Is there a way to solve an equation of the following form: $$x_i = {\bf x}^T{\bf A}_i{\bf x} + b_i$$ where $x_i$ is the ith element of vector $\bf x$.
I'm guessing this could be equivalently expressed in Einstein notation as follows $$x_i = A^{j}_{ik} x_j x^k + b_i$$ I was considering using Newton's method for each $i$ one at a time, but I'm worrying that this wouldn't converge. I also considered expressing this in matrix form using Kronecker products but couldn't quite figure it out.