2-cyclic splitting for mixed-valued least squares in engineering

We consider the least squares minimization problem in which both complex- and real-valued parameters are simultaneously present. A prominent example of this is the estimation of the frequency response function (FRF) in the presence of missing output data. In this case, the FRF parameters are complex-valued, while the missing output samples are real-valued. In the extended local polynomial method (ELPM), the missing samples are treated as global variables and are estimated simultaneously with the FRF parameters under the least squares criterion. This returns a mixed-valued least squares problem. We provide a formal setting, in which we show mixed-valued least squares problems are well-defined and have a unique solution. We introduce an iterative method for solving the resulting system of equations, which splits the complex-valued and real-valued least squares designs. We show that the resulting method is algebraically equivalent to applying a 2-cyclic matrix splitting to the mixed-valued normal equations, which ensures its convergence and provides an additional adaptive scheme to improve the convergence speed. Finally, we conduct a detailed case-study on the ELPM and compare our method to the original ad-hoc method. We present impressive improvements to the computation time by exploiting the separate physical structures within both regression matrices.