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ASYMPTOTIC WAVE-FORM EVALUATION VIA A lanczos METHOD

APPLIED MATHEMATICS LETTERS 1994年第5期7卷 75-80页

作者： GALLIVAN, K GRIMME, E VANDOOREN, P Coordinated Science Laboratory University of Illinois at Urbana-Champaign 307 N. Mathews Urbana IL 61801 USA

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

关键词： DYNAMICAL SYSTEMS MODEL REDUCTION NUMERICAL METHODS PADE APPROXIMATION lanczos algorithm

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Theoretical studies of (A)over-tilde¹A" → (X)over-tilde¹A′ resonance emission spectra of HCN/DCN using single lanczos propagation method

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JOURNAL OF THEORETICAL & COMPUTATIONAL CHEMISTRY 2003年第4期2卷 639-648页

作者： Xu, DG Guo, H Xie, DQ Univ New Mexico Dept Chem Albuquerque NM 87131 USA Nanjing Univ Dept Chem Nanjing 210008 Jiangsu Peoples R China Nanjing Univ Inst Theoret & Computat Chem Nanjing 210008 Jiangsu Peoples R China Sichuan Univ Dept Chem Chengdu Sichuan Peoples R China

Using an efficient single lanczos propagation method, we report the (A) over tilde (1)A", (X) over tilde (1)A' resonance emission spectra of HCN and DCN from a number of low-lying vibrational levels of the A-state manifold. Our calculations represent the first such undertaking in which a high-quality ab initio based potential energy surface of the excited ((a) over tilde (1)A") state and a (A) over tilde (1)A"-(X) over tilde (1)A' transition dipole surface were used. The results show a significant improvement over previous theoretical work in reproducing experimental stimulated emission pumping spectra of HCN. The improved theory-experiment agreement is attributed to the accurate (A) over tilde -state potential energy surface, while the impact of the transition dipole function was found to be relatively minor.

关键词： lanczos algorithm emission spectrum

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A COMPLETED THEORY OF THE UNSYMMETRIC lanczos PROCESS AND RELATED algorithmS .1.

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1992年第2期13卷 594-639页

作者： GUTKNECHT, MH

The theory of the "unsymmetric" lanczos biorthogonalization (BO) algorithm, which has so far been restricted to an essentially generic situation (characterized by the nonsingularity of the leading principal submatrices of the associated moment matrix or by the existence of a full set of regular formal orthogonal polynomials) is extended to the nongeneric case. The "serious" breakdowns due to the occurrence of two orthogonal right and left iteration vectors x(n) and y(n) can be overcome. For an operator of finite rank N the nongeneric BO algorithm, which generalizes the look-ahead lanczos algorithm of Parlett, Taylor, and Liu [Math. Comp., 44 (1985), pp. 105-124], terminates regularly in at most N steps, except when a very special situation depending on the initial vectors occurs;but even then the algorithm produces in at most N steps a block tridiagonal matrix whose blocks are either small or sparse and whose characteristic polynomial is the minimal polynomial of the restriction of the operator to an invariant subspace. Formulas are also derived for a nongeneric version of the corresponding linear equation solver BIORES (brief for BIORTHORES or lanczos/ORTHORES). The whole theory is developed as a consequence of known corresponding results on formal orthogonal polynomials and Pade approximants, for many of which new and simpler derivations are given.

关键词： lanczos algorithm BIORTHOGONALIZATION algorithm BO algorithm BIORES BICONJUGATE GRADIENT algorithm FORMAL ORTHOGONAL POLYNOMIAL RECURRENCE PADE APPROXIMATION CONTINUED FRACTION QUOTIENT DIFFERENCE algorithm QD algorithm

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CGS, A FAST lanczos-TYPE SOLVER FOR NONSYMMETRIC LINEAR-SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING 1989年第1期10卷 36-52页

作者： SONNEVELD, P

A lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of the conjugate gradients algorithm. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm, it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm. Numerical comparison is made with other solvers, testing the method on a family of convection diffusion equations, on various grids, and with the use of two different preconditioning methods. Upwind as well as central differencing is used in the experiments.

关键词： 65F10 65N20 preconditioned conjugate gradients lanczos algorithm iterative methods CG CGS SPD Bi-CG

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A lanczos-like method for non-autonomous linear ordinary differential equations

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BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA 2023年第1期16卷 81-102页

作者： Giscard, Pierre-Louis Pozza, Stefano Univ Littoral Cote dOpale UR 2597 LMPA Lab Math Pures & Appl Joseph Liouville F-62100 Calais France Charles Univ Prague Fac Math & Phys Prague 8 Czech Republic

The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. The *-lanczos algorithm is a (symbolic) algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. In this paper, we explain how the *-lanczos algorithm is built from a generalization of Krylov subspaces, and we prove crucial properties, such as the matching moment property. A strategy for its numerical implementation is also outlined and will be subject of future investigation.

关键词： lanczos algorithm Matrix differential equations Time-ordered exponential Matching moments Tridiagonal matrices Ordinary differential equations

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PADE APPROXIMATION OF LARGE-SCALE DYNAMIC SYSTEMS WITH lanczos METHODS

PADE APPROXIMATION OF LARGE-SCALE DYNAMIC SYSTEMS WITH LANCZ...

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33rd IEEE Conference on Decision and Control

作者： GALLIVAN, K GRIMME, E VANDOOREN, P UNIV ILLINOIS COORDINATED SCI LABURBANAIL 61801

ISBN: (纸本)0780319680

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 Pade approximants which has several advantages over more traditional explicit moment matching approaches. An extension of the lanczos algorithm is developed for computing multi-point Pade approximations of descriptor systems.

关键词： DYNAMIC SYSTEM PADE APPROXIMATION lanczos algorithm MODEL REDUCTION

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Methodes de type lanczos rationnel pour la reduction de modeles

Methodes de type Lanczos rationnel pour la reduction de mode...

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作者： Houda Barkouki Universite Cadi Ayyad

学位级别：博士

Numerical solution of dynamical systems have been a successful means for study- ing complex physical phenomena. However, in large-scale setting, the system dimension makes the computations infeasible due to memory and time limita- tions, and ill-conditioning. The remedy of this problem is model reduction. This dissertation focuses on projection methods to efficiently construct reduced order models for large linear dynamical systems. Especially, we are interesting by pro- jection onto unions of standard Krylov subspaces which lead to a class of reduced order models known as rational interpolation. Based on this theoretical framework that relate Krylov projection to rational interpolation, four rational lanczos-type algorithms for model reduction are proposed. At first, an adaptive rational block lanczos-type method for reducing the order of large scale dynamical systems is introduced, based on a rational block Lanc- zos algorithm and an adaptive approach for choosing the interpolation points. A generalization of the first algorithm is also given where different multiplicities are consider for each interpolation point. Next, we proposed another extension of the standard Krylov subspace method for Multiple-Input Multiple-Output (MIMO) systems, which is the global Krylov subspace, and we obtained also some equa- tions that describe this process. Finally, an extended block lanczos method is introduced and new algebraic properties for this algorithm are also given. The accuracy and the efficiency of all proposed algorithms when applied to model order reduction problem are tested by means of different numerical exper- iments that use a collection of well known benchmark examples.

关键词： lanczos algorithm Model reduction Moment matching Rational Krylov subspace Transfer function

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Tridiagonalization of systems of coupled linear differential equations with variable coefficients by alanczos-like method

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LINEAR ALGEBRA AND ITS APPLICATIONS 2021年 624卷 153-173页

作者： Giscard, Pierre-Louis Pozza, Stefano Univ Littoral Cote dOpale Lab Math Pures & Appl Joseph Liouville LMPA UR 2597 F-62100 Calais France Charles Univ Prague Fac Math & Phys Sokolovska 83 Prague 18675 8 Czech Republic

We show constructively that, under certain regularity assumptions, any system of coupled linear differential equations with variable coefficients can be tridiagonalized by a time-dependent lanczos-like method. The proof we present formally establishes the convergence of the so-called *-lanczos algorithm and yields a full characterization of algorithmic breakdowns. From there, the solution of the original differential system is available in a finite and treatable number of scalar integral equations. This is a key piece in evaluating the elusive ordered exponential function both formally and numerically. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Tridiagonalization Matrix differential equations lanczos algorithm Time-ordered exponential Tridiagonal matrices Distributions

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A distance-based spectral clustering approach with applications to network community detection

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JOURNAL OF INDUSTRIAL INFORMATION INTEGRATION 2017年 6卷 22-32页

作者： Shen, Gang Ye, Dongmei Huazhong Univ Sci & Technol Sch Software Engn Wuhan 430074 Hubei Peoples R China

Spectral clustering is an important unsupervised learning approach to many object partitioning and pattern analysis problems. In this paper, we present our work on a novel spectral clustering algorithm that groups a collection of objects using the spectrum of the pairwise distance matrix. If the points in a metric space can be associated with a well-defined distance, it is proven that the pairwise distance matrix is almost negative definite, and we show that the eigenvectors for its most significant negative eigenvalue can be used to approximate the solution to a quadratic binary partition problem. We define the quality measures for the one dimensional partitioning of the eigenvector entries, which are further applied to evaluate the partitioning results for the data points projected into the space spanned by the selected eigenvectors. Since the lanczos iterative algorithm may be revised to find the eigenvalues efficiently in a distributed way, we adapt this algorithm to the network community detection problem using a decentralized multi-agent framework. The performance of the proposed approach is tested with different datasets, and the empirical experiments show that this approach is able to enhance the effectiveness of clustering. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Spectral clustering Distance matrix lanczos algorithm Network community detection

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Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers

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COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING 2001年第8期17卷 521-527页

作者： Martikainen, J Rossi, T Toivanen, J Univ Jyvaskyla Dept Math Informat Technol FIN-40351 Jyvaskyla Finland

The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the lanczos algorithm that the original problem. In these experiments, the underlying Poisson and elasticity problems are solved using a standard multigrid method. Copyright (C) 2001 John Wiley & Sons, Ltd.

关键词： algebraic eigenvalue problem lanczos algorithm fast elliptic solver

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