The problem of fitting experimental data to a given model function \(f(t; p_1,p_2,\ldots ,\) \(p_N)\) can be solved numerically by methods such as Levenberg–Marquardt, which are based on the \(\chi ^2\) 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, it is common to apply some variation of the “variable projection method” of Golub-Pereyra to reduce the “nonlinear activity” to the nonlinear parameters only. We describe a method based on a notion which we call the “shortcut derivative.” 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. This notion is applied to minimization problems that 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, which permits any standard nonlinear optimization methods to be applied to the nonlinear aspect. Numerical examples are given to demonstrate the effectiveness of this approach. The method has been extensively applied to the peak-fitting analysis of photoemission spectra.
