Extended convergence regions for the AOR method

In this paper, we give sufficient conditions for the convergence of the (AOR) method, when the matrix A for Ax = b is a strictly diagonally dominant matrix. These results improve the conclusions obtained in the Theorem 4 [10].With the notion of generalized diagonal dominant matrix, we enlarge the convergence regions given in Theorem 9 [10], when A is a nonsingular H-matrix.In the last section we generalize theorem 6 of Robert [11] and we present some results which extend the convergence regions for the (AOR) method.     

A new Type of Matrix Splitting and its Applications

A possible type of the matrix splitting is introduced. Using this matrix splitting, we introduce a few properties and representations of generalized inverses as well as iterative methods for computing various solutions of singu- lar linear systems. This matrix splitting is a generalization of the known index splitting from [13] and a proper splitting from [4]. Using a generalization of the condition number and introduced representations of generalized inverses, we ob- tain several norm estimates.     

Inner Generalized Inverses with Prescribed Idempotents

We define and characterize inner generalized inverses with prescribed idempotents. These classes of generalized inverses are natural algebraic extension of generalized inverses of linear operators with prescribed range and kernel. We consider the reverse order rule for inner generalized inverses of elements of a ring, some perturbation bounds, and we construct an iterative method for computing a (p, q)-inner inverse in Banach algebras.     

求加权广义逆的矩阵方法

本文给出了应用矩阵方法求矩阵Ａ的加权广义逆Ａ＾［１，２，Ｗ３］，Ａ＾［１，２，ｗ３］和Ａ＾［１，２］［ｗ，ｗ］的充要条件。     

分支数≤7时S^2NS有向图的禁用子图刻划

实方阵A称为强符号非异阵（S~2NS阵）,若任一与A符号模式相同的矩阵非异且其逆的符号模式也一致。若一个有向图是某一S~2NS阵对应赋号有向图的基础有向图,称为S~2NS有向图。本文用禁用子图形式给出了分支数≤7时有向图成为S~2NS有向图的刻划,同时部分地解决了[2]和[4]中提出的问题。     

A numerical method for the servo constraint problem of underactuated mechanical systems

Servo constraints are used in inverse dynamics simulations of discrete mechanical systems, especially for trajectory tracking control problems [1], whose desired outputs are represented by state variables and treated as servo constraints [2]. Servo constraint problems can be classified into fully actuated and underactuated multibody systems, and the equations of motion take the form of differential algebraic equations (DAEs) including holonomic and servo constraints. For fully actuated systems, control inputs can be solved from the equations by model inversion, as the input distribution matrix is nonsingular and invertible. However, underactuated systems have more degrees of freedom than control inputs. The input distribution matrix is not invertible, and in contrast to passive constraints, the realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Therefore, it is challenging for the determination of control inputs which force the underactuated system to realize the partly specified motion. For differentially flat underactuated systems, the differentiation index of DAEs may exceed three. Hence we need to apply specific index reduction techniques, such as the projection approach applied in [3], [4], and [6]. The present work applies index reduction by minimal extension [5] to differentially flat underactuated crane systems and shows that the index can be reduced from five to three and even to one. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universal Iterative Methods for Computing Generalized Inverses

In this paper we construct a few iterative processes for computing {1,2} inverses of a linear bounded operator, based on the hyper-power iterative method or the Neumann-type expansion. Under suitable conditions these methods converge to the {1,2,3} or {1,2,4} inverses. Also, we specify conditions when the iterative processes converge to the Moore-Penrose inverse, the weighted Moore-Penrose inverse or to the group inverse. A few error estimates are derived. The advantages of the introduced methods over Tanabe's method [16] for computing reflexive generalized inverses are also investigated.     

约束矩阵方程的解的表示及行列式公式

本文利用Ｂｏｔｔ－Ｄｕｆｆｉｎ逆导出约束矩阵方程的唯一解或通解的表示及其较为简洁的行列式公式，并由之给出了某些特殊矩阵方程与约束性方程组的解以及几个重要广义逆矩阵的表示及其较是洁的行列式公式，我们的结果改进与推广了文（２－１１，１４，１５）中的有关结果。     

Perturbation analysis for oblique projection generalized inverses of closed linear operators in Banach spaces

In this paper, we study the perturbation problem for oblique projection generalized inverses of closed linear operators in Banach spaces. By the method of the perturbation analysis of linear operators, we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the T-bounded perturbation, which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in Banach spaces.     

Linear congruence relations for 2-adic L-series at integers

In the paper we find a further generalization of congruences of the K. Hardy and K. S. Williams [5] type which seems to be a full generalization of congruences of G. Gras [4]. Moreover we extend results of [5], [7], [8], [9] and in part of [6]. We apply ideas and methods of [2], [7] and [9].     

长方矩阵常见广义逆的代数扰动

本文详细讨论了长方矩阵常见广义逆的代数扰动理论,并给出了他们代数扰动的表达式,改进了文献3,4的相应结论.     

Differentiation of generalized inverses for rational and polynomial matrices

Our basic motivation is a direct method for computing the gradient of the pseudo-inverse of well-conditioned system with respect to a scalar, proposed in [13] by Layton. In the present paper we combine the Layton’s method together with the representation of the Moore-Penrose inverse of one-variable polynomial matrix from [24] and developed an algorithm for computing the gradient of the Moore-Penrose inverse for one-variable polynomial matrix. Moreover, using the representation of various types of pseudo-inverses from [26], based on the Grevile’s partitioning method, we derive more general algorithms for computing {1}, {1, 3} and {1, 4} inverses of one-variable rational and polynomial matrices. Introduced algorithms are implemented in the programming language MATHEMATICA. Illustrative examples on analytical matrices are presented.     

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore–Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore–Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore–Penrose metric generalized inverse of bounded linear operators in Banach spaces.     

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

广义严格对角占优矩阵新的判定准则

In this paper we provide the new criteria for a strictly generalized diagonally dominant matrix, and it proves by an example that the results of this paper extend the results in[6]. In addition, we obtain the criteria of the nonsingular M-matrix.     

A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper,we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturbation analysis of bounded linear operators,we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.     

关于带W权Drazin逆的表示定理及迭代方法

To study singular linear system, Cline and Greville[8] proposed the concept of W-weighted Drazin inverse for the rectangular matrices,where the properties were also discussed. The computation for the W-weighted Drazin inverse is of much interest, which is mainly divided into two kinds of methods: direct method[2,4,6] and iterative method[3,5,7,9,12,13]. In this paper, we study the iterative method and successive matrix squaring(SMS) method for the W-weighted Drazin inverse and generalize the main results in [12,13].     

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].     

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…     

Characterization of linear stationary iterative processes for solving a singular system of linear equations

There are many iterative methods for solving a consistent singular system of linear equations [2, 3, 5, 6, 8, 10–13]. In this paper we characterize the class of all singular system of linear equations using generalized inverses.