I have a linear system of equations with a variable dependent on a nonlinear system of equations:
The linear system, solving for $\mathbf{x}_{n+1}$, is:
$$ \mathbf{A}(\boldsymbol\mu_{n+1})\mathbf{x}_{n+1} = \mathbf{b} $$
The $N \times N$ matrix $\mathbf{A}$ is dependent on the $N \times 1$ vector $\boldsymbol\mu_{n+1}$. To find $\boldsymbol\mu_{n+1}$, there is a nonlinear system of equations:
$$ F(\mu_{n+1}, \mu_{n}) = \mathbf{a}(\mu_{n+1}, \mu_{n}) - \mathbf{D}(\mu_{n+1})\boldsymbol\mu_{n+1} = 0 $$
I can solve the nonlinear system of equations for $\boldsymbol\mu_{n+1}$, plug it into the linear system of equations and find $\mathbf{x}_{n+1}$. But I'd like to derive a single nonlinear system of equations to find $\mathbf{x}_{n+1}$. How do I couple these two?