User Documentation for RALFit (Fortran version)

RALFit computes a solution \vx to the non-linear least-squares problem

\min_\vx \ F(\vx) := \frac{1}{2}\| \vr(\vx) \|_{\vW}^2 + \frac{\sigma}{p}\| \vx\|_2^p,

where \vW\in\mathbb{R}^{m\times m} is a diagonal, non-negative, weighting matrix, and \vr(\vx) =(\comp[1]{r}(\vx), \comp[2]{r}(\vx),...,\comp[m]{r}(\vx))^T is a non-linear function.

A typical use may be to fit a function f(\vx,t) to the data y_i, \ t_i, weighted by the uncertainty of the data, \sigma_i:

\min_\vx \ \frac{1}{2} \sum_{i=1}^m \left(\frac{y_i - f(\vx;t_i)}{\sigma_i}\right)^2,

which corresponds to taking r_i(\vx) := y_i - f(\vx;t_i) and \vW such that \vW_{ii} = (1/{\sigma_i^2}). For this reason we refer to the function \vr as the residual function.

Various algorithms for solving this problem are implemented – see Description of the method used.