A Procedure for Nonlinear Least Squares Refinement in Adverse Practical Conditions

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Abstract
An iterative method is described for computing optimum values of a set of parameters for the best agreement in a least squares between a given set of (experimental) values and corresponding calculated (theoretical) values where these may be dependent non-linearly on the parameters. The method is for use when (a) the amount of work in the calculation of the theoretical values is large as compared with the amount required to solve a set of linear equations in the same number of unknowns; and (b) the amount of work in calculating the first order partial derivatives of all the theoretical values with respect to one parameter would be even greater, so that using finite difference approximations to the partial derivatives is preferable. In addition to setting up and solving the normal equations at each step, the latent vectors of their matrix and estimates of the errors on the current parameter values are computed and used. Also the known physical limits of the range of each parameter are used. The procedure has second order convergence.