Joint eigenvalue estimation by balanced simultaneous Schur decomposition

The problem of joint eigenvalue estimation for the non-defective commuting set of matrices A is addressed. A procedure revealing the joint eigenstructure by simultaneous diagonalization of. A with simultaneous Schur decomposition （SSD） and balance procedure alternately is proposed for performance considerations and also for overcoming the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations, it is shown that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i. e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Under mild assumptions, the convergence of the two cost functions alternately optimized, i. e., the norm of A and the norm of the left-lower part of A is proved. Numerical experiments are conducted in a multi-dimensional harmonic retrieval application and suggest that the presented method converges considerably faster than the methods based on only unitary transformation for matrices which are not near to normality.

Mode decomposition of nonlinear eigenvalue problems and application in flow stability

Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an Nth-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.

Derivative of a Determinant with Respect to an Eigenvalue in the <i>LDU</i>Decomposition of a Non-Symmetric Matrix

We demonstrate that, when computing the LDU decomposition (a typical example of a direct solution method), it is possible to obtain the derivative of a determinant with respect to an eigenvalue of a non-symmetric matrix. Our proposed method augments an LDU decomposition program with an additional routine to obtain a program for easily evaluating the derivative of a determinant with respect to an eigenvalue. The proposed method follows simply from the process of solving simultaneous linear equations and is particularly effective for band matrices, for which memory requirements are significantly reduced compared to those for dense matrices. We discuss the theory underlying our proposed method and present detailed algorithms for implementing it.

Derivative of a Determinant with Respect to an Eigenvalue in the Modified Cholesky Decomposition of a Symmetric Matrix, with Applications to Nonlinear Analysis

In this paper, we obtain a formula for the derivative of a determinant with respect to an eigenvalue in the modified Cholesky decomposition of a symmetric matrix, a characteristic example of a direct solution method in computational linear algebra. We apply our proposed formula to a technique used in nonlinear finite-element methods and discuss methods for determining singular points, such as bifurcation points and limit points. In our proposed method, the increment in arc length (or other relevant quantities) may be determined automatically, allowing a reduction in the number of basic parameters. The method is particularly effective for banded matrices, which allow a significant reduction in memory requirements as compared to dense matrices. We discuss the theoretical foundations of our proposed method, present algorithms and programs that implement it, and conduct numerical experiments to investigate its effectiveness.

Modified multiple-component scattering power decomposition for PolSAR data based on eigenspace of coherency matrix

A modified multiple-component scattering power decomposition for analyzing polarimetric synthetic aperture radar(PolSAR)data is proposed.The modified decomposition involves two distinct steps.Firstly,ei⁃genvectors of the coherency matrix are used to modify the scattering models.Secondly,the entropy and anisotro⁃py of targets are used to improve the volume scattering power.With the guarantee of high double-bounce scatter⁃ing power in the urban areas,the proposed algorithm effectively improves the volume scattering power of vegeta⁃tion areas.The efficacy of the modified multiple-component scattering power decomposition is validated using ac⁃tual AIRSAR PolSAR data.The scattering power obtained through decomposing the original coherency matrix and the coherency matrix after orientation angle compensation is compared with three algorithms.Results from the experiment demonstrate that the proposed decomposition yields more effective scattering power for different PolSAR data sets.

基于一般散射模型的Hybrid Freeman/Eigenvalue分解算法(英文)

提出了一种新的基于一般散射模型的hybrid Freeman/eigenvalue分解算法,用于分析极化合成孔径雷达(PolS AR)数据。文中,单位矩阵作为体散射模型,相干矩阵的两个较大特征值对应的特征向量作为表面散射模型和二次散射模型,并且不需要反射对称条件。新算法有三个优点:第一,表面散射和二次散射不需要反射对称条件,更符合一般散射体的建模;第二,因为散射能量是相干矩阵特征值的线性组合,所以散射能量具有旋转不变性;第三,表面散射能量和二次散射能量避免了负值现象。在San Francisco地区的AIRSAR数据上进行了实验,证明了新算法的有效性。

一种自适应的混合Freeman/Eigenvalue极化分解模型

全极化SAR数据的极化分解在土地利用分类、目标检测与识别以及地表参数反演等领域得到了广泛应用。目前,主要有基于特征值分解和基于模型分解2类极化分解方法。混合Freeman/Eigenvalue极化分解结合了两者的优势,避免了基于模型的极化分解中负功率问题并且能够利用已知的散射机制解释分解后的散射分量。为了进一步拓展该分解在不同地表类型中的应用,通过引入参数Neumann一般化体散射模型,提出了一种自适应的极化分解模型。利用德国Black Forest地区的L波段AirSAR(airborne synthetic aperture Radar)全极化数据进行实验,并与现有的Yamaguchi三分量模型和自适应非负分解(adaptive nonnegative eigenvalue decomposition,ANNED)对比分析,以验证模型的有效性。研究表明,自适应的混合Freeman/Eigenvalue极化分解模型保证了分解能量的非负性及完全分解,适应于不同类型的地表,能有效地区分不同地类。

Modified version of three-component model-based decomposition for polarimetric SAR data

A new hybrid Freeman/eigenvalue decomposition based on the orientation angle compensation and the various extended volume models for polarimetric synthetic aperture radar(PolSAR) data are presented. There are three steps in the novel version of the three-component model-based decomposition.Firstly, two special unitary transform matrices are applied on the coherency matrix for deorientation to decrease the correlation between the co-polarized term and the cross-polarized term.Secondly, two new conditions are proposed to distinguish the manmade structures and the nature media after the orientation angle compensation. Finally, in order to adapt to the scattering properties of different media, five different volume scattering models are used to decompose the coherency matrix. These new conditions pre-resolves man-made structures, which is beneficial to the subsequent selection of a more suitable volume scattering model.Fully PolSAR data on San Francisco are used in the experiments to prove the efficiency of the proposed hybrid Freeman/eigenvalue decomposition.

A NEW METHOD FOR ESTIMATING BOUNDS OF EIGENVALUES

A new method for estimating the bounds of eigenvalues ispresented. In order to show that the method proposed is as effectiveas Qiu's an undamping spring-mass system with 5 nodes and 5 degrees ofreedom is given. To illustrate that the present method can beapplied to structures which cannot be treated by non-negativedecomposition, a plane frame with 202 nodes and 357 beam elements isgiven. The results show that the present method is effective forestimating the bounds of eigenvalues and is more common than Qiu's.

A bearing fault feature extraction method based on cepstrum pre-whitening and a quantitative law of symplectic geometry mode decomposition

In order to extract the fault feature of the bearing effectively and prevent the impact components caused by bearing damage being interfered with by discrete frequency components and background noise,a method of fault feature extraction based on cepstrum pre-whitening(CPW)and a quantitative law of symplectic geometry mode decomposition(SGMD)is proposed.First,CPW is performed on the original signal to enhance the impact feature of bearing fault and remove the periodic frequency components from complex vibration signals.The pre-whitening signal contains only background noise and non-stationary shock caused by damage.Secondly,a quantitative law that the number of effective eigenvalues of the Hamilton matrix is twice the number of frequency components in the signal during SGMD is found,and the quantitative law is verified by simulation and theoretical derivation.Finally,the trajectory matrix of the pre-whitening signal is constructed and SGMD is performed.According to the quantitative law,the corresponding feature vector is selected to reconstruct the signal.The Hilbert envelope spectrum analysis is performed to extract fault features.Simulation analysis and application examples prove that the proposed method can clearly extract the fault feature of bearings.

Subspace decomposition-based correlation matrix multiplication

The correlation matrix, which is widely used in eigenvalue decomposition （EVD） or singular value decomposition （SVD）, usually can be denoted by R = E[yiy＇i]. A novel method for constructing the correlation matrix R is proposed. The proposed algorithm can improve the resolving power of the signal eigenvalues and overcomes the shortcomings of the traditional subspace methods, which cannot be applied to low SNR. Then the proposed method is applied to the direct sequence spread spectrum （DSSS） signal＇s signature sequence estimation. The performance of the proposed algorithm is analyzed, and some illustrative simulation results are presented.

Complex complete quadratic combination method for damped system with repeated eigenvalues

A new response-spectrum mode superposition method, entirely in real value form, is developed to analyze the maximum structural response under earthquake ground motion for generally damped linear systems with repeated eigenvalues and defective eigenvectors. This algorithm has clear physical concepts and is similar to the complex complete quadratic combination （CCQC） method previously established. Since it can consider the effect of repeated eigenvalues, it is called the CCQC-R method, in which the correlation coefficients of high-order modal responses are enclosed in addition to the correlation coefficients in the normal CCQC method. As a result, the formulas for calculating the correlation coefficients of high-order modal responses are deduced in this study, including displacement, velocity and velocity-displacement correlation coefficients. Furthermore, the relationship between high-order displacement and velocity covariance is derived to make the CCQC-R algorithm only relevant to the high-order displacement response spectrum. Finally, a practical step-by-step integration procedure for calculating high-order displacement response spectrum is obtained by changing the earthquake ground motion input, which is evaluated by comparing it to the theory solution under the sine-wave input. The method derived here is suitable for generally linear systems with classical or non-classical damping.

Unified parametric approaches for high-order integral observer design for matrix second-order linear systems

A type of high-order integral observers for matrix second-order linear systems is proposed on the basis of generalized eigenstructure assignment via unified parametric approaches. Through establishing two general parametric solutions to this type of generalized matrix second-order Sylvester matrix equations, two unified complete parametric methods for the proposed observer design problem are presented. Both methods give simple complete parametric expressions for the observer gain matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the fight factorization of the system, and allows eigenvalues of the error system to be set undetermined and sought via certain optimization procedures. A spring-mass-dashpot system is utilized to illustrate the design procedure and show the effect of the proposed approach.

High-Order Supervised Discriminant Analysis for Visual Data

In practical applications, we often have to deal with high-order data, for example, a grayscale image and a video clip are intrinsically a 2nd-order tensor and a 3rd-order tensor, respectively. In order to satisty these high-order data, it is conventional to vectorize these data in advance, which often destroys the intrinsic structures of the data and includes the curse of dimensionality. For this reason, we consider the problem of high-order data representation and classification, and propose a tensor based fisher discriminant analysis （FDA）, which is a generalized version of FDA, named as GFDA. Experimental results show our GFDA outperforms the existing methods, such as the 2-directional 2-dimensional principal component analysis （（2D）2pCA）, 2-directional 2-dimensional linear discriminant analysis （（2D）2LDA）, and multilinear discriminant analysis （MDA）, in high-order data classification under a lower compression ratio.

Decompositions of Some Special Block Tridiagonal Matrices

In this paper, we present a unified approach to decomposing a special class of block tridiagonal matrices K (α ,β ) into block diagonal matrices using similarity transformations. The matrices K (α ,β )∈ Rpq× pq are of the form K (α ,β = block-tridiag[β B,A,α B] for three special pairs of (α ,β ): K (1,1), K (1,2) and K (2,2) , where the matrices A and B, A, B∈ Rp× q , are general square matrices. The decomposed block diagonal matrices (α ,β ) for the three cases are all of the form: (α ,β ) = D1 (α ,β ) ⊕ D2 (α ,β ) ⊕---⊕ Dq (α ,β ) , where Dk (α ,β ) = A+ 2cos ( θk (α ,β )) B, in which θk (α ,β ) , k = 1,2, --- q , depend on the values of α and β. Our decomposition method is closely related to the classical fast Poisson solver using Fourier analysis. Unlike the fast Poisson solver, our approach decomposes K (α ,β ) into q diagonal blocks, instead of p blocks. Furthermore, our proposed approach does not require matrices A and B to be symmetric and commute, and employs only the eigenvectors of the tridiagonal matrix T (α ,β ) = tridiag[β b, a,αb] in a block form, where a and b are scalars. The transformation matrices, their inverses, and the explicit form of the decomposed block diagonal matrices are derived in this paper. Numerical examples and experiments are also presented to demonstrate the validity and usefulness of the approach. Due to the decoupled nature of the decomposed matrices, this approach lends itself to parallel and distributed computations for solving both linear systems and eigenvalue problems using multiprocessors.

Hamiltonian Polynomial Eigenvalue Problems

We present in this paper a new method for solving polynomial eigenvalue problem. We give methods that decompose a skew-Hamiltonian matrix using Cholesky like-decomposition. We transform first the polynomial eigenvalue problem to an equivalent skew-Hamiltonian/Hamiltonian pencil. This process is known as linearization. Decomposition of the skew-Hamiltonian matrix is the fundamental step to convert a structured polynomial eigenvalue problem into a standard Hamiltonian eigenproblem. Numerical examples are given.

ON SMOOTH LU DECOMPOSITIONS WITH APPLICATIONS TO SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS

We study the smooth LU decomposition of a given analytic functional A-matrix A（A） and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of A（A）, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.

可对角化矩阵特征值分解扰动问题的快速求解方法

针对特征值扰动计算的传统方法收敛速度慢的问题,提出了一种求解特征值扰动问题的快速迭代算法.首先,通过矩阵变换将初始矩阵的特征值扰动问题转化为对角矩阵的特征值扰动问题.然后,提出了一种快速迭代算法求解扰动参数,同时对算法的收敛性进行分析,并将其与基于摄动级数展开法导出的方法进行对比.再次,采用逐一求解特征值并进行矩阵降阶的策略,有效降低运算量.最后,通过2个算例分别展示算法的计算过程及其在结构模态参数追踪方面的应用效果.

基于复杂度追踪的模态参数识别方法对比研究

复杂度追踪(complexity pursuit, CP)是求解振动信号盲源分离(blind source separation, BSS)问题的一类经典方法。用复杂度追踪估计解混矩阵主要有基于源信号复杂度计算的梯度下降(complexity pursuit-gradient descent, CP-GD)算法和基于时间可预测度的广义特征值分解(temporal predictability-generalized eigenvalue decomposition, TP-GED)算法。当前,这两种算法的关联性与算法性能尚缺乏研究,因此对这两种算法的等价性和计算性能进行了研究。首先,给出CP-GD和TP-GED两种算法的具体理论及算法流程;其次,利用二、三自由度振动系统直观地展示并对比解混向量对应的源信号复杂度及可预测度的变化规律;最后,通过对多工况下多自由度系统的模态参数识别算例,对比研究两种算法的精度及计算量。研究结果表明:在低阻尼比及高信噪比条件下,两种方法得到的解混矩阵是相同的;考虑到计算信号复杂度和梯度下降较为耗时,CP-GD算法计算代价要高于TP-GED算法。

矩阵特征值和特征向量在微分方程求解中的应用

文章借助数学物理方程中线性微分方程的分解定理,将高等数学中的n阶线性齐次微分方程与线性代数中的矩阵建立了联系,得出了利用矩阵的特征值与特征向量求解线性微分方程的通解,并且通过实例加以验证。