For a problem I am working on, it would be really nice to solve the equation \begin{equation} \mathbf{c}_{i+1}^\dagger \mathbf{S}_{i+1} \mathbf{c}_{i+1} = \mathrm{const.} = \mathbf{c}_i^\dagger \mathbf{S}_i \mathbf{c}_i \end{equation} for $\mathbf{c}_{i+1}$ where $\mathbf{c}$ is a complex vector, $\mathbf{S}$ is a real symmetric matrix and the index $i$ denotes a step in some iterative process. For above equation, $\mathbf{S}_{i+1}, \mathbf{S}_i$ and $\mathbf{c}_i$ are known quantities.
Above equation thus imposes a requirement for the $\mathbf{c}_{i+1}$'s when $\mathbf{S}_i$ changes to $\mathbf{S}_{i+1}$ in the iteration. As I understand it, $\mathbf{c}_{i+1}$ is somehow an extrapolation of $\mathbf{c}_i$ in this context.
Since this is not my day-to-day business, I am quite stuck on how to approach this problem.
The equation $\vec{c}_{i+1}^\dagger S_{i+1}\vec{c}_{i+1}=C$ generally has many solutions, and your question doesn't seems to outline which solution you're interested in. If any solution "close" to $\vec{c}_i$ (in the sense that $\vec{c}_{i+1}=\vec{c}_i$ in the limit $S_{i+1}\to S_i$) will do, one simple approach is to rescale $\vec{c}_i$. $$ \vec{c}_{i+1}=\sqrt{\frac{C}{\vec{c}_{i}^\dagger S_{i+1}\vec{c}_i}}\vec{c}_i $$ If it gives any geometric intuition, the set of solutions of the equation can be thought of as a kind of "ellipsiod" in $\mathbb{C}^n$, whose shape varies one step to the next, and this procedure amounts to radially projecting onto the surface. This would be suitable for e.g. modeling a particle moving on a sphere of changing radius.
If the $S_i$ are all positive (or negative) definite, this approach shouldn't have any issues. If the $S_i$ have a nonzero kernel, there may be issues with division by zero (or even nonexistence of solutions), and it might be necessary to know a bit more about the properties of $S_i$.