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.

User Documentation for RALFit (Fortran version)

Contents

Indices and tables