Limitations of Lagrange Polynomial Interpolation

Let's look at the function Cos[x]^10 on the interval from -2 to 2.


  f[t_]:=If[t>=-2 && t<=2,Cos[t]^10,0]


  original=Plot[f[x],{x,-2,2},PlotRange->{-1,1}];

Let's test Lagrange Polynomial Interpolation by trying more and more points:


  x={-2,0,2}
  y={Cos[-2]^10,1,Cos[2]^10}


  p2[t_]:=lagrangepoly[t,x,y];
  p2[t]
  p2plot=Plot[p2[t],{t,-2,2}];
  Show[original,p2plot];


  x={-2,-1,0,1,2}
  y=Cos[x]^10


  p4[t_]:=lagrangepoly[t,x,y];
  p4[t]
  p4plot=Plot[p4[t],{t,-2,2}];
  Show[original,p4plot];


  x={-2,-2+2/3,-2+4/3,0,2-4/3,2-2/3,2}
  y=Cos[x]^10


  p6[t_]:=lagrangepoly[t,x,y];
  p6[t]
  p6plot=Plot[p6[t],{t,-2,2}];
  Show[original,p6plot];

The Lagrange polynomials are becoming better estimates, but there are oscillations at the ends. There are some remedies to this. One of the must used is Cubic Spline Interpolation.

Up to First Attempt: Lagrange Polynomial Interpolation