scholarly journals On the shift‐invert Lanczos method for the buckling eigenvalue problem

2021 ◽  
Author(s):  
Chao‐Ping Lin ◽  
Huiqing Xie ◽  
Roger Grimes ◽  
Zhaojun Bai
Keyword(s):  
Eigenvalue Problem ◽  
Lanczos Method

Analysis of vibration levels of large structural system with recursive component mode synthesis method: Theory and convergence

2006 ◽  
Vol 220 (9) ◽  
pp. 1339-1345 ◽  
Author(s):  
C W Kim
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Large Scale ◽  
Synthesis Method ◽  
Lanczos Method ◽  
Component Mode Synthesis ◽  
Structural System ◽  
Accurate Analysis ◽  
Satisfactory Performance ◽  
And Performance

The component mode synthesis (CMS) method has been extensively used in industries. However, industry finite-element (FE) models need a more efficient CMS method for satisfactory performance since the size of FE models needs to be increased for a more accurate analysis. Recently, the recursive component mode synthesis (RCMS) method was introduced to solve large-scale eigenvalue problem efficiently. This article focuses on the convergence of the RCMS method with respect to different parameters, and evaluates the accuracy and performance compared with the Lanczos method.

scholarly journals Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 211
Author(s):  
Asuka Ohashi ◽  
Tomohiro Sogabe
Keyword(s):  
Eigenvalue Problem ◽  
Large Scale ◽  
Numerical Algorithms ◽  
Direct Methods ◽  
Coefficient Matrix ◽  
Lanczos Method ◽  
Singular Value ◽  
Preconditioned Conjugate Gradient ◽  
Schur Decomposition ◽  
Pcg Method

We consider computing an arbitrary singular value of a tensor sum: T:=In⊗Im⊗A+In⊗B⊗Iℓ+C⊗Im⊗Iℓ∈Rℓmn×ℓmn, where A∈Rℓ×ℓ, B∈Rm×m, C∈Rn×n. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of (TTT−σ˜2Iℓmn)−1, where σ˜ is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix TTT−σ˜2Iℓmn. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of TTT and the PCG method for TTT−σ˜2Iℓmn using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments.

Generalizing of Numerically Solving Methods of Eigenvalue Problems to Asymmetrical, Damping Included Case

2001 ◽  
Author(s):  
Shahram Rezaei
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Lanczos Method ◽  
Subspace Method ◽  
Damping Matrix ◽  
Stiffness Matrices

Abstract In this paper, “Subspace” method is generalized to asymmetrical case. In the new algorithm described here, “Lanczos” method is used to find the first subspace and to solve the eigenvalue problem resulted in generalized subspace method. To solve the standard eigenvalue problem developed by “Lanczos” method “Jacoby” method is used. If eigenvalue problem includes damping matrix, that will be imported in new defined mass and stiffness matrices.

scholarly journals A Method of Indefinite Krylov Subspace for Eigenvalue Problem

2018 ◽  
Vol 2018 ◽  
pp. 1-5 ◽  
Author(s):  
M. Aliyari ◽  
M. Ghasemi Kamalvand
Keyword(s):  
Eigenvalue Problem ◽  
Krylov Subspace ◽  
Eigenvalue Problems ◽  
Lanczos Method ◽  
Hermitian Matrices ◽  
Better Than

We describe an indefinite state of Arnoldi’s method for solving the eigenvalues problems. In the following, we scrutinize the indefinite state of Lanczos’ method for solving the eigenvalue problems and we show that this method for the J-Hermitian matrices works much better than Arnoldi’s method.

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