  The problem of fitting experimental data to a given model function $f(t;
p_1,p_2,\dots,p_N)$ is conventionally solved numerically by methods such as
that of Levenberg-Marquardt, which are based on approximating the Chi-squared
measure of discrepancy by a quadratic function. Such nonlinear iterative
methods are usually necessary unless the function $f$ to be fitted is itself a
linear function of the parameters $p_n$, in which case an elementary linear
Least Squares regression is immediately available. When linearity is present in
some, but not all, of the parameters, we show how to streamline the
optimization method by reducing the "nonlinear activity" to the nonlinear
parameters only. Numerical examples are given to demonstrate the effectiveness
of this approach. The main idea is to replace entries corresponding to the
linear terms in the numerical difference quotients with an optimal value easily
obtained by linear regression. More generally, the idea applies to minimization
problems which are quadratic in some of the parameters. We show that the
covariance matrix of $\chi^2$ remains the same even though the derivatives are
calculated in a different way. For this reason, the standard non-linear
optimization methods can be fully applied.
