The differentiation of pseudo-inverses and non-linear least squares problems whose variables separate.

Abstract

For given data (t, Yi), l, , m, we consider the least squares fit ofnonlinear models of the form It is shown that by defining the matrix {(0t)}i, qgj(0t; ti), and the modified functional r2(0t (lY O(0t)/(0t)yl)22, it is possible to optimize first with respect to the parameters 0t, and then to obtain, a posteriori, the optimal parameters . The matrix (0t) is the Moore-Penrose generalized inverse of O(t). We develop formulas for the Fr6chet derivative of orthogonal projectors associated with and also for /(0t), under the hypothesis that O(0t) is of constant (though not necessarily full) rank. Detailed algorithms are presented which make extensive use ofwell-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).