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

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2000年第1-2期123卷 395-421页

作者： Freund, RW Bell Labs Lucent Technol Murray Hill NJ 07974 USA

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. (C) 2000 Elsevier Science B.V. All rights reserved.

关键词： lanczos algorithm Arnoldi process linear dynamical system VLSI interconnect transfer function Pade approximation stability passivity positive real function

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Efficient diagonalization of kicked quantum systems

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PHYSICA D-NONLINEAR PHENOMENA 1999年第1-4期131卷 247-253页

作者： Ketzmerick, R Kruse, K Geisel, T Univ Calif Santa Barbara Inst Theoret Phys Santa Barbara CA 93106 USA Max Planck Inst Stromungsforsch D-37073 Gottingen Germany Univ Gottingen Inst Nichtlineare Dynam D-37073 Gottingen Germany

We show that the time-evolution operator of kicked quantum systems, although a full matrix of size N x N, can be diagonalized with the help of a new method based on a suitable combination of fast Fourier transform and lanczos algorithm in just O(N-2 ln N) operations. It allows the diagonalization of matrices of sizes up to N approximate to 10(6) going far beyond the possibilities of standard diagonalization techniques which need O(N-3) operations. We have applied this method to the kicked Harper model revealing its intricate spectral properties. (C) 1999 Elsevier Science B.V All rights reserved.

关键词： quantum chaos diagonalization kicked Harper model lanczos algorithm

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A Krylov subspace method for information retrieval

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2004年第2期26卷 566-582页

作者： Blom, K Ruhe, A Chalmers Dept Comp Sci SE-41296 Gothenburg Sweden Univ Gothenburg SE-41296 Gothenburg Sweden Royal Inst Technol Dept Numer Anal & Comp Sci NADA SE-10044 Stockholm Sweden

A new algorithm for information retrieval is described. It is a vector space method with automatic query expansion. The original user query is projected onto a Krylov subspace generated by the query and the term-document matrix. Each dimension of the Krylov space is generated by a simple vector space search, using first the user query and then new queries generated by the algorithm and orthogonal to the previous query vectors. The new algorithm is closely related to latent semantic indexing (LSI), but it is a local algorithm that works on a new subspace of very low dimension for each query. This makes it faster and more flexible than LSI. No preliminary computation of the singular value decomposition (SVD) is needed, and changes in the data base cause no complication. Numerical tests on both small (Cranfield) and larger (Financial Times data from the TREC collection) data sets are reported. The new algorithm gives better precision at given recall levels than simple vector space and LSI in those cases that have been compared.

关键词： information retrieval vector space model query expansion latent semantic indexing singular value decomposition lanczos algorithm Krylov subspace

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An adjoint network approach for RLCG interconnect model order reductions

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IEICE TRANSACTIONS ON FUNDAMENTALS OF ELECTRONICS COMMUNICATIONS AND COMPUTER SCIENCES 2006年第2期E89A卷 439-447页

作者： Chu, CC Lee, HJ Lai, MH Feng, WS Department of Electrical Engineering Chang Gung University Taiwan Taiwan EverCAD Software Corp. Taiwan Taiwan Chang Gung University Tao-Tuan Taiwan EverCad Software Corp. Hsin-Chu Taiwan Department of Electrical Engineering Chang Gung University Tao-Yuan Taiwan IEEE Society United States Chinese Institute of Engineers China

This work proposes a new method for RLCG interconnect model-order reductions in consideration with the adjoint network. Relationships between an original MNA network and its corresponding adjoint MNA network will be explored first. It will be shown that the congruence transformation matrix used in the one-sided projection can be constructed by using the bi-orthogonal bases developed from the Lanezos-type algorithms. In particular, if the multi-port driving-point impedance of RLCG interconnect circuits is the main concern, the transfer functions and system moments of the adjoint network can be directly calculated from those of the original RLCG interconnect network by exploring symmetric properties of the MNA formulation. Therefore, the cost of constructing the congruence transformation matrix can be simplified by up to 50% of the previous methods. Comparative studies among various standard methods and the proposed methods are also investigated. Experimental results on large-scale RLCG interconnect circuits will demonstrate the accuracy and the efficiency of the proposed method.

关键词： RLCG interconnects model-order reduction Arnoldi algorithm lanczos algorithm adjoint network

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IRR: An algorithm for computing the smallest singular value of large scale matrices

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2001年第1期77卷 89-104页

作者： Guo, HB Fudan Univ Inst Math Shanghai 200433 Peoples R China

For large-scale matrices, there is no practical algorithm to compute the smallest singular value with a satisfied relative accuracy. The widely used bidiagonalization lanczos method can compute the largest singular value with good relative accuracy, but not the smallest one. In this paper we transform the smallest singular value of matrix A to the largest eigenvalue of (A(T)A)(-1), and use Rayleigh-Ritz method, which is referred as Inverse-Rayleigh-Ritz (IRR) method. The technique computing quadratic Form plays an: important role in IRR. IRR takes no more hop cost and storage than lanczos-like Krylov methods on A and gives more accurate results.

关键词： quadratic form lanczos algorithm Inverse-Rayleigh-Ritz the smallest singular values large matrix

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A KRYLOV STABILITY-CORRECTED COORDINATE-STRETCHING METHOD TO SIMULATE WAVE PROPAGATION IN UNBOUNDED DOMAINS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第2期35卷 B376-B400页

作者： Druskin, Vladimir Remis, Rob Schlumberger Doll Res Ctr Cambridge MA 02139 USA Delft Univ Technol Fac Elect Engn Math & Comp Sci Circuits & Syst Grp NL-2628 CD Delft Netherlands

The Krylov subspace projection approach is a well-established tool for the reduced-order modeling of dynamical systems in the time domain. In this paper, we address the main issues obstructing the application of this powerful approach to the time-domain solution of exterior wave problems. We use frequency-independent perfectly matched layers to simulate the extension to infinity. Pure imaginary stretching functions based on Zolotarev's optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest. A new Krylov-based solution method via stability-corrected operator exponents is presented which allows us to construct reduced-order models (ROMs) that respect the delicate spectral properties of the original scattering problem. The ROMs are unconditionally stable and are based on a renormalized bi-lanczos algorithm. We give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two-dimensional electromagnetic wave propagation in unbounded domains, including a photonic waveguide example. The new algorithm outperforms the conventional finite-difference time-domain method for problems on large time intervals.

关键词： model-order reduction lanczos algorithm hyperbolic problems wave propagation PML scattering poles resonances photonic crystals stability correction

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An orthogonal similarity reduction of a matrix into semiseparable form

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第1期27卷 176-197页

作者： Van Barel, M Vandebril, R Mastronardi, N Katholieke Univ Leuven Dept Comp Sci B-3001 Louvain Heverlee Belgium CNR Sez Bari Ist Applicaz Calcolo M Picone I-70126 Bari Italy

An algorithm to reduce a symmetric matrix to a similar semiseparable one of semiseparability rank 1, using orthogonal similarity transformations, is proposed in this paper. It is shown that, while running to completion, the proposed algorithm gives information on the spectrum of the similar initial matrix. In fact, the proposed algorithm shares the same properties of the lanczos method and the Householder reduction to tridiagonal form. Furthermore, at each iteration, the proposed algorithm performs a step of the QR method without shift to a principal submatrix to retrieve the semiseparable structure. The latter step can be considered a kind of subspace-like iteration method, where the size of the subspace increases by one dimension at each step of the algorithm. Hence, when during the execution of the algorithm the Ritz values approximate the dominant eigenvalues closely enough, diagonal blocks will appear in the semiseparable part where the norm of the corresponding subdiagonal blocks goes to zero in the subsequent iteration steps, depending on the corresponding gap between the eigenvalues. A numerical experiment is included, illustrating the properties of the new algorithm.

关键词： similarity transformation semiseparable matrix lanczos algorithm Ritz values

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Theoretical study of the rovibrational spectrum of He₂-OCS

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CANADIAN JOURNAL OF CHEMISTRY-REVUE CANADIENNE DE CHIMIE 2010年第8期88卷 779-786页

作者： Wang, Xiao-Gang Carrington, Tucker, Jr. Queens Univ Dept Chem Kingston ON K7L 3N6 Canada

We report calculated microwave and infrared rovibrational transitions of the van der Waals complex He-2 - OCS. The calculations were done using a product basis, a lanczos eigensolver, and potentials built from He-OCS, and He-He potential functions taken from the literature. All five of the large amplitude coordinates are treated exactly and calculations are done for J values up to five. All rovibrational levels are converged to 0.001 cm(-1) by using basis sets with as many as 87 million funcions. Good agreement is found with previously reported experimental results. Although we assume that the dipole moment is along the OCS axis, we find transitions with appreciable intensity between different torsion states.

关键词： rovibrational spectroscopy quantum dynamics van der Waals clusters doped helium clusters lanczos algorithm

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Exact numerical methods for a many-body Wannier-Stark system

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COMPUTER PHYSICS COMMUNICATIONS 2015年 186卷 19-30页

作者： Parra-Murillo, Carlos A. Madronero, Javier Wimberger, Sandro Heidelberg Univ Inst Theoret Phys D-69120 Heidelberg Germany Univ Valle Dept Fis Cali Colombia Univ Parma Dipartimento Fis & Sci Terra I-43124 Parma Italy Ist Nazl Fis Nucl Grp Collegato Parma Sez Milano Bicocca Milan Italy

We present exact methods for the numerical integration of the Wannier-Stark system in a many-body scenario including two Bloch bands. Our ab initio approaches allow for the treatment of a few-body problem with bosonic statistics and strong interparticle interaction. The numerical implementation is based on the lanczos algorithm for the diagonalization of large, but sparse symmetric Floquet matrices. We analyze the scheme efficiency in terms of the computational time, which is shown to scale polynomially with the size of the system. The numerically computed eigensystem is applied to the analysis of the Floquet Hamiltonian describing our problem. We show that this allows, for instance, for the efficient detection and characterization of avoided crossings and their statistical analysis. We finally compare the efficiency of our lanczos diagonalization for computing the temporal evolution of our many-body system with an explicit fourth order Runge-Kutta integration. Both implementations heavily exploit efficient matrix vector multiplication schemes. Our results should permit an extrapolation of the applicability of exact methods to increasing sizes of generic many-body quantum problems with bosonic statistics. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Floquet problem Bose-Hubbard model Wannier-Stark system lanczos algorithm Quantum chaos Ultracold atoms Optical lattices

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A renormalization scheme for calculating the spectrum of a vibronic system occurring in molecules or impurities in insulators

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PHYSICA B-CONDENSED MATTER 2008年第18期403卷 3093-3096页

作者： Baher, S. Mehrjoo, S. Lorestan Univ Dept Phys Khorramabad *** Iran

An iteration scheme which makes use of a numerical renormalization group approach is used to calculate the spectrum of vibronic levels. This spectrum resulted from dynamic effects occurring in certain molecules or impurities in insulators. The Hamiltonian of these systems is expressed in the matrix form, using products of suitable electron-phonon states as a basis. In applying this method to multimode electron-phonon systems, phonon modes are coupled in a chain-like fashion. Then a finite chain calculation in terms of Hubbard X-operators is explored by setting up the vibronic Hamiltonian. Calculations are based on lanczos algorithm, in which only the nearest neighbor matrix elements along the chain need to be taken into account. The iterative scheme is then applied to a two-level electronic system coupled to phonons. A single-particle Green's function corresponding to a two-level system is applied to calculate the spectral density of states, which, coupled to single mode is carried out. The strength of lines in density of states is affected by the coupling constant as well as the temperature dependence of some measurable quantities. (C) 2008 Elsevier B.V. All rights reserved.

关键词： Hubbard X-operators Green's function density of states lanczos algorithm two-level system

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