I have a function that is not only infinitely differentiable, but it is also very cheap to calculate any of those derivatives. It looks like:$f(\boldsymbol{C}, \boldsymbol{x})=\sum_{i} C_{i} \prod_{j} a_{j}^{x_{i,j}} -\sum_{i}C_{i}  $Where both i run over the same length, and where $a_{j}$ are constants. This function is easily differentiable because after the first, it's either just removing a $C_{i}$ or multiplying with $ln(a_{j})$.I'm looking for the most efficient method of finding the zeroes of such a function, that takes full advantage of the differentiability. Are there any methods that are preferred in this scenario?
Finding zeroes of an infinitely differentiable function of ~100 to ~1000 variables
optimization
Your function can be written as$$f(C, x) = C^T d(x), \quad\text{with } (d(x))_i = \prod_j a_j^{x_{i,j}} - 1,$$so for any fixed $x$, the roots of $f$ are exactly the vectors $C$ in the subspace orthogonal to $d(x)$.The reverse is also true: For any given $C$, find an orthogonal vector $d$ and solve$$d_i = \prod_j a_j^{x_{i,j}} - 1$$for $x$. One obvious solution would be$$x_{i,1} = \ln(d_i+1) / \ln(a_1),\\x_{i,j} = 0, j > 1.$$