The optimisation space

The optimisation of the parameters of an arbitrary model is dependent on a function f which takes the current parameter values θ$\mathbb {R}$n and returns a single real value f (θ)∈$\mathbb {R}$ corresponding to position θ in the n-dimensional space. For it is that single value which is minimised as

$\displaystyle \hat{\theta}$ = arg$\displaystyle \min_{\theta}^{}$f (θ), (14.1)

where $\hat{\theta}$ is the parameter vector which is equal to the argument which minimises the function f (θ). In most analyses in relax, f (θ) is the chi-squared equation

χ2(θ) = $\displaystyle \sum_{{i=1}}^{n}$$\displaystyle {\frac{{(y_i - y_i(\theta))^2}}{{\sigma_i^2}}}$, (14.2)

where i is the summation index over all data, yi is the experimental data, yi(θ) is the back calculated data, and σi is the experimental error.

The relax user manual (PDF), created 2024-06-08.