Problem generating list of functions

Question

My code runs well up to the final two lines:

analytic[n_, m_, t_] := 
 1/2 (KroneckerDelta[n, m] + (-1)^m *(I)^(n - m)*BesselJ[n - m, 2 t])

matrix[l_, t_] := Table[analytic[n, m, t], {n, l}, {m, l}]

diagonal[l_, t_] := Eigenvalues[matrix[l, t]]

S[l_, t_] := -(Sum[
    diagonal[l, t][[j]] Log[
       diagonal[l, t][[j]]] + (1 - diagonal[l, t][[j]])* 
      Log[1 - diagonal[l, t][[j]]], {j, 1, l}])

xdata = Range[0.1, 1, 0.1]

ydata[l_] := S[l, xdata]

plot[l_] := ListPlot[Transpose[{xdata, ydata[l]}]]

It has a real problem with ydata. I can't even get the dimensions of this list (or at least what should be a list), let alone transpose the two lists. However, a simple scatter graph like

xdata = Range[0, 2 Pi, Pi/6]

ydata[x_, y_] := Sin[xdata + x] + y

plot[x_, y_] := ListPlot[Transpose[{xdata, ydata[x, y]}]]

plot[1, 1]

works no problem and I can't see the difference with what I've done here. I'm new to Mathematica and would really appreciate some help with this!

Answer 1

Probem. The code

xdata = Range[0.1, 1, 0.1]
ydata[l_] := S[l, xdata]

feeds a list into the second argument of S. But the function S is not designed to deal with that.

Solution. One can add

S[l_,t_List] := Map[S[l,#]&,t];

as a second definition for S. When calling S with a list as the second argument, this new definition is used. For example, using OP's code plus this new definition for S, we can do

plot[5]

Comments. I propose simplifying the definition for S a little bit:

S[l_,t_] := diagonal[l,t] // -#.Log[#]-(1-#).Log[1-#]&;
S[l_,t_List] := Map[S[l,#]&,t];

This will call diagonal[l,t] only once, rather than 4*l many times in OP's code.

Another observation is that for larger values of l, we get Indeterminate results. Compare

ydata[5]
(* {0.0348764,0.111092,0.211485,0.326061,0.447528,
    0.570372,0.690543,0.805309,0.913158,1.01367} *)

ydata[7]
(* {Indeterminate,Indeterminate,0.211485 +6.97574*10^-16 I,Indeterminate,0.447528,
    0.570372,0.690543,0.80531,0.91316,1.01367} *)

The problem is that when calling say diagonal[7,0.1], there are several eigenvalues very close to $1$. But this is a separate problem.