The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate

For given data (tᵢ, yᵢ), i = 1, , m, we consider the least squares fit of nonlinear models of the form \ (a, ;t) = ₉ = ₁ⁿ aⱼ ⱼ ({ ;t), a Rⁿ, Rᵏ. } \ For this purpose we study the minimization of the nonlinear functional \ r (a, ) = ₈ = ₁ᵐ ({yᵢ - ({{ a, , tᵢ }) }) ² }. \ It is shown that by defining the matrix \ { () \} ₈, ₉ = ⱼ (;tᵢ), and the modified functional r₂ () = \| y - () ^ + () y \|₂², it is possible to optimize first with respect to the parameters, and then to obtain, a posteriors, the optimal parameters a. The matrix ^ + () is the Moore–Penrose generalized inverse of (). We develop formulas for the Frechet derivative of orthogonal projectors associated with () and also for ^ + (), under the hypothesis that () is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use of well-known reliable linear least squares techniques, and numerical results and comparisons are given. These results are generalizations of those of H. D. Scolnik 20 and Guttman, Pereyra and Scolnik 9.