The fit equation is

$Y = f(A) + \epsilon$

We assume near $Y$ , the curvy subspace of $f(A)$ can be approximated by a plane.  This, using Taylor series,

$Y = f(A_0) + F(A_0) \cdot (A - A_0) + \cdots$,

where $F(A_0)$ is divergence of $f(A)$ at $A_0$.

Using same technique in linear regression,

$A - A_0 = (F(A_0)^T \cdot F(A_0))^{-1} \cdot F(A_0) \cdot ( Y-f(A_0))$

With an initial guess, the interaction should approach to the best estimated parmeter $\hat{A}$.

The covariance is

$Var(A) = \sigma^2 (F(A)^T \cdot F(A))^{-1}$

The above method is also called Gauss-Newton method.