Multi input minimax adaptive Antoulas-Anderson algorithm for rational approximation with stable poles

Abstract

This thesis details the development of the 'symmetric stable multi-input multioutput Adaptive Antoulas Anderson' algorithm, we call this algorithm symmetric smiAAA. The symmetric smiAAA algorithm builds rational approximations, for multiple inputs. The approximations share a common set of parameters called the poles. The primary goal of this algorithm is to address shortcomings in multi-input multi-output rational approximation algorithms currently used in electro-magnetic transients programs. All state of the art algorithms currently follow a similar methodology: The user selects the number of poles to use and supplies an initial guess for their values. The algorithms optimize the shared poles and return the best approximation they found. The user is not guaranteed a specific accuracy in the approximations. If the results returned are not sufficiently accurate, the algorithm must be run again with additional poles. Symmetric smiAAA is designed with the goal of achieving user-defined accuracy, with no information about the number of poles. The user selects the desired accuracy of the approximations and the algorithm does the rest. Symmetric smiAAA returns approximations with the desired accuracy by finding the number of shared poles needed for the desired accuracy, and their values. This work introduces the following three features to the 'Adaptive Antoulas Anderson' algorithm. First, we extend the ideas from the single-input Adaptive Antoulas Anderson algorithm, to multi-input multi-output problems. Second, we introduce enforcement of constraints on the values of the poles. Lastly, we extend a single input post-processing optimization based upon the Lawson method, to multi-input multi-output problems. The symmetric smiAAA algorithm combines these three features with the symmetry enforcement introduced in the FastAAA algorithm. In order to test it against the current industry standards, we compare the symmetric smiAAA algorithm with Vector Fitting and the recently published RKFIT algorithms. These comparisons demonstrate that symmetric smiAAA produces approximations with similar accuracy and running time, while allowing the user to select only the desired accuracy. Symmetric smiAAA is a robust and powerful algorithm which provides the user full control over the final accuracy of the approximations.