A Variable Projection Method for Solving Separable Nonlinear Least Squares Problems

Consider the separable nonlinear least squares problem of finding ~a in R^n and ~alpha in R^k which, for given data (y_i, t_i) i=1,...,m and functions varphi_j(~alpha,t) j=1,2,...n (m>n), minimize the functionalr(~a,~alpha) = ||~y - Phi(~alpha)~a||_(2)^(2)where Phi(~alpha)_(i,j) = varphi_(j)(~alpha,t_j). This problem can be reduced to a nonlinear least squares problem involving $\mathovd{\mathop{\alpha}\limits_{\textstyle\tilde{}}}$ only and a linear least squares problem involving ~a only. the reduction is based on the results of Colub and Pereyra, SIAM J. Numerical Analysis, April 1973, and on the trapezoidal decomposition of Phi, in which an orthogonal matrix Q and a permutation matrix P are found such that\begin{displaymath} Q Phi R = R & S 0 & 0 \end{array}\right) \begin{array}{l} \rbrace\, r \\ \mbox{} \end{array} \end{displaymath}where R is nonsingular and upper trianular. To develop an algorithm to solve the nonlinear least squares probelm a formula is proposed for the Frechet derivation D(Phi_(2) (~alpha)) where Q i partioned int