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作者： Garza-Vargas, Jorge University of California Berkeley

学位级别：Ph.D., Doctor of Philosophy

We rigorously analyze numerical methods for (approximately) computing the eigenvalues and eigenvectors of a matrix. Our main results are the following:• Spectral bisection. We show that a backward-stable solution to the eigenvalue problem can be obtained by a randomized algorithm in nearly matrix-multiplication time on any input. More specifically, we show that for every δ > 0, there is a randomized version of the spectral bisection algorithm that, on any input A ∈ C n×n , with probability 1 − O(1/n), finds matrices V, D ∈ C n×n with V invertible and D diagonal such that ∥A − V DV −1∥ ≤ δ∥A∥, using O(TMM(n)log2 (n/δ)) arithmetic operations, in finite arithmetic with O(log4 (n/δ)log n) bits of precision. Here TMM(n) is the number of arithmetic operations required to multiply two n × n complex matrices numerically stably, known to satisfy TMM(n) = O(n ω+η ) for every η > 0 where ω is the exponent of matrix multiplication.• The Hessenberg shifted QR algorithm. We introduce a randomized shifting strategy for the Hessenberg QR algorithm, for which we prove (in the finite precision arithmetic model) rapid global convergence. It follows from our results, that for any δ, ϕ > 0, a randomized version of the shifted Hessenberg QR algorithm can be used to compute, on any input A ∈ C n×n , with probability at least 1 − ϕ, matrices U, T ∈ C n×n with U unitary and T upper triangular, such that ∥A − UT U∗∥ ≤ δ∥A∥, using O(n3log(n/(δϕ))2 log log(n/(δϕ))) arithmetic operations performed using O(log2 (n/(δϕ))log log(n/(δϕ))) bits of precision. This provides the first rapidly globally convergent shifting strategy in the nonsymmetric case, which was an open problem even under the assumption of exact arithmetic.• The lanczos algorithm. We analyze the lanczos algorithm where the initial vector u is sampled uniformly at random from the unit sphere Sn−1 , and show that when run on a Hermitian input A ∈ Cn×n for c1 log n iterations (where c1 depends on some coarse global property of the spec

关键词： algorithmic analysis Diagonalization lanczos algorithm QR algorithm Spectral bisection algorithm

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Effect of the CHS on Seismic Responses of the Single-Layer Spherical Reticulated Shell under Vertical Seismic Motions

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ADVANCES IN CIVIL ENGINEERING 2023年第1期2023卷

作者： Xu, Jie Liu, Renjie Ji, Ce Wang, Chao Wang, Guangyong Shandong Jianzhu Univ Sch Civil Engn Jinan 250101 Peoples R China Yantai Univ Sch Civil Engn Yantai 264005 Peoples R China

Single-layer spherical reticulated shells are typical roof structures for the gymnasiums. The center-hung scoreboard (CHS) is large weight display device which is usually suspended on the roof of the gymnasium. The effect of the CHS on the dynamic characteristics and seismic responses of the single-layer spherical reticulated shell (SPRS) is not fully clear. In this paper, the effect of the CHS on the SPRS under vertical seismic action is investigated. Two kinds of FE models are built with Abaqus software, including the flexibly suspended model and the simplified model. In a simplified model, the CHS is simplified as four fixed masses on the four CHS suspension nodes. The dynamic explicit method is used for the seismic responses, and the lanczos method is used for the dynamic characteristics. The influence of the CHS weight and the sling length of on dynamic characteristics and seismic responses are analysed. It turns out that in the flexibly suspended model, the first three vibration modes are free swing of the CHS, and the CHS weight and the sling length have a significant impact on the fourth and subsequent modes. The length of the sling has a large impact on some low-order frequencies, but has little impact on the high-order frequencies. Compared with the simplified model, the axial forces of some structural members and some nodal acceleration in the flexibly suspended model under vertical seismic motions would increase by as high as 523% and 564%, respectively. It turns out that the seismic responses of the SPRS would be underestimated if a simplified model is used for analysis. The region in the central of the SPRS, the hoop members of the SPRS, and the support platform are the most affected regions in terms of both axial force and nodal acceleration.

关键词： lanczos algorithm

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Computational study of the rovibrational spectrum of H2O-HF

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JOURNAL OF MOLECULAR SPECTROSCOPY 2022年 384卷 111587-111587页

作者： Viglaskan, Dominika Wang, Xiao-Gang Carrington, Tucker, Jr. Tew, David P. UFR Sci UMR CNRS 7331 Grp Spectrometrie Mol & Atmospher BP 1039 F-51687 Reims 2 France Queens Univ Chem Dept Kingston ON K7L 3N6 Canada Univ Oxford Phys & Theoret Chem Lab South Parks Rd Oxford OX1 3QZ England

In this paper we report rovibrational energy levels, transition frequencies, and intensities computed for H2O-HF using a new ab initio potential energy surface and compare with available experimental data. We use the rigid monomer approximation. A G(4) symmetry-adapted lanczos algorithm and an uncoupled product basis are employed. The rovibrational levels are computed up to J = 4. The new analytic 9-D potential is fit to 39 771 counterpoise corrected CCSD(T)(F12*)/aug-cc-pVTZ energies and reduces to the sum of uncoupled H2O and HF potentials in the dissociation limit. On the new potential better agreement with experiment is obtained by re-assigning the R(1) transitions of two vibrational states.

关键词： Van der Waals molecule lanczos algorithm Ro-vibrational spectroscopy

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Joint Doppler and DOA estimation using (Ultra-)Wideband FMCW signals

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SIGNAL PROCESSING 2020年 168卷 107259-000页

作者： Xu, Shengzhi Kooij, Bert Jan Yarovoy, Alexander Delft Univ Technol NL-2628 CD Delft Netherlands

The joint Doppler and Direction-of-Arrival (DOA) estimation of moving targets using an (Ultra-)Wideband (UWB) frequency modulated continuous-wave (FMCW) antenna array radar is investigated. Besides the well-known range migration problem, another concern for wideband signals is the DOA estimation problem. For the first time, both problems are considered in this paper simultaneously, where the wideband DOA is transformed into a second-order coupling system similar to the range migration problem by using the property of the FMCW signal. A novel embedded compensation approach to eliminate the coupling terms caused by range migration and wideband DOA is proposed and 2D multiple signal classification (2D MUSIC) algorithm is subsequently applied with dynamic noise subspace to joint estimation of Doppler and DOA. Further, to reduce the computational load caused by multiple eigendecompositions of large matrices, efficient implementation methods are proposed and their performance in speed, accuracy and robustness is compared. The performance of the proposed methods is validated by the numerical simulations and is compared with Keystone MUSIC. Finally, it is shown that for a small number of targets, the Rayleigh-Ritz is the most efficient approach among them. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Ultra-Wideband (UWB) Direction-of-Arrival (DOA) Doppler 2 Dimensional multiple signal classification (2D MUSIC) lanczos algorithm Rayleigh-Ritz step

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Model Order Reduction for Linear Time-Invariant System With Symmetric Positive-Definite Matrices: Synthesis of Cauer-Equivalent Circuit

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IEEE TRANSACTIONS ON MAGNETICS 2020年第3期56卷 1-8页

作者： Hiruma, Shingo Igarashi, Hajime Hokkaido Univ Grad Sch Informat Sci & Technol Sapporo Hokkaido 0600814 Japan JSPS Tokyo 1020083 Japan

This article introduces a new model order reduction method for a linear time-invariant system with symmetric positive-definite matrices. The proposed method allows the construction of a reduced model, represented by a Cauer-equivalent circuit, from the original system. The method is developed by extending the Cauer ladder network method for the quasi-static Maxwell's equations, which is shown to be regarded as the lanczos algorithm with respect to a self-adjoint matrix. As a numerical example, a Cauer-equivalent circuit is generated from a simple mathematical model as well as the finite-element (FE) model of a magnetic reactor that is driven by a pulsewidth modulation voltage wave. The instantaneous power obtained from the circuit analysis is shown to be in good agreement with that obtained from the original FE model.

关键词： Continued fraction Krylov subspace methods lanczos algorithm Pade approximation self adjoint matrix transfer function

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SoftNet: A Package for the Analysis of Complex Networks

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algorithmS 2022年第9期15卷 296-296页

作者： Fenu, Caterina Reichel, Lothar Rodriguez, Giuseppe Univ Cagliari Dept Math & Comp Sci Via Osped 72 I-09124 Cagliari Italy Kent State Univ Dept Math Sci Kent OH 44242 USA

Identifying the most important nodes according to specific centrality indices is an important issue in network analysis. Node metrics based on the computation of functions of the adjacency matrix of a network were defined by Estrada and his collaborators in various papers. This paper describes a MATLAB toolbox for computing such centrality indices using efficient numerical algorithms based on the connection between the lanczos method and Gauss-type quadrature rules.

关键词： complex network analysis centrality measure matrix function lanczos algorithm

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A Matrix-Free Likelihood Method for Exploratory Factor Analysis of High-Dimensional Gaussian Data

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JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS 2020年第3期29卷 675-680页

作者： Dai, Fan Dutta, Somak Maitra, Ranjan Iowa State Univ Dept Stat Ames IA 50011 USA

This technical note proposes a novel profile likelihood method for estimating the covariance parameters in exploratory factor analysis of high-dimensional Gaussian datasets with fewer observations than number of variables. An implicitly restarted lanczos algorithm and a limited-memory quasi-Newton method are implemented to develop a matrix-free framework for likelihood maximization. Simulation results show that our method is substantially faster than the expectation-maximization solution without sacrificing accuracy. Our method is applied to fit factor models on data from suicide attempters, suicide ideators, and a control group. for this article are available online.

关键词： EM algorithm fMRI lanczos algorithm L-BFGS-B Profile likelihood

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PairDiag: An exact diagonalization program for solving general pairing Hamiltonians

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COMPUTER PHYSICS COMMUNICATIONS 2021年 259卷 107349-107349页

作者： Liu, Xiao-Yu Qi, Chong Royal Inst Technol Dept Phys S-10405 Stockholm Sweden Chinese Acad Sci Inst Modern Phys Lanzhou 730000 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China

We present a program for solving exactly the general pairing Hamiltonian based on diagonalization. The program generates the seniority-zero shell-model-like basis vectors via the '01' inversion algorithm. The Hamiltonian matrix is constructed in this seniority-zero space. The program evaluates all non-zero elements of the Hamiltonian matrix "on the fly" using the scattering operator and a search algorithm. The matrix is diagonalized by using the iterative lanczos algorithm. The OpenMP parallel program thus developed, PairDiag, can efficiently calculate the ground-state eigenvalue and eigenvector of the general pairing Hamiltonian for both the even-mass and the odd-mass system. The program is packaged in a Fortran module, which makes it easy to use the program to replace the BCS approximation in standard self-consistent mean field calculations. For systems with dimension around 10(8), the calculation can be done within hours on standard desktop computers. (C) 2020 The Authors. Published by Elsevier B.V.

关键词： General pairing Hamiltonian Diagonalization lanczos algorithm

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Application of a Krylov subspace method for an efficient solution of acoustic transfer functions

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2021年 148卷 107135-107135页

作者： Sittl, Christopher Marburg, Steffen Wagner, Marcus OTH Regensburg Lab Finite Element Method Fac Mech Engn D-93053 Regensburg Germany Tech Univ Munich Dept Mech Engn Chair Vibroacoust Vehicles & Machines D-85748 Garching Germany

Solving acoustic radiation problems, arising from systems including fluid-structure interaction, is of interest in many engineering applications. Computing frequency response functions over a large frequency range is a concern in such applications. A method which solves the Helmholtz equation for multiple frequencies in one step is the matrix-Pade-via-lanczos connection for unsymmetric systems, as presented by Wagner et al. [1]. The present work is based on Ref. [1] and presents a method for efficiently computing frequency responses over a frequency range for coupled structural-acoustic problems, where the structure and the acoustic near field are discretized with finite elements and an analytical Dirichlet-to-Neumann map approximates the far field. The method is based on a Krylov-subspace projection technique which derives a matrix-valued Pade approximation for a restricted area in the near field and the pressure field on a spherical boundary. On the spherical boundary, where the finite domain is truncated, the non-local modified Dirichlet-to-Neumann operator is applied as a low-rank update matrix. The present contribution extends this method and incorporates new techniques for a more stable model reduction through the lanczos algorithm and a novel weighted adaptive windowing technique. Further, structural damping is incorporated, for computing the acoustic radiation of a harmonically excited plate. These computed results are compared with acoustic measurements in an anechoic chamber and verified with computational results obtained with a commercial code that uses the perfectly matched layer method. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： lanczos algorithm Krylov-subspace projection Dirichlet-to-Neumann map Pade approximation Fluid-structure interaction Acoustics

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lanczos-type algorithms with embedded interpolation and extrapolation models for solving large-scale systems of linear equations

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INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND MATHEMATICS 2019年第5期10卷 429-442页

作者： Maharani, Maharani Larasati, Niken Salhi, Abdellah Khan, Wali Mashwani Univ Jenderal Soedirman Dept Math Purwokerto Jawa Tengah Indonesia Univ Essex Dept Math Sci Wivenhoe Pk Colchester CO4 3YS Essex England Univ Sci & Technol Dept Math Kohat Pakistan

The new approach to combating instability in lanczos-type algorithms for large-scale problems is proposed. It is a modification of so called the embedded interpolation and extrapolation model in lanczos-type algorithms (EIEMLA), which enables us to interpolate the sequence of vector solutions generated by a lanczos-type algorithm entirely, without rearranging the position of the entries of the vector solutions. The numerical results show that the new approach performs more effectively than the EIEMLA. In fact, we extend this new approach on the use of a restarting framework to obtain the convergence of lanczos algorithms accurately. This kind of restarting challenges other existing restarting strategies in lanczos-type algorithms.

关键词： lanczos algorithm interpolation extrapolation systems of linear equations embedded interpolation and extrapolation model in lanczos-type algorithms EIEMLA restarting strategy

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