On search for an optimal parameter for a smoothing spline

We study the problem of choosing an optimal parameter for smoothing an abstract smoothing spline for which the norm of the deviation in the nodes of the mesh (the norm of the error) must coincide with the prescribed level of data error. The obtained equation is nonlinear in the smoothing parameter, and it can be solved iteratively, for example, by the Newton method. In using the Newton method, at each step of the iterative process, two problems of smoothing with the same smoothing parameter but with different vectors of approximated data must be solved.
We propose an algorithm for solving this equation using representations of the error operator of the smoothing spline and the complementary operator in the form of the sum of power series. Its novelty consists in applying a hybrid approach depending on the relation of the next approximation of the smoothing parameter and its optimal value. In the algorithm, we use approximations of the error operator and the complementary operator as a partial sum of a series, the algorithm of linear-fractional approximation of error functions, and a refinement of approximations of the error functions by extrapolation with respect to the length of the partial sums of the series. The proposed algorithm enables us to achieve an optimal value of the smoothing parameter in fewer iterations (in practical computations, in two iterations) by solving several smoothing problems at each step.