k$ we find the geodesic from $A$ to $B$ is given by $P(t), 0 \leq t \leq 1$, where $P(t)$ is the diagonal matrix with entries $e^{(\beta_i-\alpha_i) t + \alpha_i}, i=0,\ldots,k$. Now we can calculate the distance between $A$ and $B$ as $\int_0^1 \left| \frac{dP}{dt} \right|_P \, dt = \sqrt{ \sum (\beta_i - \alpha_i)^2 }$ or $$\label{eqn:  distance formula}
    \mathrm{dist}(A,B)  =
    \sqrt{ \sum_{i=0}^k \left( \log \frac{b_i}{a_i} \right)^2 }\ .$$

One goal is then to understand how well the $r$th iteration of $F = T, T_{\nu}, T_K$ applied to a Hermitian metric $G$ approximates the limiting balanced metric $B := F^{(\infty)}(G).$ That is we wish to understand the function $$\mathrm{err}_{F, k} (G, r) = \mathrm{dist}\left( F^{(r)}(G),\ F^{(\infty)}(G) \right).$$ In particular we would like to give an effective bound: $$\mathrm{err}_{F, k} (G, r) < \mathrm{bnd}_{F, k} (d, r)$$ where $d = \mathrm{dist}(G,B).$ We propose such a bound in section \[subsection: The effect on distance\].

The balanced metrics {#subsection: The balanced metrics}
--------------------

The metrics obtained by taking the coefficients of the polynomial $\alpha(1+cX)^k,$ i.e. $a_q = \alpha c^q{k \choose q},$ for any $\alpha, c>0,$ are fixed for both the $T$ and the $T_K$ maps; it is not for $T_{\nu}$ unless $c=1,$ in 