Question

I am having some slight difficulty with the following code:

Lagrange[list_] := 
  Module[{points = list, length, k, j, m, x, g}, 
    length = Length[points]; 
    k = length - 1; 
    f = Sum[points[[j + 1,2]]*Product[If[j != m, (x - points[[m + 1,1]])/
          (points[[j + 1,1]] - points[[m + 1,1]]), 1], {m, 0, k}], {j, 0, k}]; 
    g = FullSimplify[Expand[f]]; 
    Return[f]]

The output I get is:

Out[101]= 0. -1.85698 (-1.5+x$26810) (-0.75+x$26810) (0. +x$26810) (0.75+x$26810)
         +0.490717 (-1.5+x$26810) (-0.75+x$26810) (0. +x$26810) (1.5 +x$26810)
         -0.490717 (-1.5+x$26810) (0. +x$26810) (0.75 +x$26810) (1.5 +x$26810)
         +1.85698 (-0.75+x$26810) (0. +x$26810) (0.75 +x$26810) (1.5 +x$26810)

My concern is with these “$” symbols. I don’t know what they mean, I can’t find documentation on them, and they are preventing the plotting of this polynomial.

Answer 1

The $ in your output is from the unique variable generated by the lexical scoping of Module (see the More Information part of mathematica/ref/Module). This is why I made my LagrangePoly function accept the symbol that the polynomial is meant to be in. I used LagrangePoly[list_, var_:x], which defaults to the global symbol x.

A simple example of the problem is

In[1]:= Module[{x}, x]    
Out[1]= x$583

The number in the “local” variable x$nnn comes from the global $ModuleNumber.

If you don’t understand this, then you should probably read the tutorial Blocks Compared with Modules.