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
Answers
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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