Gauss Interpolation Formula

Gauss Interpolation Formula :

             The general Newton formula or Newton's Divided Difference Interpolation Formula gives 

 y = f(x) = f(x₀)+(x-x₀)f(x₀,x₁)+(x-x₀)                                         (x-x₁)f(x₀,x₁,x₂)+(x-x₀)(x-x₁)                 (x-x₂)f(x₀,x₁,x₂,x₃)+(x-x₀)(x-x₁)(x-x₂)           (x-x₃)f(x₀,x₁,x₂,x₃,x₄)+(x-x₀)(x-x₁)(x-x₂)(x-x₃)(x-x₄)f(x₀,x₁,x₂,x₃,x₄,x₅)+(x-x₀)(x-x₁) (x-x₂)(x-x₃)(x-x₄)                                                              ×(x-x₅)f(x₀,x₁,x₂,x₃,x₄,x₅,x₆)+....

Now putting x₀=x₀ , x₁= x₀+h ,x₂= x₀-h,             x₃= x₀+2h ,x₄=x₀-2h ,x₅=x₀+3h ,x₆=x₀-3h

We immediately get 

f(x) = f(x₀)+(x-x₀)f(x₀,x₀+h)+(x-x₀)(x-x₀-h)

           f(x₀,x₀+h,x₀-h)+(x-x₀)(x-x₀-h)(x-x₀+h)

         f(x₀,x₀+h,x₀-h,x₀+2h)+(x-x₀)(x-x₀-h)                 (x-x₀+2h)f(x₀,x₀+h,x₀-h,x₀+2h,x₀-2h)

     + (x-x₀)(x-x₀-h)(x-x₀+h)(x-x₀-h)(x-x₀+2h)

       f(x₀,x₀+h,x₀-h,x₀+2h,x₀-2h,x₀+3h)+....

                                    ............(1)

Further , let u= x-x₀ /h ⇒x-x₀ = hu

when 

 f(x) = f(x₀)+hu f(x₀,x₀+h)

            + hu(hu-h) f(x₀-h,x₀+h

      +hu(hu-h)(hu+h) f(x₀-h,x₀,x₀+h,x₀+2h)

   +hu(hu-h)(hu+h)(hu-2h) 

                  f(x₀-2h,x₀-h,x₀,x₀+h,x₀+2h)

   +hu(hu-h)(hu+h)(hu-2h)(hu+2h)

        f(x₀-2h,x₀-h,x₀,x₀+h,x₀+2h,x₀+3h)+......

But , we also know that 

   f(x₀,x₀+h) = Δy₀/h 

 f(x₀-h,x₀,x₀+h) =Δ²y₋₁/2h²

f(x₀-h,x₀x₀+h,x₀+2h) = Δ³y₋₁/3!h³

f(x₀-2h,x₀-h,x₀,x₀+h,x₀+2h)= Δ⁴y₋₂/4!h⁴

f(x₀-2h,x₀-h,x₀,x₀+h,x₀+2h,x₀+3h)=

                                               Δ⁵y₋₂/5!h⁵ etc.

Now ,substituting these values in equation(1) , we readily obtain 

y= y₀+hu Δy₀/h + h²u(u-1) Δ²y₋₁/2h² +

      h³u(u-1)(u+1) Δ³y₋₁/3!h³ 

      + h⁴u(u-1(u+1)(u-2) Δ⁴y₋₂/4!h⁴

     + h⁵u(u-1)(u+1)(u-2)(u+2) Δ⁵y₋₂/5!h⁵+.....

⇒y= y₀+uc₁Δy₀ +uc₂Δ²y₋₁+(u+1)c₃Δ³y₋₁

                 +(u+1)c₄ Δ⁴y₋₂ +..........

This result is known as Gauss Forward Formula for equal intervals.