Speeding and optimizing the Graphics Plot

Question

I am having a Hermitian Matrix(HCF[k]) depending on a parameter $k$.

My job is to plot the Eigenvalues of the matrix as the parameter $k$ is changed.

But I am facing some problems.
(i) My Mathematica file size, which is of around whopping 90Mb, when HCF[k] is around $284\times284$.
(ii) This part of code is very slow as the size of HCF[k] is increasing.

kdel = 0.00001; (* offset *)
kIn = -π - kdel;  (*Initial value of k *)
kFin = π + kdel;  (*Final value of k *)
kInc = 0.001; (*Increment of k*)

eigeng = 
 ParallelTable[Eigenvalues[HCF[k]], {k, kIn, kFin, kInc}];


kList = ParallelTable[k, {k, kIn, kFin, kInc}];


kFList = ParallelTable[kList, {i, Transpose@eigeng}];

dataToPlot = Flatten[{kFList\[Transpose], eigeng}\[Transpose], {{1, 3}, {2}}];

Graphics[{Point[{#1, #2}]} & @@@ dataToPlot, Frame -> True,
...(* for the aesthetic of plot, i.e. axis title, range and bla bla*)]

• Is there a way the size issue can be overcome?
• Is there a way the code can be sped up a little bit?(I checked my rest of the code, it is very fast, which I did by breaking and evaluating it into small cells).

Addendum

• Form of HCF[k], as it is huge I will try to show you how it looks for small size say
  \left( \begin{array}{cccccc} -20 \pi & -\frac{e^{i k}}{2} & 0 & -\frac{1}{2} & 0 & 0 \\ -\frac{1}{2} e^{-i k} & -20 \pi & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & -\frac{1}{2} & 0 & -\frac{e^{i k}}{2} & 0 & -\frac{1}{2} \\ -\frac{1}{2} & 0 & -\frac{1}{2} e^{-i k} & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{1}{2} & 20 \pi & -\frac{e^{i k}}{2} \\ 0 & 0 & -\frac{1}{2} & 0 & -\frac{1}{2} e^{-i k} & 20 \pi \\ \end{array} \right)

Answer 1

The following uses your definitions of the matrix from pastebin, and of starting points and increments.

Parallelization does help to speed up the calculation of the eigenvalues, but you have to distribute the definitions of your helper variables to the parallel kernels using DistributeDefinitions first:

AbsoluteTiming[Table[Eigenvalues[mat], {k, kIn, kFin, kInc}];]
(* Out: {142.851, Null} *)

DistributeDefinitions[kdel, kIn, kFin, kInc]
AbsoluteTiming[ParallelTable[Eigenvalues[mat], {k, kIn, kFin, kInc}];]
(* Out: {90.5666, Null} *)

This is on a two-core machine, so the benefits from parallelization may be even better on higher core-count machines.

---

Having said that, the definitions of kList and kFList can be simplified:

kList = Range[kIn, kFin, kInc];
kFList = ConstantArray[kList, Length@Transpose@eigeng];