ON THE APPROXIMATE COMPUTATION OF EXTREME EIGENVALUES AND THE CONDITION NUMBER OF NONSINGULAR MATRICES

From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.

Generation of Complex Matrices Belonging to Matrix Groups with Controlled Condition Number

The authors study the generation of matrices with complex entries belonging to some matrix groups, mainly those that are defined by a scalar product space. These matrices have useful applications in quantum mechanical problems and complex control problems. In this work, the authors try to generate matrices such that： （1） the condition number of these types of matrices is controlled and （2） The algorithm used to generate these matrices preserves their structure.

COMPONENTWISE CONDITION NUMBERS FOR GENERALIZED MATRIX INVERSION AND LINEAR LEAST SQUARES

We present componentwise condition numbers for the problems of MoorePenrose generalized matrix inversion and linear least squares. Also, the condition numbers for these condition numbers are given.

On the Preconditioning Properties of RHSS Preconditioner for Saddle-Point Linear Systems

In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.

Application of uncertain type of AHP to condition assessment of cable-stayed bridges

In order to guarantee the safety service and life-span of long-span cable-stayed bridges, the uncertain type of analytic hierarchy process （AHP） method is adopted to access the bridge condition. The correlative theory and applied objects of uncertain type of AHP are introduced, and then the optimal transitive matrix method is chosen to calculate the interval number judgment matrix, which makes the weights of indices more reliable and accurate. Finally, with Harbin Songhua River Cable-Stayed Bridge as an example, an index system and an assessment model are proposed for the condition assessment of this bridge, and by using uncertain type of AHP, the weights of assessment indices are fixed and the final assessment results of the bridge are calculated, which proves the feasibility and practicability of this method. The application of this assessment method can provide the scientific basis for maintenance and management of long-span cable-stayed bridges.

Geometric Condition of 3UPS-S Parallel Mechanism in Singular Configuration

The existing researches on singularity of parallel mechanism are mostly limited to the property and regularity of singularity locus and there is no further research into the geometric relationship between uncontrolled kinematic screw and parallel mechanism in singularity. A 3UPS-S parallel mechanism is presented which fulfils 3-DOF in rotation. The regularity of nutation angle singularity is analyzed based on the Jacobian matrix, and the singularity surface of 3UPS-S parallel mechanisms is obtained. By applying the concept of reciprocal product in screw theory, the singular kinematic screw is derived when 3UPS-S parallel mechanism is in singularity. The geometric relationship between singular kinematic screw and singular configuration of 3UPS-S parallel mechanism is investigated by using programs in MATLAB. It is revealed that there are two kinds of situation. Firstly, the three limbs of 3UPS-S parallel mechanism intersect the singular kinematic screw in space simultaneously; Secondly, two limbs cross the singular kinematic screw while the third limb parallels with that screw. It is concluded that the nutation angle singularity of 3UPS-S parallel mechanism belongs to the singular linear complexes. This paper sheds light into and clarifies the geometric relationship between singular kinematic screw and singular configuration of 3UPS-S parallel mechanism.

THE PROPERTIES OF A KIND OF RANDOM SYMPLECTIC MATRICES

Several important properties of a kind of random symplectic matrix used by A. Bunse-Gerstner and V. Mehrmann are studied and the following results are obtained: 1) It can be transformed to Jordan canonical form by orthogonal similar transformation; 2) Its condition number is a constant; 3) The condition number of it is about 2.618.

The Significance of Solutions Obtained from Ill-Posed Systems of Linear Equations Constituted by Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering

Synchrotron radiation based experimental techniques known as Anomalous Small-Angle X-ray Scattering (ASAXS) provide deep insight into the nanostructure of uncountable material systems in condensed matter research i.e. solid state physics, chemistry, engineering and life sciences thereby rendering the origin of the macroscopic functionalization of the various materials via correlation to its structural architecture on a nanometer length scale. The techniques constitute a system of linear equations, which can be treated by matrix theory. The study aims to analyze the significance of the solutions of the stated matrix equations by use of the so-called condition numbers first introduced by A. Turing, J. von Neumann and H. Goldstine. Special attention was given for the comparison with direct methods i.e. the Gaussian elimination method. The mathematical roots of ill-posed ASAXS equations preventing matrix inversion have been identified. In the framework of the theory of von Neumann and Goldstine the inversion of certain matrices constituted by ASAXS gradually becomes impossible caused by non-definiteness. In Turing’s theory which starts from more general prerequisites, the principal minors of the same matrices approach singularity thereby imposing large errors on inversion. In conclusion both theories recommend for extremely ill-posed ASAXS problems avoiding inversion and the use of direct methods for instance Gaussian elimination.

Perturbation Analysis for the Matrix-Scaled Total Least Squares Problem

In this paper, we extend matrix scaled total least squares (MSTLS) problem with a single right-hand side to the case of multiple right-hand sides. Firstly, under some mild conditions, this paper gives an explicit expression of the minimum norm solution of MSTLS problem with multiple right-hand sides. Then, we present the Kronecker-product-based formulae for the normwise, mixed and componentwise condition numbers of the MSTLS problem. For easy estimation, we also exhibit Kronecker-product-free upper bounds for these condition numbers. All these results can reduce to those of the total least squares (TLS) problem which were given by Zheng et al. Finally, two numerical experiments are performed to illustrate our results.

Force/Moment Isotropy of 8/4-4 Parallel Six-Axis Force Sensor Based on Performance Atlases

A six-axis force sensor with parallel 8/4-4 structure is introduced and its measurement principle is analyzed.Based on condition numbers of Jacobian matrix spectral norm of the sensor,the relationship between the force and moment isotropy and some structural parameters is deduced.Orthogonal test methods are used to determine the degree of primary and secondary factors that have significant effect on sensor characteristics.Furthermore,the relationship between each performance index and the structural parameters of the sensor is analyzed by the method of the atlas,which lays a foundation for structural optimization design of the force sensor.

Stability of GM(1,1) power model on vector transformation

The morbidity problem of the GM（1,1） power model in parameter identification is discussed by using multiple and rotation transformation of vectors. Firstly we consider the morbidity problem of the special matrix and prove that the condition number of the coefficient matrix is determined by the ratio of lengths and the included angle of the column vector, which could be adjusted by multiple and rotation transformation to turn the matrix to a well-conditioned one. Then partition the corresponding matrix of the GM（1,1） power model in accordance with the column vector and regulate the matrix to a well-conditioned one by multiple and rotation transformation of vectors, which completely solve the instability problem of the GM（1,1） power model. Numerical results show that vector transformation is a new method in studying the stability problem of the GM（1,1） power model.

Novel high-performance element in the electromagnetic finite-element method——node-edge element

It is known in the computational electromagnetics （CEM） that the node element has a relative wellconditioned matrix, but suffers from the spurious solution problem; whereas the edge element has no spurious solutions, but usually produces an ill-conditioned matrix. Particularly, when the mesh is over dense, the iterative solution of the matrix equation from edge element converges very slowly. Based on the node element and edge element, a node-edge element is presented, which has no spurious solutions and better-conditioned matrix. Numerical experiments demonstrate that the proposed node-edge element is more efficient than now-widely used edge element.

Capacity analysis of high-rank line-of-sight MIMO channels

Matrix perturbation theory is utilized to investigate high-rank line of sight multiple input multiple output channels in a microwave relay system. The upper and lower bounds of channel capacity are derived based on space time block codes technique and singular values decomposition. A useful constraint for designing LOS MIMO channels is developed by the use of the condition number of the MIMO channel matrix. The theoretical analysis of channel capacity is confirmed by the simulation. The results show that the proposed method is able to give a physical explanation of the high-rank LOS MIMO channel matrix characteristics.

DOMAIN DECOMPOSITION PRECONDITIONERS FOR SECOND-ORDER HYPERBOLIC EQUATIONS ON L-SHAPED REGIONS

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations on L-shaped region are considered. We analyse the use of domain deocmposilion preconditioner.s for the solution of linear systems via the preconditioned conjugate gradient method. For the constant-coefficient second-order hyperbolic equaions with initial and Dirichlet boundary conditions,we prove that the conditionnumber of the preconditioned interface system is bounded by 2+x2 2+0.46x2 where x is the quo-tient between the lime and space steps. Such condition number produces a convergence rale that is independent of gridsize and aspect ratios. The results could be extended to parabolic equations.

考虑激发充分性的机械臂动力学最小参数集辨识

在考虑粗大误差和随机误差情况下,为了提高机械臂系统动力学参数辨识的精确性和鲁棒性,文章提出了基于鲁棒粒子群算法的参数辨识方法,建立了机械臂动力学模型;筛选出了动力学最小参数集,用于消除动力学参数间的冗余性。为了实现参数充分激励,基于有穷傅里叶级数构造了激励轨迹,并以最小化矩阵条件数为目标实现了轨迹优化。为了提高参数辨识性能,在粒子群算法中引入了“向其他个体学习”的新型学习方式;在适应度函数设计中加入了粗大误差剔除功能,有效提高了算法的抗干扰能力。经机械臂系统加噪实验验证,鲁棒粒子群算法优化的驱动力矩残差均值为1.1975N·m,比量子粒子群算法减少了75.29%;力矩实际值与计算值相关度为0.9593,比量子粒子群算法提高了11.73%。实验结果表明,鲁棒粒子群算法在机械臂动力学参数辨识中具有更高的精度和更强的鲁棒性。

一类三阶耗散微分算子边界条件的分类

本文对导致三阶微分算子为耗散的边界条件进行了分类,为了得到这些分类,将边界条件分为两种类型讨论.对于耦合型边界条件即rank(A)=rank(B)=3时,得到了两种使三阶微分算子为耗散的边界条件;对于混合型边界条件即rank(A)=rank(B)=2时,得到了十二种使三阶微分算子为耗散的边界条件.

核燃料导热方程数值求解算法研究

针对核燃料子通道分析方法中导热方程矩阵维度较大的问题,引入多种三对角矩阵数值求解方案,研究不同矩阵维度下数值求解方法的适用性。通过推导核燃料导热方程,对方程的特点进行分析,引入条件数判别法对矩阵的病态程度进行量化,使用列选主元三角分解法、超松弛迭代法和共轭梯度法对不同维度大型稀疏试算矩阵进行求解,比对各数值求解方法的残差平方和,并在一定精度下比对迭代求解方法的收敛步数。在综合考虑计算精确度和速度后,给出不同条件下核燃料导热方程的数值求解方案。

ON MATRIX UNITARILY INVARIANT NORM CONDITION NUMBER

In this paper, the unitarily invariant norm \\.\\ on C-mxn is used. We first discuss the problem under what case, a rectangular matrix A has minimum condition number K(A) = \\A\\ \\A(+)\\, where A(+) designates the Moore-Penrose inverse of A; and under what condition, a square matrix A has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of K(A) = \\A\\ \\A(-1)\\(2) in error estimation.

一种改进的光流算法

光流法是运动图像序列分析的一种重要方法。该文通过引入前向-后向光流方程,计算其Hessian矩阵,把Hessian矩阵条件数的倒数作为Lucas-Kanade光流法的加权阵,可有效消除局部邻域中不可靠约束点,同时提高基本约束方程解的稳定性。实验表明该方法相对于其它梯度约束光流法具有更好的可靠性。

用矩阵条件数求解相移最佳采样方式

应用相移干涉术测量物体相位时 ,相移量与相移帧数存在着最佳取值问题 ,即最佳采样方式 ,它决定着系统抗扰动和降低误差的能力 用系数矩阵的条件数衡量相位测量误差矩阵与强度探测误差矩阵方程的性态 ,即光强方程组对扰动的敏感程度 ,经数学推导及论证 ,得出结论 :当系数矩阵为正交矩阵时 ,相位计算的误差最小 ,或者说在等间距满周期采样情况下 ,具有最强的抗噪音能力 在推导出最佳采样方式后 ,对系数矩阵的条件数与采样方式之间的关系进行了计算机模拟 。