A method for the solution of certain non-linear problems in least squares

Abstract

The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method.Let the function to be approximated be h{x, y, z, and the least squares criterion requires the minimization of n s(a, 0, 7. * * * ) = £ /<- A different kind of approach, not based upon the standard method, is given by Cauchy, MSthode generate pour la resolution des systemes d'equations simultanees, C. R. Acad. Sci. Paris, 25, 536-538 (1847).

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