Lagrange Interpolation

Lagrange Interpolation is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The task is to implement the Legrange Interpolation formula and use it to solve the example problem to find a polynomial P of degree<4 satisfying P(1)=1 P(2)=4 P(3)=1 P(4)=5 as described at [1]

Related task Curve that touches three points

F#

// Lagrange Interpolation. Nigel Galloway: September 5th., 2023
let symbol=MathNet.Symbolics.SymbolicExpression.Variable
let qi=MathNet.Symbolics.SymbolicExpression.FromInt32
let eval (g:MathNet.Symbolics.SymbolicExpression) x=let n=Map["x",MathNet.Symbolics.FloatingPoint.Real x] in MathNet.Symbolics.Evaluate.evaluate n g.Expression
let fN g=let x=symbol "x" in g|>List.fold(fun z c->(x-c)*z)(qi 1)
let fG(n,g)=let n,g=n|>List.map qi,g|>List.map qi in List.map2(fun i g->i,g,n|>List.except [i]) n g
let LIF n=fG n|>List.sumBy(fun(ci,bi,c)->bi*(fN c)/(c|>List.fold(fun z c->(ci-c)*z)(qi 1)))
printfn $"%s{LIF([1;2;3;4],[1;4;1;5]).Expand().ToString()}"

Output:

-21 + 215/6*x - 16*x^2 + 13/6*x^3