Acceleration of Vector Fitting by Reusing the Householder Reflectors in Multiple QR Factorization

The classic method of accelerating vector fitting (VF) for a multiport network is to do several small QR factorizations to extract the R22 matrices before solving the least-square system. In the literature and some open-source VF implementations, each QR factorization is performed separately. Taking a closer look at the theory, however, we can see that the first block of the matrices being factorized are the same, which means the computational cost can be reduced if the factorization of this part is skipped. To achieve this goal, however, we cannot simply call the high-level QR functions offered in many computational packages; instead, we must go down to the bottom level of QR factorization and reuse the Householder reflectors directly. In this paper, the theory and implementation of this idea is presented in detail. The theoretic flop reduction is roughly 25%, while in actual tests the time reduction may reach 60%.