A new method for finding the optimal smoothing parameter for the abstract smoothing spline

In this paper, we present a new method for finding the optimal smoothing parameter αα for the abstract smoothing spline σασα. The latter is defined as the solution of the minimization problem σα≔argminx{α‖Tx‖2+‖Ax−z‖2}, and, given an estimate ϵϵ for the noise, the standard choice for αα is that σασα should satisfy the residual equation ‖Aσα−z‖=ϵ.‖Aσα−z‖=ϵ. Whereas the classical algorithm for solving this equation based on Newton’s method converges quadratically, our new method can have any given order of convergence.