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.     

Spectral analysis for a three-dimensional superradiance problem

In this paper three-dimensional superradiance problem is considered and a differential operator that commutes with the integral operator related to the problem is obtained. All the eigenfunctions of the differential operator are found and a complete set of eigensolutions for the three-dimensional superradiance problem is obtained.     

The calculation of the distance to a nearby defective matrix

The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill‐conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter‐dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

A filtering method for the interval eigenvalue problem

We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices.     

An inversion formula for a matrix polynomial about a (unit) root

A solution to the problem of a closed-form representation for the inverse of a matrix polynomial about a unit root is provided by resorting to a Laurent expansion in matrix notation, whose principal-part coefficients turn out to depend on the non-null derivatives of the adjoint and the determinant of the matrix polynomial at the root. Some basic relationships between principal-part structure and rank properties of algebraic function of the matrix polynomial at the unit root as well as informative closed-form expressions for the leading coefficient matrices of the matrix-polynomial inverse are established.     

A computational matrix method for solving systems of high order fractional differential equations

In this paper, we introduced an accurate computational matrix method for solving systems of high order fractional differential equations. The proposed method is based on the derived relation between the Chebyshev coefficient matrix A of the truncated Chebyshev solution u(t)$\mathbf{u}(\mathbf{t})$ and the Chebyshev coefficient matrix A^(ν)${\mathbf{A}}^{(\nu )}$ of the fractional derivative u^(ν)${\mathbf{u}}^{(\nu )}$. The fractional derivatives are presented in terms of Caputo sense. The matrix method for the approximate solution for the systems of high order fractional differential equations (FDEs) in terms of Chebyshev collocation points is presented. The systems of FDEs and their conditions (initial or boundary) are transformed to matrix equations, which corresponds to system of algebraic equations with unknown Chebyshev coefficients. The remaining set of algebraic equations is solved numerically to yield the Chebyshev coefficients. Several numerical examples for real problems are provided to confirm the accuracy and effectiveness of the present method.     

Approximate inverses of multidiagonal matrices and application to the block PCG method

A fast method to compute high-order approximate inverses based on truncated elimination is constructed for multidiagonal matrices of diagonal dominance. Together with the block preconditioned conjugate gradient method, it can be used for the numerical solution of elliptic partial differential equations and related problems. Through numerical experiments it is shown that the method is robust and efficient.     

Fast sensitivity analysis of defective system

Sensitivity analysis of linear vibration system is of wide interest. In this paper, sensitivity analysis based on non-defective system and defective system is summarized in all cases. Specially, for the defective systems, a fast method for the perturbation problem of state vectors is constructed in terms of the theories of generalized eigenvectors and adjoint matrices. By this method, the state vector derivatives can be expressed by a linear combination of generalized eigenvectors. The expansion coefficients can be obtained without solving large-scale equations based on eigensolutions of original system. Numerical results demonstrate the effectiveness and the stability of the method.     

Mapped barycentric Chebyshev differentiation matrix method for the solution of regular Sturm-Liouville problems

In this paper, using spectral differentiation matrix and an elimination treatment of boundary conditions, Sturm-Liouville problems (SLPs) are discretized into standard matrix eigenvalue problems. The eigenvalues of the original Sturm-Liouville operator are approximated by the eigenvalues of the corresponding Chebyshev differentiation matrix (CDM). This greatly improves the efficiency of the classical Chebyshev collocation method for SLPs, where a determinant or a generalized matrix eigenvalue problem has to be computed. Furthermore, the state-of-the-art spectral method, which incorporates the barycentric rational interpolation with a conformal map, is used to solve regular SLPs. A much more accurate mapped barycentric Chebyshev differentiation matrix (MBCDM) is obtained to approximate the Sturm-Liouville operator. Compared with many other existing methods, the MBCDM method achieves higher accuracy and efficiency, i.e., it produces fewer outliers. When a large number of eigenvalues need to be computed, the MBCDM method is very competitive. Hundreds of eigenvalues up to more than ten digits accuracy can be computed in several seconds on a personal computer.     

小波算子矩阵法求定积分的近似值

结合Haar小波和算子矩阵的思想,给出一种新的Haar小波积分算子矩阵.利用所得小波积分算子矩阵来求定积分的近似值,将求定积分的问题转化为算子矩阵相乘,从而更容易计算机求解.特别是对于无法求得原函数的定积分,采用本文方法可以有效的求其近似值.最后数值算例验证了方法的可行性和有效性.     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

Aggregating nonnegative eigenvectors of the adjacency matrix as a measure of centrality for a directed graph

Eigenvector centrality is a popular measure that uses the principal eigenvector of the adjacency matrix to distinguish importance of nodes in a graph. To find the principal eigenvector, the power method iterating from a random initial vector is often adopted. In this article, we consider the adjacency matrix of a directed graph and choose suitable initial vectors according to strongly connected components of the graph instead so that nonnegative eigenvectors, including the principal one, can be found. Consequently, for aggregating nonnegative eigenvectors, we propose a weighted measure of centrality, called the aggregated-eigenvector centrality. Weighting each nonnegative eigenvector by the reachability of the associated strongly connected component, we can obtain a measure that follows a status hierarchy in a directed graph.     

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton-Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X+A^∗X^-1A=I, where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA^∗. We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.     

On the jackknife statistics for eigenvalues and eigenvectors of a correlation matrix

This paper deals with some problems of eigenvalues and eigenvectors of a sample correlation matrix and derives the limiting distributions of their jackknife statistics with some numerical examples.     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

Approximate inverse-free preconditioners for Toeplitz matrices

In this paper, we propose approximate inverse-free preconditioners for solving Toeplitz systems. The preconditioners are constructed based on the famous Gohberg-Semencul formula. We show that if a Toeplitz matrix is generated by a positive bounded function and its entries enjoys the off-diagonal decay property, then the eigenvalues of the preconditioned matrix are clustered around one. Experimental results show that the proposed preconditioners are superior to other existing preconditioners in the literature.