A COCR method for solving complex symmetric linear systems

The Conjugate Orthogonal Conjugate Gradient (COCG) method has been recognized as an attractive Lanczos-type Krylov subspace method for solving complex symmetric linear systems; however, it sometimes shows irregular convergence behavior in practical applications. In the present paper, we propose a Conjugate A$A$-Orthogonal Conjugate Residual (COCR) method, which can be regarded as an extension of the Conjugate Residual (CR) method. Numerical examples show that COCR often gives smoother convergence behavior than COCG.     

环Z/p~kZ上的两类矩阵方程

利用环上矩阵方法,研究了环Z/pkZ上的两类矩阵方程,确定了这两类矩阵方程的解数.     

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

We prove an invariant Harnack’s inequality for operators in non-divergence form structured on Heisenberg vector fields when the coe?cient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli to obtain Harnack’s inequality.     

A Formula for the Exponential of a Real Skew-Symmetric Matrix of Order 4

In this short paper the formula of the exponential matrix e ^A when A is a kew-symmetric real matrix of order 4 is derived. The formula is a generalization of the well known Rodrigues formula for skew-symmetric matrices of order 3.This revised version was published online in October 2005 with corrections to the Cover Date.     

Linear recurrence relations in the algebra of matrices and applications

We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for Aⁿ (nr) and e^tA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices A^j, where 0jr−1.     

Optimal accumulation of Jacobian matrices by elimination methods on the dual computational graph

The accumulation of the Jacobian matrix F of a vector function can be regarded as a transformation of its linearized computational graph into a subgraph of the directed complete bipartite graph K_n,m. This transformation can be performed by applying different elimination techniques that may lead to varying costs for computing F. This paper introduces face elimination as the basic technique for accumulating Jacobian matrices by using a minimal number of arithmetic operations. Its superiority over both edge and vertex elimination methods is shown. The intention is to establish the conceptual basis for the ongoing development of algorithms for optimizing the computation of Jacobian matrices.     

利用有限域上2-幂零矩阵构作Cartesian认证码

给出 2 -幂零矩阵的Jordan标准型在相似变换下的稳定群的结构 ,利用有限域上 2 -幂零矩阵构作了一个Cartesian认证码 ,计算出了该认证码的参数 .在假定编码规则按等概率分布选取的条件下 ,计算出了该认证码被成功的模仿攻击的最大概率PI 及被成功的替换攻击的最大概率PS.     

Nonhomogeneous markov chains with desired properties - the nearest markovian model

A new class of operators performing an optimization (optimization operators or, simply, optimators) which generate transition matrices with required properties such as ergodicity, recurrence etc., is considered and their fundamental features are described. Some criteria for comparing such operators by taking into account their strenght are given and sufficient conditions for both weak and strong ergodicity are derived. The nearest Markovian model with respect to a given set of observed probability vectors is then defined as a sequence of transition matrices satisfying certain constraints that express our prior knowledge about the system. Finally, sufficient conditions for the existence of such a model are given and the related algorithm is illustrated by an example.     

Weighted FOM and GMRES for solving nonsymmetric linear systems

This paper presents two new methods called WFOM and WGMRES, which are variants of FOM and GMRES, for solving large and sparse nonsymmetric linear systems. To accelerate the convergence, these new methods use a different inner product instead of the Euclidean one. Furthermore, at each restart, a different inner product is chosen. The weighted Arnoldi process is introduced for implementing these methods. After describing the weighted methods, we give the relations that link them to FOM and GMRES. Experimental results are presented to show the good performances of the new methods compared to FOM(m) and GMRES(m). This revised version was published online in August 2006 with corrections to the Cover Date.