Noniterative estimation of a nonlinear parameter

ABSTRACTAn algorithm is described which solves the parameters x = (x 1 , x 2 , ..., x m ) andp in an approximation problem Ax*«y(p), where the parameter p occurs non-linearly in y. Instead of linearization methods, which require an approximatevalue of p to be supplied as a priori inf ormation, and which may lead to the find-ing of local minima, the proposed algorithm finds the global minimum bypermitting the use of series expansions of arbitrary order, exploiting an a pri-ori knowledge that the addition of a particular function, corresponding to anew column in A, will not improve the goodness of the approximation.I. INTRODUCTIONIn the present paper the following approximationproblem will be studied. Given(1) a function y(s, p) of an independent variable s (theextension to functions of several variables is trivial) andcontaining a parameter 5? which occurs nonlinear ly, an d(2) a set of functions fi(s), / 2 (s), •••, f m ( s ) °f s >determine a linear combination g(s, x) =x 1 f^s) + x