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.