# Looking for tighter bounds

I have to solve an equation which is $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where
$$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$

The $z_i$ are all positive and add to $1$; the $K_i$ are positive but range from very small to very large (at least, one is strictly greater than $1$ and one is strictly smaller than $1$).

The solution ($w$) looked for is between two vertical asymptotes corresponding to $-1 / (K_{\text{max}} - 1)$ (which is negative) and $1 / (1 - K_{\text{min}})$ (which is greater than $1$). Considering the fact that, at solution, all $x_i$ and $y_i$ must be positive and smaller than $1$ (since they both must add to $1$), I have been able to find a left bound corresponding to the maximum value of $\dfrac{K_i z_i - 1}{K_i-1}$ (considering only the $K_i > 1$) and a right bound corresponding to the minimum value of $\dfrac{1 - z_i}{1 - K_i}$ (considering only the $K_i < 1$). These bounds are inside the asymptotes but I need a tighter interval.

I have not been able to go further. Any help will be more than welcome.

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Why must $\sum x_i$ equal 1? I get that it equals $1/(2-w)$. Note that $y_i = K_i x_i$, then sum the equation $x_i(1+(K_i-1)w)=z_i$ over $i$ and apply $\sum x_i=\sum y_i$. –  Brendan McKay Nov 1 '13 at 13:41
@BrendanMcKay. At solution, the sum of the x[i]'s and the sum of the y[i]'s must be equal to 1. Each of these quantities will be computable and computed when the solution (w) looked for will be obtained. –  Claude Leibovici Nov 2 '13 at 9:08
Sorry, I miscalculated. –  Brendan McKay Nov 2 '13 at 10:46