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39.1 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, All algorithms proceed from an initial guess using the linearization, where x is the initial point, \delta 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.

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.