Fast generalized cross validation using Krylov subspace methods

The task of fitting smoothing spline surfaces to meteorological data such as temperature or rainfall observations is computationally intensive. The generalized cross validation (GCV) smoothing algorithm, if implemented using direct matrix techniques, is O(n ³) computationally, and memory requirements are O(n ²). Thus, for data sets larger than a few hundred observations, the algorithm is prohibitively slow. The core of the algorithm consists of solving series of shifted linear systems, and iterative techniques have been used to lower the computational complexity and facilitate implementation on a variety of supercomputer architectures. For large data sets though, the execution time is still quite high. In this paper we describe a Lanczos based approach that avoids explicitly solving the linear systems and dramatically reduces the amount of time required to fit surfaces to sets of data.