Efficient implementation of a variable projection algorithm for nonlinear least squares problems
Communications of the ACM
Transmission line design of clock trees
Proceedings of the 2002 IEEE/ACM international conference on Computer-aided design
Solving rank-deficient separable nonlinear equations
Applied Numerical Mathematics
Algorithms for separable nonlinear least squares with application to modelling time-resolved spectra
Journal of Global Optimization
Computer oriented methods for fitting tabular data in the linear and nonlinear least squares sense
AFIPS '72 (Fall, part II) Proceedings of the December 5-7, 1972, fall joint computer conference, part II
Solving separable nonlinear equations with jacobians of rank deficiency one
CIS'04 Proceedings of the First international conference on Computational and Information Science
Variable projection for nonlinear least squares problems
Computational Optimization and Applications
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For given data ($t_i\ , y_i), i=1, \ldots ,m$ , we consider the least squares fit of nonlinear models of the form F($\underset ~\to a\ , \underset ~\to \alpha\ ; t) = \sum_{j=1}^{n}\ g_j (\underset ~\to a ) \varphi_j (\underset ~\to \alpha\ ; t) , \underset ~\to a\ \epsilon R^s\ , \underset ~\to \alpha\ \epsilon R^k\ $. For this purpose we study the minimization of the nonlinear functional r($\underset ~\to a\ , \underset ~\to \alpha ) = \sum_{i=1}^{m} {(y_i - F(\underset ~\to a , \underset ~\to \alpha , t_i))}^2$. It is shown that by defining the matrix ${ \{\Phi (\underset ~\to \alpha\} }_{i,j} = \varphi_j (\underset ~\to \alpha ; t_i)$ , and the modified functional $r_2(\underset ~\to \alpha ) = \l\ \underset ~\to y\ - \Phi (\underset ~\to \alpha )\Phi^+(\underset ~\to \alpha ) \underset ~\to y \l_2^2$, it is possible to optimize first with respect to the parameters $\underset ~\to \alpha$ , and then to obtain, a posteriori, the optimal parameters $\overset ^\to {\underset ~\to a}$. The matrix $\Phi^+(\underset ~\to \alpha$) is the Moore-Penrose generalized inverse of $\Phi (\underset ~\to \alpha$), and we develop formulas for its Frechet derivative under the hypothesis that $\Phi (\underset ~\to \alpha$) is of constant (though not necessarily full) rank. From these formulas we readily obtain the derivatives of the orthogonal projectors associated with $\Phi (\underset ~\to \alpha$), and also that of the functional $r_2(\underset ~\to \alpha$). 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 [1971].