By Chang P.-R.

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4. The minimum norm error approach: Determine xk in AT Yk for which the Euclidean norm ||xk − x||2 is minimal. We will now focus our attention to the important aspect of the construction of an efﬁcient orthogonal basis. This is the key element in any Model Order Reduction approach. After that we will give an example on how to use the orthogonal basis in order to obtain a smaller to solve system that replaces the given system Ax = b. 3 The Krylov Subspace We now derive an orthogonal basis that, in exact arithmetic, spans the Krylov subspace.

We refer to the work of Benner [3]. Another remark: since the matrices are projected by a similarity transform, preservation of passivity is not guaranteed in this method. In [27] a Balanced Truncation method is presented which is provably passive. Here also Poor Man’s TBR [28] should be mentioned as a fruitful approach to implement TBR is a more efﬁcient way. We refer to a later chapter in this book for more information on this topic. 6 Optimal Hankel Norm Reduction Closely related to Balanced Truncation is Optimal Hankel Norm reduction [18].

7. E. S. Nakhla. Asymptotic Waveform Evaluation and moment matching for interconnect analysis. Kluwer Academic Publishers, 1994. 8. S. L. Wachspress. Alternating direction implicit iteration for systems with complex spectra. SIAM J. Numer. 870, June 1991. 9. P. Feldmann and R. Freund. Efﬁcient linear circuit analysis by Pad´e approximation via the Lanczos process. IEEE Trans. Computer-Aided Design, 14:137-158, 1993. 10. P. Feldmann and R. Freund. Reduced-order modeling of large linear subcircuits via a block Lanczos algorithm.