Module rgsl::types::multifit_solver [−][src]
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Nonlinear Least-Squares Fitting
This chapter describes functions for multidimensional nonlinear least-squares fitting. The library provides low level components for a variety of iterative solvers and convergence tests. These can be combined by the user to achieve the desired solution, with full access to the intermediate steps of the iteration. Each class of methods uses the same framework, so that you can switch between solvers at runtime without needing to recompile your program. Each instance of a solver keeps track of its own state, allowing the solvers to be used in multi-threaded programs.
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).
High Level Driver
These routines provide a high level wrapper that combine the iteration and convergence testing for easy use.