x: 1 2 3 4
f(x): 1 8 27 64
Using Lagranges interpolation formula​

Answer 1

Given :  f(1) = 1,f(2) = 8 , f(3) = 27 and f(4) = 64

To find : value of f(2.5) using lagrange's interpolation formula

Solution:

f(1) =  1    => x₀ = 1     y₀ = 1

f(2)  = 8   => x₁ = 2     y₁ = 8

f(3) = 27   => x₂ = 3     y₂  = 27

f(4) = 64   => x₃ = 4     y₃  = 64

Using lagrange's interpolation formula

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)

x= 2.5

= 1(2.5 - 2)(2.5 - 3)(2.5-4 )/(1 - 2)(1-3)(1 - 4)  + 8(2.5 - 1)(2.5 - 3)(2.5-4 )/(2 - 1)(2-3)(2 - 4)  + 27(2.5 - 1)(2.5 - 2)(2.5-4 )/(3 - 1)(3-2)(3 - 4)  + 64(2.5 - 1)(2.5 - 2)(2.5-3 )/(4 - 1)(4-2)(4- 3)

= (0.5)(-0.5)(-1.5)/(-1)(-2)(-3)  +  8(1.5)(-0.5)(-1.5)/1(-1)(-2)  + 27(1.5)(0.5)(-1.5)/2(1)(-1) + 64(1.5)(0.5)(-0.5)/3(2)(1)

= -0.0625  + 4.5 + 15.1875 - 4

= 15.625

f(2.5) = 15.625

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