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Question: Reference for the derivation of the interpolation polynomial

Question:Reference for the derivation of the interpolation polynomial

Adam Ledger 360 Maple
polynomial interpolation
May 20 2018
0

I figured out what the interpolation is, but can someone please give me a reference for the publication of the person that discovered that this works thankyou. 

 


 

> 

PolyI1 := proc (N, n) options operator, arrow; (-1)^N*(sum(n*factorial(n-1)*(-1)^k*a[k]/(factorial(n-N-1)*(n-k)*factorial(k)*factorial(N-k)), k = 0 .. N)) end proc

proc (N, n) options operator, arrow; (-1)^N*(sum(n*factorial(n-1)*(-1)^k*a[k]/(factorial(n-N-1)*(n-k)*factorial(k)*factorial(N-k)), k = 0 .. N)) end proc

 (1)
> 

PolyI2 := proc (N, n) options operator, arrow; CurveFitting[PolynomialInterpolation]([seq([k, a[k]], k = 0 .. N)], n) end proc

proc (N, n) options operator, arrow; CurveFitting[PolynomialInterpolation]([seq([k, a[k]], k = 0 .. N)], n) end proc

 (2)
> 

collect(expand(PolyI1(2, n)), n); PolyI2(2, n)

((1/2)*a[2]-a[1]+(1/2)*a[0])*n^2+(-(1/2)*a[2]+2*a[1]-(3/2)*a[0])*n+a[0]

  

((1/2)*a[2]-a[1]+(1/2)*a[0])*n^2+(-(1/2)*a[2]+2*a[1]-(3/2)*a[0])*n+a[0]

 (3)
> 

collect(expand(PolyI1(3, n)), n); PolyI2(3, n)

((1/6)*a[3]-(1/2)*a[2]+(1/2)*a[1]-(1/6)*a[0])*n^3+(-(1/2)*a[3]+a[0]-(5/2)*a[1]+2*a[2])*n^2+(-(3/2)*a[2]+(1/3)*a[3]-(11/6)*a[0]+3*a[1])*n+a[0]

  

((1/6)*a[3]-(1/2)*a[2]+(1/2)*a[1]-(1/6)*a[0])*n^3+(-(1/2)*a[3]+a[0]-(5/2)*a[1]+2*a[2])*n^2+(-(3/2)*a[2]+(1/3)*a[3]-(11/6)*a[0]+3*a[1])*n+a[0]

 (4)
> 

collect(expand(PolyI1(4, n)), n); PolyI2(4, n)

(-(1/6)*a[1]+(1/4)*a[2]-(1/6)*a[3]+(1/24)*a[0]+(1/24)*a[4])*n^4+(-(5/12)*a[0]+(3/2)*a[1]-2*a[2]+(7/6)*a[3]-(1/4)*a[4])*n^3+((35/24)*a[0]-(13/3)*a[1]+(19/4)*a[2]-(7/3)*a[3]+(11/24)*a[4])*n^2+(-(25/12)*a[0]-(1/4)*a[4]+4*a[1]-3*a[2]+(4/3)*a[3])*n+a[0]

  

(-(1/6)*a[1]+(1/4)*a[2]-(1/6)*a[3]+(1/24)*a[0]+(1/24)*a[4])*n^4+(-(5/12)*a[0]+(3/2)*a[1]-2*a[2]+(7/6)*a[3]-(1/4)*a[4])*n^3+((35/24)*a[0]-(13/3)*a[1]+(19/4)*a[2]-(7/3)*a[3]+(11/24)*a[4])*n^2+(-(25/12)*a[0]-(1/4)*a[4]+4*a[1]-3*a[2]+(4/3)*a[3])*n+a[0]

 (5)
> 

collect(expand(PolyI1(5, n)), n); PolyI2(5, n)

((1/12)*a[3]-(1/120)*a[0]+(1/24)*a[1]-(1/12)*a[2]-(1/24)*a[4]+(1/120)*a[5])*n^5+(-a[3]+(11/24)*a[4]+(1/8)*a[0]-(7/12)*a[1]+(13/12)*a[2]-(1/12)*a[5])*n^4+((49/12)*a[3]-(41/24)*a[4]+(7/24)*a[5]-(17/24)*a[0]+(71/24)*a[1]-(59/12)*a[2])*n^3+((15/8)*a[0]-(13/2)*a[3]+(61/24)*a[4]-(5/12)*a[5]-(77/12)*a[1]+(107/12)*a[2])*n^2+(-(137/60)*a[0]+5*a[1]+(10/3)*a[3]-(5/4)*a[4]+(1/5)*a[5]-5*a[2])*n+a[0]

  

((1/12)*a[3]-(1/120)*a[0]+(1/24)*a[1]-(1/12)*a[2]-(1/24)*a[4]+(1/120)*a[5])*n^5+(-a[3]+(11/24)*a[4]+(1/8)*a[0]-(7/12)*a[1]+(13/12)*a[2]-(1/12)*a[5])*n^4+((49/12)*a[3]-(41/24)*a[4]+(7/24)*a[5]-(17/24)*a[0]+(71/24)*a[1]-(59/12)*a[2])*n^3+((15/8)*a[0]-(13/2)*a[3]+(61/24)*a[4]-(5/12)*a[5]-(77/12)*a[1]+(107/12)*a[2])*n^2+(-(137/60)*a[0]+5*a[1]+(10/3)*a[3]-(5/4)*a[4]+(1/5)*a[5]-5*a[2])*n+a[0]

 (6)
> 

NULL


 

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