Appendix A: A non-integral method, and its problems

It follows that, in principle, rotation curves can be folded by minimizing the functional

with respect to variations in and .

However, this method makes *direct* use of velocity and radial displacement measurements which
are both intrinsically very noisy.
By contrast, the Fourier method described in this paper is based on minimizing a functional
defined over the Fourier coefficients *A*(*m*) which are estimated by integrating over the
noisy velocity data - in other words, the method has an intrinsic smoothing mechanism incorporated
within it.
A recognized consequence of defining functions directly over noisy data is that they frequently
fail to exhibit the mathematical properties of minimum points, even where such points are
known to exist.
In such a situation any minimizing algorithm designed to locate minimum points will fail.
In the present case it is therefore to be expected that the functional *F* will
frequently fail to exhibit the mathematical properties of a minimum point in the region of
the actual
solution.

In practice, when applied to the PS sample, the algorithm failed to find a sensible solution about 30% of the time. To illustrate this point, Fig. A1 shows the results of applying the algorithm to the ten rotation curves shown in Figs. 7 and 8. There are three total failures and one (71-G14) very poor solution. The remaining six solutions compare favourably with both the PS and the MK IV auto-folder solutions.

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