Nonlinear regression and the solution of simultaneous equations

If one has a set of observables (z1, ··· , zm) which are bound in a relation with certain parameters (a1, ··· , an) by an equation &zgr;(z1, ··· , a1, ···) = 0, one frequently has the problem of determining a set of values of the ai which minimizes the sum of squares of differences between observed and calculated values of a distinguished observable, say zm. If the solution of the above equation for zm, zm = &eegr;(z1, ··· ; a1, ···) gives rise to a function &eegr; which is nonlinear in the ai, then one may rely on a version of Gaussian regression [1, 2] for an iteration scheme that converges to a minimizing set of values. It is shown here that this same minimization technique may be used for the solution of simultaneous (not necessarily linear) equations.

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