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