Roland W. Freund

Efficient linear circuit analysis by Pade approximation via the Lanczos process

In this paper, we introduce PVL, an algorithm for computing the Pad6 approximation of Laplace-domain transfer functions of large linear networks via a Lanczos process. The PVL algorithm has significantly superior numerical stability, while retaining the same efficiency as algorithms that compute the Pad6 approximation directly through moment matching, such as AWE (l), (2) and its derivatives. As a consequence, it produces more accurate and higher-order approximations, and it renders unnecessary many of the heuristics that AWE and its derivatives had to employ. The algorithm also computes an error bound that permits to identify the true poles and zeros of the original network. We present results of numerical experiments with the PVL algorithm for several large examples.

QMR: a quasi-minimal residual method for non-Hermitian linear systems

SummaryThe biconjugate gradient (BCG) method is the “natural” generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.

A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems

The biconjugate gradient method (BCG) for solving general non-Hermitian linear systems

Iterative solution of linear systems

Ax = b

Krylov-subspace methods for reduced-order modeling in circuit simulation

and its transpose-free variant, the conjugate gradients squared algorithm (CGS), both typically exhib...

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

Recent advances in the field of iterative methods for solving large linear systems are reviewed. The main focus is on developments in the area of conjugate gradient-type algorithms and Krylov subspace methods for non-Hermitian matrices.

Model reduction methods based on Krylov subspaces

The simulation of electronic circuits involves the numerical solution of very large-scale, sparse, in general nonlinear, systems of differential-algebraic equations. Often, the size of these systems can be reduced considerably by replacing the equations corresponding to linear subcircuits by approximate models of much smaller state-space dimension. In this paper, we describe the use of Krylov-subspace methods for generating such reduced-order models of linear subcircuits. Particular emphasis is on reduced-order modeling techniques that preserve the passivity of linear RLC subcircuits.

Reduced-Order Modeling of Large Linear Subcircuits via a Block Lanczos Algorithm

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation of a look-ahead version of the Lanczos algorithm is presented that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix–vector products and inner products as the standard Lanczos process without look-ahead.

An implementation of the QMR method based on coupled two-term recurrences

In recent years, reduced-order modelling techniques based on Krylov-subspace iterations, especially the Lanczos algorithm and the Arnoldi process, have become popular tools for tackling the large-scale time-invariant linear dynamical systems that arise in the simulation of electronic circuits. This paper reviews the main ideas of reduced-order modelling techniques based on Krylov subspaces and describes some applications of reduced-order modelling in circuit simulation.

SPRIM: structure-preserving reduced-order interconnect macromodeling

A method for the efficient computation of accurate reduced-order models of large linear circuits is described. The method, called MPVL, employs a novel block Lanczos algorithm to compute matrix Padé approximations of matrix-valued network transfer functions. The reduced-order models, computed to the required level of accuracy, are used to speed up the analysis of circuits containing large linear blocks. The linear blocks are replaced by their reduced-order models, and the resulting smaller circuit can be analyzed with general-purpose simulators, with significant savings in simulation time and, practically, no loss of accuracy.