A New Nonlinear Least Square Algorithm for Voigt Spectral Lines

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Abstract
In this paper a new fitting algorithm which works with Voigt functions is discussed. The fitting algorithm used is an extension of the rapidly convergent gradient method of Fletcher and Powell, who claim faster convergence than the Newton-Raphson method which has been used by Chang and Shaw for fitting Lorentz line widths. The Fletcher and Powell algorithm involves the effects of second derivatives although second derivatives are not explicitly calculated. In our algorithm, first and second derivatives are computed not numerically, but analytically via a modification to Drayson's Voigt function subroutine. This algorithm provides rapid convergence even when there are few data points. Profiles have been fitted with as few as five data points. Our typical line fits involve 40 points. The run time of the algorithm has been compared with the shrinking cube algorithm of Hillman and found to be at least 10 times faster under identical starting conditions. Sample single line and single line plus background are shown illustrating the speed and efficiency of the new algorithm, as well as the importance of good zero-order estimates to start the iterations.

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