Condition number of a 5-diagonal matrix

Question

I'm trying to compute/estimate the condition number of the $N\times N$ matrix \begin{align*} \begin{pmatrix} 1 & 0 & -1 & & \\ 1 & 4 & 1 & & \\ & 1 & 4 & 1 & \\ & & \ddots & \ddots & \ddots \\ & & & 1 & 4 & 1 \\ & & & & 1 & 4 & 1 \\ & & & & 1 & 0 & -1 \end{pmatrix} \end{align*} as $N \to \infty$.

Numerically I have computed the condition number for as many of these as I could stand to type into Octave ($N < {\sim}8)$, but I was unable to deduce any conclusions from it. However, I suspect that this matrix is simple enough that someone on Math.SE might be able to get an expression/estimate for it's eigenvalues, and hence get the condition number. Help me out!

Answer 1

Here is the condition number versus $N$, computed numerically:

It appears that the asymptote is roughly 6.9.

Hope this helps!

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Mathematica code (non-optimized):

Clear[n, substitutions];

basicMatrix[n_]:= SparseArray[
         {Band[{1,1}]->4,
          Band[{2,1}]->1, 
          Band[{1,2}]->1},
        {n,n}];

substitutions[n_] := 
   {{1, 1} -> 1, 
    {1, 2} -> 0, 
    {1, 3} -> -1, 
    {n, n - 2} -> 1, 
    {n, n - 1} -> 0, 
    {n, n} -> -1};

conditionnumbers = 
  Table[{i, 
      (Last[mydone = 
        Sort@N@Abs@
           Eigenvalues[
            ReplacePart[basicMatrix[i], substitutions[i]]]]/
      First[mydone])}, 
       {i, 4, 40}];

ListPlot[conditionnumbers]