  In this paper we propose an approach for solving systems of nonlinear
equations without computing function derivatives. Motivated by the application
area of tomographic absorption spectroscopy, which is a highly-nonlinear
problem with variables coupling, we consider a situation where straightforward
translation to a fixed point problem is not possible because the operators that
represent the relevant systems of nonlinear equations are not self-mappings,
i.e., they operate between spaces of different dimensions. To overcome this
difficulty we suggest an "alternating common fixed points algorithm" that acts
alternatingly on the different vector variables. This approach translates the
original problem to a common fixed point problem for which iterative algorithms
are abound and exhibits a viable alternative to translation to an optimization
problem, which usually requires derivatives information. However, to apply any
of these iterative algorithms requires to ascertain the conditions that appear
in their convergence theorems. To circumvent the need to verify conditions for
convergence, we propose and motivate a derivative-free algorithm that better
suits the tomographic absorption spectroscopy problem at hand and is even
further improved by applying to it the superiorization approach. This is
presented along with experimental results that demonstrate our approach.
