关于解线性系统的一个有效的贝叶斯迭代方法

This paper concerns with the statistical methods for solving general linear systems. After a brief review of Bayesian perspective for inverse problems,a new and efficient iterative method for general linear systems from a Bayesian perspective is proposed.The convergence of this iterative method is proved,and the corresponding error analysis is studied.Finally, numerical experiments are given to support the efficiency of this iterative method,and some conclusions are obtained.     

A NOTE ON THE DRAZIN INVERSE OF A MODIFIED MATRIX

In this article,the expression for the Drazin inverse of a modified matrix is considered and some interesting results are established.This contributes to certain recent results obtained by Y.Wei [9].     

OPERATOR AND MATRIX REPRESENTATION FOR THE GENERALIZED INVERSE A_(T,S)~(2)

. IntroductionIn this paper we adopt the same notations on generalized inverses of matrices andprojectors as those in [1].Several rePresentations for the generalized inverses of matrices, for example, those forthe Moors-Penrose inverse A and the Drazin inverse A(d), have led to[1--4]:where k = index (A).In order to study the matrix form of the above-mentioned operator represelltations,we denote linear operators by A, B,' ', and their matrices by the corresponding A, B,' '.All of the li…     

A hybrid method for inverse cavity scattering problem for shape

In this paper, we give a hybrid method to numerically solve the inverse open cavity scattering problem for cavity shape, given the scattered solution on the opening of the cavity. This method is a hybrid between an iterative method and an integral equations method for solving the Cauchy problem. The idea of this hybrid method is simple, the operation is easy, and the computation cost is small. Numerical experiments show the feasibility of this method, even for cases with noise.     

关于Drazin逆的刻划

Let ■ be a pre-additive category. Assume that ψ:X→X is a morphism of (?). In this paper, we give the necessary and sufficient conditions for ψ to have the Drazin inverse by using the von Neumann regular inverse for the ψk, and extend a result by Puystjens and Hartwig from the group inverse to Drazin inverse.     

The Core-EP,Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

We study the constrained systemof linear equations Ax=b,x∈R(A^k)for A∈C^n×nand b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^Db.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_x∈R(A^k)||b?Ax||2,where||·||2 is the 2-norm.For the inconsistent system with a matrix A of index one,it was proved recently that the solution is A^■b using the core inverse A^■of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A^■b where A^■is the core-EP inverse of A.We establish two Cramer’s rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer’s rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.     

MATRIX ALGORITHMS AND ERROR FORMULA FOR BIVARIATE THIELE-TYPE RECTANGULAR MATRIX VALUED RATIONAL INTERPOLATION

A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.     

Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.     

AN ITERATIVE BEM FOR THE INVERSE PROBLEM OF DETECTING CORROSION IN A PIPE

In this paper, we consider an inverse problem of determining the corrosion occurring in an inaccessible interior part of a pipe from the measurements on the outer boundary. The problem is modelled by Laplace's equation with an unknowm term γ in the boundary condition on the inner boundary. Based on the Maz'ya iterative algorithm, a regularized BEM method is proposed for obtaining approximate solutions for this inverse problem. The numerical results show that our method can be easily realized and is quite effective.     

New perturbation bounds for the W-weighted Drazin inverse

The constructive perturbation bounds for the W-weighted Drazin inverse are derived by two approaches in this paper. One uses the matrixG = [(A+E)W]^l?(AW)^l, whereA, E ∈ C ^mxn ,W ∈ C ^nxm ,l = max Ind(AW), Ind[(A + E)W]. The other uses a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces of a matrix. The new approaches to develop perturbation bounds for W-weighted Drazin inverse of a matrix extend the previous results in [19, 29, 31, 36, 42, 44]. Several examples which indicate the sharpness of the new perturbation bounds are presented.     

Hilbert空间上算子W-加权Drazin逆的刻画及表示

该文研究了Hilbert空间上线性算子的W-加权Drazin逆,利用算子的分块矩阵表示,给出了W-加权Drazin逆的刻画及表示,所获结果推广了魏益民等的相关结果.     

The representation and perturbation of theW-weighted Drazin inverse

LetA andE bem x n matrices andW an n xm matrix, and letA _d,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseA _d,W and B_d,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥B_d,W∥ and ∥B_{d, W}, -A_d,W∥/∥A_d,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

Further results on the Drazin inverse of even‐order tensors

The notion of the Drazin inverse of an even‐order tensors with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402‐3413]. In this article, we further elaborate this theory by establishing a few characterizations of the Drazin inverse and $\mathrm{??}$‐weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors using different types of generalized inverses and full rank decomposition of tensors. We also address the solution of multilinear systems by using the Drazin inverse and iterative (higher order Gauss‐Seidel) method of tensors. Besides these, the convergence analysis of the iterative technique is also investigated within the framework of the Einstein product.     

带W权Drazin逆的一种新算法

利用矩阵A的带W权Drazin逆的一个性质特征,对任意的矩阵A∈Cm×n,W∈Cn×m,建立了带W权的Drazin逆Ad,w的一种新的表示式,给出了具体的算法步骤,并且在文末给出了算例.     

广义逆A_(T,S)~(2)的子式

１．引 言 设Ａ∈Ｃｍ×ｎ,Ｍ和Ｎ分别为ｍ和ｎ阶Ｈｅｒｍｉｔｅ正定阵,考虑下列方程 （１） ＡＸＡ＝Ａ （２） ＸＡＸ＝Ｘ （３）（ＡＸ）＊＝ＡＸ （４）（ＸＡ）＊＝ＸＡ （３Ｍ）（ＭＡＸ）＊＝ＭＡＸ （４Ｎ）（ＮＸＡ）＊＝ＮＸＡ 如果Ｘ∈Ｃｎ×ｍ满足条件（１）和（２）,则称Ｘ为Ａ的自反广义逆,记作Ｘ＝Ａ（１,２）;如果 Ｘ满足条件（２）,则称Ｘ为 Ａ的｛２｝逆,记作 Ｘ＝Ａ（２）;如果Ｘ满足（１）－（４）,则称Ｘ为 Ａ的 Ｍ－Ｐ逆,记作Ｘ＝Ａ＋;如果Ｘ满足（１）、（２）、（３Ｍ）、（４Ｎ）,则称 Ｘ为 Ａ的加权…     

广义逆A_(T,S)~(2)的子式

The explicit expression for the generalized inverse A_(T,S)~2 in [6] is utilized in presenting the minors of the generalized inverse A_(T,S)~(2). Thus, without calculating M-P inverse, weighted M-P inverse, group inverse and Drazin inverse, we are able to find the minors of them. The main results are also the generalization of the results proposed by [5] and [8].     

The Algebraic Perturbation Method for Generalized Inverses

Algebraic perturbation methods were first proposed for the solution of nonsingular linear systems by R. E. Lynch and T. J. Aird [2]. Since then, the algebraic perturbation methods for generalized inverses have been discussed by many scholars [3]-[6]. In [4], a singular square matrix was perturbed algebraically to obtain a nonsingular matrix, resulting in the algebraic perturbation method for the Moore-Penrose generalized inverse. In [5], some results on the relations between nonsingular perturbations and generalized inverses of $m\times n$ matrices were obtained, which generalized the results in [4]. For the Drazin generalized inverse, the author has derived an algebraic perturbation method in [6]. In this paper, we will discuss the algebraic perturbation method for generalized inverses with prescribed range and null space, which generalizes the results in [5] and [6]. We remark that the algebraic perturbation methods for generalized inverses are quite useful. The applications can be found in [5] and [8]. In this paper, we use the same terms and notations as in [1].     

修正矩阵A-CB的Drazin逆(英文)

本文研究了修正矩阵Drazin逆的表示形式.利用k次幂等矩阵和可对角化矩阵的性质,减弱了文献[4]中的条件,获得了新的Drazin逆的表示形式.