Algorithm 315: The damped Taylor's series method for minimizing a sum of squares and for solving systems of nonlinear equations
Helmut Spath · Communications of the ACM · 1967
commentThis procedure performs inverse interpolation in n dimensions, i.e., it will find a set of values for n variables x, such that n functions f(x) are zero.A more sophisticated technique, suitable for large values of n, has been developed by S. M. Robinson (Interpolative Solution of Systems of Nonlinear Equations, SIAM Journal of Numerical Analysis, 3 (1966), 650-658).It can also be used to fit a curve with n arbitrary parameters to a set of points, the n functions being formed, in this case, by equating to zero the differential of the sum of the squares of the residues with respect to each parameter in turn.The functions required are specified by a procedure of the form functions (f, x) where f and x are declared as arrays from 1 to n: This procedure should calculate the n functions from a set of values given in x, placing the results in f.The first step is made by forming partial derivatives over an interval initstep.h0 --6 should be suitable for values of x of the order 1 to 10. Exit from the procedure will occur if: (i) the root sum square of the x increments is less than