A model order reduction technique for systems depending on two parameters is developed. Given a large system model, the method generates the descriptor matrices of a system model of lower order that is a rational interpolant of the transfer function of the large system—the transfer functions have identical values and derivatives for a finite set of parameter values. The new technique is a generalization of recently developed algorithms for one-parameter systems that are based on projections onto Krylov subspaces defined by the descriptor matrices.

In this paper, we describe some recent developments in the use of projection methods to produce a reduced-order model for a linear time-invariant dynamical system which approximates its frequency response. We give an overview of the family of Rational Krylov methods and compare them with "near-optimal" approximation methods based on balancing transformations.

In this paper, we describe some recent developments in the use of projection methods to produce reduced-order models for linear time-invariant dynamic systems. Previous related efforts in model reduction problems from various applications are also discussed. An overview is given of the theory governing the definition of the family of Rational Krylov methods, the practical heuristics involved and the important future research directions.

Rational Krylov methods are potentially one of the most robust and efficient algorithms to compute lower order approximations to linear time invariant dynamical systems that match a specified number of moments of the transfer function at multiple points in the complex plane. The characterization of multipoint rational interpolation in terms of bases of multiple Krylov spaces was developed recently. In this paper, we summarize some results concerning the use of approximate solutions to the linear systems of equations that arise on each step of the method. Such approximations are used to reduce the space and time required to produced the reduced order system

In part I of this work 10], a rational Lanczos algorithm was developed which led to rational inter-polants of dynamical systems. In this sequel, the important implementational issue of interpolation point selection is analyzed in detail. A residual expression is derived for the rational Lanczos algorithm and is used to govern the placement and type of the interpolation points. Algorithms are developed and applied to a problem arising from circuit interconnect modeling.

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. We also explore the use of variants of the Lanczos method in order to obtain approximations with better properties than the ones resulting from the standard Lanczos algorithm. 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 iGamma1 b ...

The utility of Lanczos methods (1950) for the approximation of large-scale dynamical systems is considered. In particular, it is shown that the Lanczos method is a technique for yielding Pade approximants which has several advantages over more traditional explicit moment matching approaches. An extension of the Lanczos algorithm is developed for computing multipoint Pade approximations of descriptor systems.< >

In this paper we show that the two-sided Lanczos procedure combined with implicit restarts, offers significant advantages over Padé approximations used typically for model reduction in circuit simulation.

Iterative methods based on Lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large-scale discrete ill-posed linear least-squares problems of the form min x ‖Ax−b‖2. Methods for regularization in the Krylov subspaces are discussed which use generalized cross validation (GCV) for determining the regularization parameter. These methods have the advantage that no a priori information about the noise level is required. To improve convergence of the Lanczos process we apply a variant of the implicitly restarted Lanczos algorithm by Sorensen using zero shifts. Although this restarted method simply corresponds to using LBDR with a starting vector (AA T) p b, it is shown that carrying out the process implicitly is essential for numerical stability. An LBDR algorithm is presented which incorporates implicit restarts to ensure that the global minimum of the CGV curve corresponds to a minimum on the curve for the truncated SVD solution. Numerical results are given comparing the performance of this algorithm with non-restarted LBDR.

The nonsymmetric Lanczos method has recently received attention as a model reduction technique for large-scale systems. However, the Lanczos method may generate an unstable partial realization for a given stable system. To remedy this situation, inexpensive implicit restarts are developed to stabilize a Lanczos generated model.