the entries of the metric; the second to last gives the distance from the balanced metric, or $\mathrm{err}_{T_K, k}(r, G)$; and the last column gives the bound $\mathrm{bnd}_{T_K,k}(d,r) = \log(1+e^{kd}\sigma_{T_K,k}^r)$ proposed in secion \[subsection: The effect on distance\].

$$\begin{array}{c|ccc|c|c}
r & a_0 & a_1 & a_2 & \mathrm{dist}(-,B) & \mathrm{bnd} \\
\hline
  0  &     0.8826  &    15.0043  &    31.7738  &  0.2848  &  1.0180  \\
  1  &     0.9738  &    12.6377  &    35.0561  &  0.0640  &  0.3027  \\
  2  &     0.9946  &    12.1292  &    35.8067  &  0.0131  &  0.0683  \\
  3  &     0.9989  &    12.0259  &    35.9612  &  0.0026  &  0.0140  \\
  4  &     0.9998  &    12.0052  &    35.9922  &  0.0005  &  0.0028  \\
  5  &     1.0000  &    12.0010  &    35.9984  &  0.0001  &  0.0006
\end{array}$$

We consider another non-palindromic metric, proportional to $G=(1, 25, 0.07, 13),$ with $k=3.$ We use the $T_{\nu}$ operator and list the results of the first few iterations $T_{\nu}^{(r)}(G)$ below. We note that the limiting metric is $B = (1, 3, 3, 1).$

$$\begin{array}{c|cccc|c|c}
r & a_0 & a_1 & a_2 & a_3 & \mathrm{dist}(-,B) & \mathrm{bnd} \\
\hline
  0  &    0.20720  &    5.18011  &    0.01450  &    2.69366  & 5.67338  & 17.02014  \\
  1  &    0.57206  &    