Limitations of Lagrange Polynomial Interpolation
Limitations of Lagrange Polynomial Interpolation
In[20]:=
f[t_]:=If[t>=-2 && t<=2,Cos[t]^10,0]
In[21]:=
original=Plot[f[x],{x,-2,2},PlotRange->{-1,1}];
Let's test Lagrange Polynomial Interpolation by trying more and more points:
In[22]:=
x={-2,0,2}
y={Cos[-2]^10,1,Cos[2]^10}
Out[22]=
{-2, 0, 2}
Out[23]=
10 10
{Cos[2] , 1, Cos[2] }
In[24]:=
p2[t_]:=lagrangepoly[t,x,y];
p2[t]
p2plot=Plot[p2[t],{t,-2,2}];
Show[original,p2plot];
Out[24]=
10
-((-2 + t) (2 + t)) (-2 + t) t Cos[2]
------------------- + ------------------- +
4 8
10
t (2 + t) Cos[2]
------------------
8
In[25]:=
x={-2,-1,0,1,2}
y=Cos[x]^10
Out[25]=
{-2, -1, 0, 1, 2}
Out[26]=
10 10 10 10
{Cos[2] , Cos[1] , 1, Cos[1] , Cos[2] }
In[27]:=
p4[t_]:=lagrangepoly[t,x,y];
p4[t]
p4plot=Plot[p4[t],{t,-2,2}];
Show[original,p4plot];
Out[27]=
(-2 + t) (-1 + t) (1 + t) (2 + t)
--------------------------------- -
4
10
(-2 + t) (-1 + t) t (2 + t) Cos[1]
------------------------------------ -
6
10
(-2 + t) t (1 + t) (2 + t) Cos[1]
----------------------------------- -
6
10
(-1 - t) (-2 + t) (-1 + t) t Cos[2]
------------------------------------- +
24
10
(-1 + t) t (1 + t) (2 + t) Cos[2]
-----------------------------------
24
In[28]:=
x={-2,-2+2/3,-2+4/3,0,2-4/3,2-2/3,2}
y=Cos[x]^10
Out[28]=
4 2 2 4
{-2, -(-), -(-), 0, -, -, 2}
3 3 3 3
Out[29]=
10 4 10 2 10 2 10 4 10
{Cos[2] , Cos[-] , Cos[-] , 1, Cos[-] , Cos[-] ,
3 3 3 3
10
Cos[2] }
In[30]:=
p6[t_]:=lagrangepoly[t,x,y];
p6[t]
p6plot=Plot[p6[t],{t,-2,2}];
Show[original,p6plot];
Out[30]=
4 2 2 4
(-81 (-2 + t) (-(-) + t) (-(-) + t) (- + t) (- + t)
3 3 3 3
4 2
(2 + t)) / 256 + (243 (-2 + t) (-(-) + t) (-(-) + t)
3 3
4 2 10
t (- + t) (2 + t) Cos[-] ) / 1024 +
3 3
4 2 4
(243 (-2 + t) (-(-) + t) t (- + t) (- + t) (2 + t)
3 3 3
2 10
Cos[-] ) / 1024 -
3
4 2 2
(243 (-2 + t) (-(-) + t) (-(-) + t) t (- + t) (2 + t)
3 3 3
4 10
Cos[-] ) / 2560 -
3
2 2 4
(243 (-2 + t) (-(-) + t) t (- + t) (- + t) (2 + t)
3 3 3
4 10
Cos[-] ) / 2560 +
3
4 2 2 4
(81 (-2 + t) (-(-) + t) (-(-) + t) t (- + t) (- + t)
3 3 3 3
10
Cos[2] ) / 5120 +
4 2 2 4
(81 (-(-) + t) (-(-) + t) t (- + t) (- + t) (2 + t)
3 3 3 3
10
Cos[2] ) / 5120
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.