I'm trying to find the best way to solve this equation for the vector $x$.
\begin{equation} Ax+b-ty=0 \end{equation}
Where $x=[x_1, x_2, x_3, ... , x_n]^T$ and $y = [\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3} , ... , \frac{1}{x_n}]^T$.
$A \in \mathbb{R}^{n\times n}$ is a symmetric and positive definite matrix
$b \in \mathbb{R}^{n\times 1}$
$t \in \mathbb{R}^{+}$
My unsuccessful attempt so far:
- Expand out each row, $i$ and multiply both sides by $x_i$
- Now you'll have a linear set of equations with coefficients of $x_1^2$, $x_2^2$, $x_1x_2$ ...
- Create the vector $X$ containing all quadratic permutations of $x$: $X = [x_1^2, x_2^2, x_1x_2, ... ]^T$
- Rewrite the original equation as $\tilde A X = \tilde b$ and solve for $X$ using the pseudo inverse
- Somehow get $x$ from the values of $X$
If this problem is too hard to solve. Let a simplified version be the case where $A$ is circulant.