A method for the e�cient computation of accu� rate reduced�order models of large linear circuits is de� scribed. The method � called MPVL � employs a novel block Lanczos algorithm to compute matrix Pad�e ap� proximations of matrix�valued network transfer func� tions. The reduced�order models � computed to the re� quired level of accuracy � are used tospeed up the anal� ysis of circuits containing large linear blocks. The lin� ear blocks are replaced by their reduced�order models� and the resulting smaller circuit can be analyzed with general�purpose simulators � with signi�cant savings in simulation time and � practically � no loss of accuracy. 1

This paper discusses the analysis of large linear electrical networks consisting of passive components, such as resistors, capacitors, inductors, and trans-formers. Such networks admit a symmetric formula-tion of their circuit equations. We introduce SyPVL, an eficient and numerically stable algorithm for the computation of reduced-order models of large, linear, passive networks. SyPVL represents the specializa-tion of the more general PVL algorithm, to symmetric problems. Besides the gain in eficiency over PVL, SyPVL also preserves the symmetry of the problem, and, as a consequence, can often guarantee the sta-bility of the resulting reduced-order models. Moreover, these reduced-order models can be synthesized as actual physical circuits, thus facilitating compatibility with existing analysis tools. The application of SyPVL is illustrated with two interconnect-analysis examples. 1

The utility of Lanczos methods for the approximation of large-scale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pad'e approximants which has several advantages over more traditional explicit moment matching approaches. An extension of the Lanczos algorithm is developed for computing multi-point Pad'e approximations of descriptor systems. Keywords: Dynamic system, Pad'e approximation, Lanczos algorithm, model reduction. 1. Introduction This paper explores the use of Lanczos techniques for the reduced-order modeling and simulation of large-scale, SISO dynamical systems. One can define such a system through the set of state space equations ae E x(t) = Ax(t) + bu(t) y(t) = cx(t) + du(t): (1) The scalar functions u(t) and y(t) are the system's input and output while x(t) is the state vector of dimension n. For simplicity, the direct-coupling term, d, will be assumed to be zero. The system matrix, A 2 R n\Thetan ...

. We describe the application of the PVL algorithm to the small-signal analysis of circuits, including sensitivity computations. The PVL algorithm is based on the efficient computation of the Pad'e approximation of the network transfer function via the Lanczos process. The numerical stability of the algorithm permits the computation of the Pad'e approximation to any accuracy over a certain frequency range. We extend the algorithm to compute sensitivities of network transfer functions, their poles, and their zeros, with respect to arbitrary circuit parameters, with minimal additional computational cost. We demonstrate the implementation of our algorithm on circuit examples. 1 Introduction The process of analyzing analog circuits with full accounting of parasitic elements, interconnect analysis at the board or chip level, and numerous other circuit-simulation tasks often require the analysis of large linear networks. These networks can become extremely large, especially when circuits ar...

In this paper we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. 1 1 Introduction For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a n-th order dynamical system be described by x = Ax + bu (1.1) y = cx + du (1.2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is well-known that the transfer function of this system : h(s) = c(sI \Gamma A) \Gamma1 b + d has a Taylor expansion around s = 1 that looks like : h(s) = d + cbs \Gamma1 + cAbs \Gamma2 + cA 2 bs \Gamma3 + cA 3 bs \Gamma4 + : : : The coefficients m \Gammai of the powers of s \Gammai satisfy thus m 0 = d ; m \Gammai = cA i\Gamma1 b ; i 1: For i 1 these are also called moments or Markov parameters of the system fA; b; cg. It follows already from the work of Hankel that the first 2n moments 1 To appea...

Introduction In these two lectures we try to show the relations between the Lanczos algorithm and Pad'e approximations as used e.g. in identification and model reduction of dynamical systems. These notes are based on material in the papers [10, 17, 11, 12] for which a lot of credit ought to be given to the respective coauthors. For simplicity we assume here that all systems are SISO, although some results do extend to the MIMO case. Let a n-th order dynamical system be described by x = Ax + bu (1) y = cx + du (2) where A is a square, b is a column vector, c is a row vector, and d is a scalar. It is well-known that the transfer function of this system : h(s) = c(sI \Gamma A) \Gamma1 b +<F29

this paper, it will be demonstrated that a Pad'e approximation of the original circuit can be obtained without explicitly passing via the moments. Through the nonsymmetric Lanczos method [9, 14], one can realize the reduced-order system f