A variable projection method for solving separable nonlinear least squares problems

Abstract

Consider the separable nonlinear least squares problem of findinga εR ⁿ and α εR ^k which, for given data (y _i ,t _i ),i=1,2,...m, and functions ϕ _j (α,t),j=1,2,...,n(m>n), minimize the functional

$$r(a,\alpha ) = \left\| {y - \Phi (\alpha )a} \right\|_2^2$$

where θ(α) _ij =ϕ _j (α,t _i ). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involvingα only, and a linear least squares problem involvinga only. In this paper we propose a new method for determining the optimalα which computationally has proved more efficient than the Golub-Pereyra scheme.