For the testing of a program of mine, I was trying to generate poorly-conditioned matrices. The context requires symmetric, real, Toeplitz, and positive definite matrices; moreover, these matrices must be decreasing as we move away from the diagonal.
A symmetric Toeplitz matrix $T$ of size $n$ is defined by a vector $\textbf{t}$ in $\mathbb{R}^n$ (0-indexed) with entries: $$ T_{ij}=t_{\left|{i-j}\right|} $$
So far, after much fiddling, the only ones I was able to find are $n-t_i=\Theta(i^k)$ for $k\in\mathbb{N}$, which exhibits exponentially growing condition numbers.
However, as soon as I try $n-t_i=\Theta(e^{-i})$, this becomes very well behaved.
Is there a theorem characterizing the tightness of my observations for such matrices? What if we have "constant sections", i.e., $t_k=t_{k-1}$ for some range of $k$?
Condition number plots:
Mathematica code link for images
Results from Böttcher et al 2010 may explain this phenomenon in terms of asymptotic growth of eigenvalues, but I am not familiar enough with the subject to understand how.