Question

Using lagranges interpolation formula find u3 if u0 = 580, u1 = 556, u2 = 520, u4 = 385.

Answer 1

Answer:

u3 = 465.25

Explanation:

x              0              1               2              4

u(x)          580          556          520         385

Here,

x₀ = 0 , x₁= 1 , x₂= 2 , x₄= 3.

f(x₀) = f(0) = u₀ = 580;

f(x₁) = f(1) = u₁ = 556;

f(x₂) = f(2) = u₂ = 520;

f(x₄) = f(4) = u₄ = 385;

By using Lagrange's formula,

f(x) = (x-x₁)(x-x₂)(x-x₄) /(x₀-x₁)(x₀-x₂)(x₀-x₄) f(x₀) + (x-x₀)(x-x₂)(x-x₄) /(x₁-x₀)(x₁-x₂)(x₁-x₄) f(x₁) + (x-x₀)(x-x₁)(x-x₄) /(x₂-x₀)(x₂-x₁)(x₂-x₄) f(x₂) +  (x-x₀)(x-x₁)(x-x₂) /(x₄-x₀)(x₄-x₁)(x₄-x₂) f(x₄)

f(3) = (3-1)(3-2)(3-4) / (0-1)(0-2)(0-4) * 580 + (3-0)(3-2)(3-4) / (1-0)(1-2)(1-4) * 556 + (3-0)(3-1)(3-4) / (2-0)(2-1)(2-4) * 520 +  (3-0)(3-1)(3-2) / (4-0)(4-1)(4-2) * 385

= (2*1*-1) / (-1*-2*- 4) * 580  + (3-*1*-1) / (1*-1*-3) * 556 + (3*2*-1) / (2*1*-2) * 520 + (3*2*1) / (4*3*2) *385

= 145-556+780+96.25

= 465.25