GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 37.1 Overview

The problem of multidimensional nonlinear least-squares fitting requires the minimization of the squared residuals of n functions, f_i, in p parameters, x_i,

\Phi(x) = (1/2) || F(x) ||^2 = (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2

All algorithms proceed from an initial guess using the linearization,

\psi(p) = || F(x+p) || ~=~ || F(x) + J p ||

where x is the initial point, p is the proposed step and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies are used to enlarge the region of convergence. These include requiring a decrease in the norm ||F|| on each step or using a trust region to avoid steps which fall outside the linear regime.

To perform a weighted least-squares fit of a nonlinear model Y(x,t) to data (t_i, y_i) with independent gaussian errors \sigma_i, use function components of the following form,

f_i = (Y(x, t_i) - y_i) / \sigma_i

Note that the model parameters are denoted by x in this chapter since the non-linear least-squares algorithms are described geometrically (i.e. finding the minimum of a surface). The independent variable of any data to be fitted is denoted by t.

With the definition above the Jacobian is J_{ij} =(1 / \sigma_i) d Y_i / d x_j, where Y_i = Y(x,t_i).

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |