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This paper describes the optimisation theory on which VPFIT, a non-linearleast-squares program for modelling absorption spectra, is based. Particularattention is paid to precision. Voigt function derivatives have previously beencalculated using numerical finite difference approximations. We show how thesecan instead be computed analytically using Taylor series expansions and look-uptables. We introduce a new optimisation method for an efficient descent path tothe best-fit, combining the principles used in both the Gauss-Newton andLevenberg-Marquardt algorithms. A simple practical fix for ill-conditioning isdescribed, a common problem when modelling quasar absorption systems. We alsosummarise how unbiased modelling depends on using an appropriate informationcriterion to guard against over- or under-fitting.  The methods and the new implementations introduced in this paper are aimed atoptimal usage of future data from facilities such as ESPRESSO/VLT andHIRES/ELT, particularly for the most demanding applications such as searchesfor spacetime variations in fundamental constants and attempts to detectcosmological redshift drift.

Precision in high resolution absorption line modelling, analytic Voigt derivatives, and optimisation methods