### 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,

\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(\delta) = || f(x+\delta) || ~=~ || f(x) + J \delta ||


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.