Lagrangian Interpolation Formula

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Lagrangian Interpolation Formula :

       Let y = f(x) be a real valued function which is defined in an interval [a,b] . Let 
x₀ , x₁ ,x₂ ,............xₙ be n+1 distinct points in that interval at which the respective values y₀ ,y₁,y₂ ............yₙ are tabulated.

     Now our aim is to construct a polynomial Φ(x) of degree ≤ n , which interpolates f(x) such that 

     Φ(xᵢ) = y(xᵢ) , i = 1,2,3,.......,n .........(1)

   Let us suppose that the polynomial Φ(x)
                                  n
be given by Φ(x) = Σ  lᵢ(x) y(xᵢ)  .........(2)
                                 i= 0

     where each lᵢ(x) is a polynomial of degree ≤n in xᵢ , called  Lagrangian function.

     The function given in equation (1) if each lᵢ (x) satisfied lᵢ(xⱼ) = 0 where i ≠ j  ,lᵢ(xⱼ)=1  when i=j  .........(3)
Now ,
  the polynomial lᵢ(x) vanishes at the (n+1) points xₒ , x₁ , ........., xₙ therefore , we can write this polynomial in the following form

lᵢ(x) = cᵢ (x-x₀)(x-x₁)........(x-xᵢ₋₁)(x-xᵢ₊₁)....(x-xₙ)
                                                              ........(4)

where cᵢ's are constant co_efficients

putting x = xᵢ in equation (4) , we get

lᵢ(xᵢ) =cᵢ(xᵢ-x₀)(xᵢ-x₁).....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)

or, 1 = cᵢ(xᵢ-x₀)(xᵢ-x₁).....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)
                                       ......(xᵢ-xₙ)     as lᵢ(xᵢ) =1

or,cᵢ = 1/(xᵢ-x₀)(xᵢ-x₁)...(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)
                                                         ..........(5)

putting the value of cᵢ from equation (5) in equation (4) , we get

lᵢ(x) = (x-x₀)(x-x₁)...(x-xᵢ₋₁)(x-xᵢ₊₁) /
             (xᵢ-x₀)(xᵢ-x₁)....(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁).....(xᵢ-xₙ)
                                                               .......(6)

putting the values of lᵢ(x) from equation (6) in equation (2) we get ,
         
Φ(x) =
  n
  Σ (x-x₀)(x-x₁)...(x-xᵢ₋₁)(x-xᵢ₊₁)....(x-xₙ)   y(xᵢ)  /
 i=1    (xᵢ-x₀)(xᵢ-x₁)...(xᵢ-xᵢ₋₁)(xᵢ-xᵢ₊₁)...(xᵢ-xₙ)
                                                  .......(8)

Equation (8) is known as Lagrangian Interpolation Formula.

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About Scientist :


Joseph-Louis Lagrange was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. 

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