Generalized Runge-Kutta methods for coupled systems of hyperbolic differential equations

Runge-Kutta formulas are discussed for the integration of systems of differential equations. The parameters of these formulas are square matrices with component-dependent values. The systems considered are supposed to originate from hyperbolic partial differential equations, which are coupled in a special way. In this paper the discussion is concentrated on methods for a class of two coupled systems. For these systems first and second order formulas are presented, whose parameters are diagonal matrices. These formulas are further characterized by their low storage requirements, by a reduction of the computational effort per timestep, and by their relatively large stability interval along the imaginary axis. The new methods are compared with stabilized Runge-Kutta methods having scalar-valued parameters. It turns out that a gain factor of 2 can be obtained.     

Asymptotic Formulas for Determinants of a Sum of Finite Toeplitz and Hankel Matrices

The purpose of this paper is to describe asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices with singular generating functions. The formulas are similar to those of the analogous problem for finite Toeplitz matrices for a certain class of symbols. However, the appearance of the Hankel matrices changes the nature of the asymptotics in some instances depending on the location of the singularities. Several concrete examples are also described in the paper.     

Generalized cauchy-vandermonde matrices

Matrices of the form [C V] consisting of a generalized Cauchy matrix and a generalized Vandermonde matrix are considered. Using the displacement structure of these matrices, inversion formulas and criteria are presented. The interpretation of linear systems with such a coefficient matrix as tangential interpolation problems leads to the concept of fundamental matrix, which is basic in this approach. For fundamental matrices recursion formulas are established. From them, fast inversion algorithms emerge that work for arbitrary nonsingular matrices of this kind.     

分块带状矩阵的逆

1引言如果分块矩阵A=(A_(ij))_(n×n)满足A_(ij)=O(j-i＞p且i-j＞q),其中A_(ij)为m阶矩阵,则称A为(p,q)-分块带状矩阵．分块带状矩阵在一些实际问题中经常出现,例如在量子场论中用途很广的非线性Schr(?)dinger方程的差分离散问题,解热传导问题等,都会遇到分块带状矩阵．常见的分块三对角矩阵,分块五对角矩阵都是特殊的分块带状矩阵．采用通常的方法求解分块带状矩阵的逆矩阵时,需要进行O(n~3)次m阶矩阵的运算．本文首先将分块带状矩阵扩充成可逆的分块上(下)三角矩阵,利用其逆矩阵导出了分块带状矩阵的逆矩阵表达式;进而利用所得到的公式分别推导了分块三对角矩阵及分块五对角矩阵的逆矩阵的快速算法,所需运算量为O(n~2)次m阶矩阵的运算．本文的结果扩充了文[1]等关于分块三对角阵求逆的相关结果．     

Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.     

Triangular factors of Cauchy and Vandermonde matrices

Explicit formulas for triangular factors of Cauchy and Vandermonde matrices and their inverses in terms of entries of these matrices are presented.     

Grey dynamic input-output analysis

A system whose information is partially known and partially unknown is named as a grey system and it is general situation in reality. On the basis of grey number/matrix and their covered operation, some definitions of grey functions are enumerated. Combining grey system theory with traditional dynamic input-output analysis, we propose the grey dynamic input-output analysis. It has two types, i.e., the continuous and the disperse. The most important results are the computational formulas of the matrix-covered set of inverse grey investment coefficient matrix. By using them, we can forecast and control the economic system under the uncertain situation. For the calculation of the matrix-covered set of inverse grey matrices, we obtain the desired formulas, which can be applied to general grey matrices. The formulas can greatly promote the development of grey theory and enrich the applied fields of uncertain mathematics. The modified case illustrates our method.     

Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols

The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.     

Displacement structures and fast inversion formulas for confluent polynomial Vandermonde-like matrices

In the present paper, confluent polynomial Vandermonde-like matrices with general recurrence structure are introduced. Three kinds of displacement structure equations and two kinds of fast inversion formulas for this class of matrices are derived by using displacement structure matrix method. A relationship between confluent polynomial Vandermonde-like matrices and confluent Cauchy-like matrices is pointed out.     

基于两类切比雪夫多项式的循环矩阵行列式的计算

利用多项式因式分解的逆变换,结合循环矩阵和切比雪夫多项式的特殊结构,首先研究第三类和第四类切比雪夫多项式的通项公式,并给出第三类、第四类切比雪夫多项式的关于行首加r尾r右循环矩阵和行尾加r首r左循环矩阵的行列式的显式表达式,最后给出算法实施步骤.     

Spectral Density of Jacobi Matrices with Small Deviations

We present several new asymptotic trace formulas for Jacobi matrices whose coefficients satisfy a small deviation condition. Our results extend most of the existing trace formulas for Jacobi matrices.     

Torsion formulas for signed graphs

Following our recent exposition on the algebraic foundations of signed graphs, we introduce bond (circuit) basis matrices for the tension (flow) lattices of signed graphs, and compute the torsions of such matrices and Laplacians. We present closed formulas for the torsions of the incidence matrix, the Laplacian, bond basis matrices, and circuit basis matrices. These formulas show that the torsions of all such matrices are powers of 2, and so imply that the matroids of signed graphs are representable over any field of characteristic not 2. A notable feature of using torsion is that the Matrix-Tree formula for ordinary graphs and Zaslavsky’s formula for unbalanced signed graphs are unified into one Matrix-Basis formula in terms of the torsion of its Laplacian matrix, rather than in terms of its determinant, which vanishes for an ordinary graph unless one row is deleted from the incidence matrix.     

Computing permanents via determinants for some classes of sparse matrices

Starting from recent formulas for calculating the permanents of some sparse circulant matrices, we obtain more general formulas expressing the permanents of a wider class of matrices as a linear combination of appropriate determinants.     

k-途径正则有向图

In this paper, we define a class of strongly connected digraph, called the k-walk- regular digraph, study some properties of it, provide its some algebraic characterization and point out that the 0-walk-regular digraph is the same as the walk-regular digraph discussed by Liu and Lin in 2010 and the D-walk-regular digraph is identical with the weakly distance-regular digraph defined by Comellas et al in 2004.     

Extremal Ranks of Some Nonlinear Matrix Expressions with Applications

In this paper, we establish a group of closed-form formulas for the maximal and minimal ranks of a nonlinear matrix expression with respect to two variant matrices by using a linearization method and some known formulas for extremal ranks of linear matrix expressions. In addition, by using some pure algebraic operations of matrices and their generalized inverses, we derive the maximal and minimal ranks of the above nonlinear matrix expression, where the two variant matrices are any solutions of two consistent matrix equations. As an application, we derive some sufficient and necessary conditions for the existence of the solution of a nonlinear matrix function.     

On certain decompositions of complex inverse Toeplitz matrices and related fast algorithms for solving linear systems with Toeplitz coefficient matrices

Formulas for inverting nonsingular Toeplitz matrices with complex entries are derived. These formulas either refine known ones or are new. They make it possible to develop economical algorithms for calculating products of inverse Toeplitz matrices with vectors.     

Inversion and canonization of block matrices

The paper is concerned with the problem of inverting block matrices to which the well-known Frobenius— Schur formulas are not applicable. These can be square matrices with four noninvertible square or rectangular blocks as well as square or rectangular matrices with two blocks. With regard to rectangular matrices, the results obtained are a further step in the development of the canonization method, which is used for solving arbitrary matrix equations.     

Inverses of -distance matrices of a tree

The determinant and the inverse of the distance matrix of a tree have been investigated in the literature, following the classical formulas due to Graham and Pollak for the determinant, and due to Graham and Lovász for the inverse. We consider two q-analogs of the distance matrix of a tree and obtain formulas for the inverses of the two distance matrices. Yan and Yeh have previously obtained expressions for the determinants of the two distance matrices. Some related results are proved.     

Completely normal matrices

Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well.     

On Singular Values Related to DAEs in Kronecker Canonical Form

The index and the structural properties of differential algebraic equations (DAEs) are often determined by rank considerations of the derivative array. Since the Kronecker canonical form is a well-understood standard form that permits deep insight into the properties of DAEs, in this contribution we undertake an analysis of the singular values of this specific derivative array. To this end, the special structure of the obtained block matrices is pointed out, such that some formulas for the computation and estimation of eigenvalues and singular values can be applied. Actually, we explore the relationship between the spectra of particular block tridiagonal matrices and some perturbed Jacobi matrices.