Computation of eigenpair partial derivatives by Rayleigh–Ritz procedure

Based on Rayleigh–Ritz procedure, a new method is proposed for a few eigenpair partial derivatives of large matrices. This method simultaneously computes the approximate eigenpairs and their partial derivatives. The linear systems of equations that are solved for eigenvector partial derivatives are greatly reduced from the original matrix size. And the left eigenvectors are not required. Moreover, errors of the computed eigenpairs and their partial derivatives are investigated. Hausdorff distance and containment gap are used to measure the accuracy of approximate eigenpair partial derivatives. Error bounds on the computed eigenpairs and their partial derivatives are derived. Finally numerical experiments are reported to show the efficiency of the proposed method.     

IRAM‐based method for eigenpairs and their derivatives of large matrix‐valued functions

Based on the implicitly restarted Arnoldi method for eigenpairs of large matrix, a new method is presented for the computation of a few eigenpairs and their derivatives of large matrix‐valued functions. Eigenpairs and their derivatives are calculated simultaneously. Equation systems that are solved for eigenvector derivatives are greatly reduced from the original matrix size. The left eigenvectors are not required. Hence, the computational cost is saved. The convergence theory of the proposed method is established. Finally, numerical experiments are given to illustrate the efficiency of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Symmetry preserving partial eigenvalue assignment for undamped structural systems

In this paper, the partial eigenvalue assignment problem for undamped structural systems by output feedback control where the output matrix is also a designing parameter is considered. We propose a method to solve this problem in which the unwanted eigenvalues are move to desired values and all other eigenpairs remain unchanged. In addition, our method can preserve symmetry of the systems. Numerical example shows that the proposed method is effective.     

振动杆有限差分模型的一类特征值反问题

1引言杆是重要的工程构件之一,具有分布质量的杆的纵向振动由下面的偏微分方程描述:     

非对称阻尼系统特征对一阶导数与二阶导数的计算

提出了一种计算非对称阻尼系统特征对一阶、二阶导数的方法.该方法利用阻尼系统的特征向量计算特征对的导数,避免了状态空间中特征向量的使用,节省了计算量,且不要求系统所有特征值的互异性.最后以两个非对称阻尼系统进行数值试验,数值结果表明提出的方法是有效的.     

DERIVATIVES OF EIGENPAIRS OF SYMMETRIC QUADRATIC EIGENVALUE PROBLEM

Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

An iterative method for partial derivatives of eigenvectors of quadratic eigenvalue problems

An iterative method is proposed to compute partial derivatives of eigenvectors of quadratic eigenvalue problems with respect to system parameters. Convergence theory of the proposed method is established. Numerical experiments demonstrate that the proposed method can be used efficiently for partial derivatives of eigenvectors corresponding to dominant eigenvalues.     

Verifying all eigenpairs in symmetric positive definite generalized eigenvalue problem

A fast method for enclosing all eigenpairs in symmetric positive definite generalized eigenvalue problem is proposed. Firstly theorems on verifying all eigenvalues are presented. Next a theorem on verifying all eigenvectors is presented. The proposed method is developed based on these theorems. Numerical results are presented showing the efficiency of the proposed method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simple application of the homotopy method to symmetric eigenvalue problems

The homotopy method is used to find all eigenpairs of symmetric matrices. A special homotopy is constructed for Jacobi matrices. It is shown that there are exactly n distinct smooth curves connecting trivial solutions to desired eigenpairs. These curves are solutions of a certain ordinary differential equation with different initial values. Hence, they can be followed numerically. Incorporated with sparse matrix techniques, this method might be used to solve eigenvalue problems for large scale matrices.     

Shifted power method for computing tensor H‐eigenpairs

In this paper, we propose a shifted symmetric higher‐order power method for computing the H‐eigenpairs of a real symmetric even‐order tensor. The local convergence of the method is proved. In addition, by utilizing the fixed‐point analysis, we can characterize exactly which H‐eigenpairs can be found and which cannot be found by the method. Numerical examples are presented to illustrate the performance of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

A modified harmonic block Arnoldi algorithm with adaptive shifts for large interior eigenproblems

The harmonic block Arnoldi method can be used to find interior eigenpairs of large matrices. Given a target point or shift τ$\tau$ to which the needed interior eigenvalues are close, the desired interior eigenpairs are the eigenvalues nearest τ$\tau$ and the associated eigenvectors. However, it has been shown that the harmonic Ritz vectors may converge erratically and even may fail to do so. To do a better job, a modified harmonic block Arnoldi method is coined that replaces the harmonic Ritz vectors by some modified harmonic Ritz vectors. The relationships between the modified harmonic block Arnoldi method and the original one are analyzed. Moreover, how to adaptively adjust shifts during iterations so as to improve convergence is also discussed. Numerical results on the efficiency of the new algorithm are reported.     

An interval uncertainty analysis method for structural response bounds using feedforward neural network differentiation

This paper proposes a new interval uncertainty analysis method for structural response bounds with uncertain‑but-bounded parameters by using feedforward neural network (FNN) differentiation. The information of partial derivative may be unavailable analytically for some complicated engineering problems. To overcome this drawback, the FNNs of real structural responses with respect to structure parameters are first constructed in this work. The first-order and second-order partial derivative formulas of FNN are derived via the backward chain rule of partial differentiation, thus the partial derivatives could be determined directly. Especially, the influences of structures of multilayer FNNs on the accuracy of the first-order and second-order partial derivatives are analyzed. A numerical example shows that an FNN with the appropriate structure parameters is capable of approximating the first-order and second-order partial derivatives of an arbitrary function. Based on the parameter perturbation method using these partial derivatives, the extrema of the FNN can be approximated without requiring much computational time. Moreover, the subinterval method is introduced to obtain more accurate and reliable results of structural response with relatively large interval uncertain parameters. Three specific examples, a cantilever tube, a Belleville spring, and a rigid-flexible coupling dynamic model, are employed to show the effectiveness and feasibility of the proposed interval uncertainty analysis method compared with other methods.     

PRECONDITIONING BLOCK LANCZOS ALGORITHM FOR SOLVING SYMMETRIC EIGENVALUE PROBLEMS

1. Introduction;The Lanczos process is an effective method [1, 2, 14, 21] for computing a feweigenValues and corresponding eigenvectors of a large sparse symmetric matrix A ERnxn. If it is practical to factor the matrix A -- PI for one or more values of p near thedesired eigenvalues, the Lanczos method can be used with the inverted operator andconvergence will be very rapid[5,10,22]. In practical applications, however, the matrixA is usually large and sparse, so factoring A is either impos…     

Partial pole assignment for the quadratic pencil by output feedback control with feedback designs

In this paper we study the partial pole assignment problem for the quadratic pencil by output feedback control where the output matrix is also a designing parameter. In addition, the input matrix is set to be the transpose of the output matrix. Under certain assumption, we give a solution to this partial pole assignment problem in which the unwanted eigenvalues are moved to desired values and all other eigenpairs remain unchanged. Copyright © 2005 John Wiley & Sons, Ltd.     

Jacobi–Davidson methods for cubic eigenvalue problems

Several Jacobi–Davidson type methods are proposed for computing interior eigenpairs of large‐scale cubic eigenvalue problems. To successively compute the eigenpairs, a novel explicit non‐equivalence deflation method with low‐rank updates is developed and analysed. Various techniques such as locking, search direction transformation, restarting, and preconditioning are incorporated into the methods to improve stability and efficiency. A semiconductor quantum dot model is given as an example to illustrate the cubic nature of the eigenvalue system resulting from the finite difference approximation. Numerical results of this model are given to demonstrate the convergence and effectiveness of the methods. Comparison results are also provided to indicate advantages and disadvantages among the various methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Vertex singularities associated with conical points for the 3D laplace equation

The solution u to the Laplace equation in the neighborhood of a vertex in a three‐dimensional domain may be described by an asymptotic series in terms of spherical coordinates $$u = \sum\nolimits_i {A_i}{\rho ^{{\nu _i}}}{f_i}(\theta ,\phi )$$ . For conical vertices, we derive explicit analytical expressions for the eigenpairs ν_i and f_i(θ, φ), which are required as benchmark solutions for the verification of numerical methods. Thereafter, we extend the modified Steklov eigen‐formulation for the computation of vertex eigenpairs using p/spectral finite element methods and demonstrate its accuracy and high efficiency by comparing the numerically computed eigenpairs to the analytical ones. Vertices at the intersection of a crack front and a free surface are also considered and numerical eigenpairs are provided. The numerical examples demonstrate the efficiency, robustness, and high accuracy of the proposed method, hence its potential extension to elasticity problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

This paper is concerned with computing $\mathrm{??}$‐eigenpairs of symmetric tensors. We first show that computing $\mathrm{??}$‐eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex‐valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.     

An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam

This paper presents a transfer matrix expression that can be used to determine the eigenpairs of a rotating beam with a cross section height that linearly decreases along the length of the beam element. The proposed method considers the effect of centrifugal force, including the effects of the axial force, hub radius, and taper ratio. Differential equations are solved for the in-plane bending vibration using the Frobenius method for a power series. The effect of the rotational speed on the eigenpairs of a rotating tapered beam is first investigated, followed by an examination of the contribution rates of the bending strain and additional strain energies generated by centrifugal forces for each mode by analyzing the variation of the energies computed from the strain and kinetic energies. To compute these contribution rates, we used a shape function that was defined by the displacement at both ends of the beam elements. The effect of tapering on the eigenfrequencies of the transverse vibration of rotating beams is analyzed by using various examples, and the contribution rates are examined by using taper ratios of 0 and 0.5.     

Backward error analysis for linearizations in heavily damped quadratic eigenvalue problem

Heavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize the original problem via linearizations. Previous work on the accuracy of eigenpairs of not heavily damped QEP focuses on analyzing the backward error of eigenpairs relative to linearizations. The objective of this paper is to explain why different linearizations lead to different errors when computing small and large eigenpairs. To obtain this goal, we bound the backward error of eigenpairs relative to the linearization methods. Using these bounds, we build upper bounds of growth factors for the backward error. We present results of numerical experiments that support the predictions of the proposed methods.