The Continuous Analogy of Newton’s Method for Solving a System of Linear Algebraic Equations

We propose a continuous analogy of Newton’s method with inner iteration for solving a system of linear algebraic equations. Implementation of inner iterations is carried out in two ways. The former is to fix the number of inner iterations in advance. The latter is to use the inexact Newton method for solution of the linear system of equations that arises at each stage of outer iterations. We give some new choices of iteration parameter and of forcing term, that ensure the convergence of iterations. The performance and efficiency of the proposed iteration is illustrated by numerical examples that represent a wide range of typical systems.

THE SOLUTION FOR THE GENERALIZED RICCATIALGEBRAIC EQUATIONS OF LINEAR EQUALITY CONSTRAINT SYSTEM

Based on the dynamic equation, the performance functional and the system constraint equation of time-invariant discrete LQ control problem, the generalized Riccati equations of linear equality constraint system are obtained according to the minimum principle, then a deep discussion about the above equations is given, and finally numerical example is shown in this paper.

The Application of Linear Ordinary Differential Equations

In this article, we will explore the applications of linear ordinary differential equations (linear ODEs) in Physics and other branches of mathematics, and dig into the matrix method for solving linear ODEs. Although linear ODEs have a comparatively easy form, they are effective in solving certain physical and geometrical problems. We will begin by introducing fundamental knowledge in Linear Algebra and proving the existence and uniqueness of solution for ODEs. Then, we will concentrate on finding the solutions for ODEs and introducing the matrix method for solving linear ODEs. Eventually, we will apply the conclusions we’ve gathered from the previous parts into solving problems concerning Physics and differential curves. The matrix method is of great importance in doing higher dimensional computations, as it allows multiple variables to be calculated at the same time, thus reducing the complexity.

Cryptanalysis of Cryptosystems Based on General Linear Group

Advances in quantum computers threaten to break public key cryptosystems such as RSA, ECC, and EIGamal on the hardness of factoring or taking a discrete logarithm, while no quantum algorithms are found to solve certain mathematical problems on non-commutative algebraic structures until now. In this background, Majid Khan et al.proposed two novel public-key encryption schemes based on large abelian subgroup of general linear group over a residue ring. In this paper we show that the two schemes are not secure. We present that they are vulnerable to a structural attack and that, it only requires polynomial time complexity to retrieve the message from associated public keys respectively. Then we conduct a detailed analysis on attack methods and show corresponding algorithmic description and efficiency analysis respectively. After that, we propose an improvement assisted to enhance Majid Khan's scheme. In addition, we discuss possible lines of future work.

On the Solutions of the Matrix Equations in Optimal Stochastic Control

In this paper, the matrix algebraic equations involved in the optimal control problem of time-invariant linear Ito stochastic systems, named Riccati- Ito equations in the paper, are investigated. The necessary and sufficient condition for the existence of positive definite solutions of the Riccati- Ito equations is obtained and an iterative solution to the Riccati- Ito equations is also given in the paper thus a complete solution to the basic problem of optimal control of time-invariant linear Ito stochastic systems is then obtained. An example is given at the end of the paper to illustrate the application of the result of the paper.

Perturbation Analysis of Continuous-Time Linear Time-Invariant Systems

In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.

Integral Operator Solving Process of the Boundary Value Problem of Abstract Kinetic Equation with the First Kind of Critical Parameter and Generalized Periodic Boundary Conditions

In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.

On the Measurement of Lower Solution Bounds of the Discrete Algebraic Lyapunov Equation

In this paper, estimations of the lower solution bounds for the discrete algebraic Lyapunov Equation (the DALE) are addressed. By utilizing linear algebraic techniques, several new lower solution bounds of the DALE are presented. We also propose numerical algorithms to develop sharper solution bounds. The obtained bounds can give a supplement to those appeared in the literature.

Robust H_∞ Control for Uncertain Markovian Jump Linear Time-Delay Systems

This paper studies the robust stochastic stabilization and robust H∞ control for linear time-delay systems with both Markovian jump parameters and unknown norm-bounded parameter uncertainties. This problem can be solved on the basis of stochastic Lyapunov approach and linear matrix inequality (LMI) technique. Sufficient conditions for the existence of stochastic stabilization and robust H∞ state feedback controller are presented in terms of a set of solutions of coupled LMIs. Finally, a numerical example is included to demonstrate the practicability of the proposed methods.

Design of Continuous and Discrete LQI Control Systems with Stable Inner Loops

Design approaches were proposed for both continuous and discrete LQI (linear quadratic with integral) controllers, if exist, to stabilize the inner loops in addition to stabilizing the closed loops and minimizing the quadratic cost functionals. Derived from the algebraic Riccati equations involved in continuous and discrete LQI control, the design approaches were straightforward obtained by setting the quadratic cost functionals to decoupled forms with positive definite weighting matrices. Examples were provided to verify the effectiveness of the approaches.

Exact Solution of a Linear Difference Equation in a Finite Number of Steps

An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.

On the Deepest Fallacy in the History of Mathematics: The Denial of the Postulate about the Approximation Nature of a Simple-Iteration Method and Iterative Derivation of Cramer’s Formulas

Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonstrated. We present a theorem and its proof that confirms the possibility to obtain the finite process and imposes the requirement for the matrix of SLAE. This matrix must be unipotent, i.e. all its eigenvalues to be equal to 1. An example of transformation of SLAE given analytically to the form with a unipotent matrix is presented. It is shown that splitting the unipotent matrix into identity and nilpotent ones results in Cramer’s analytical formulas in a finite number of iterations.

Algebraic Cryptanalysis of GOST Encryption Algorithm

This paper observes approaches to algebraic analysis of GOST 28147-89 encryption algorithm (also known as simply GOST), which is the basis of most secure information systems in Russia. The general idea of algebraic analysis is based on the representation of initial encryption algorithm as a system of multivariate quadratic equations, which define relations between a secret key and a cipher text. Extended linearization method is evaluated as a method for solving the nonlinear sys- tem of equations.

Gauge Formulation of Heaviside’s Equations

A primordial field Self-interaction Principle, analyzed in Hestenes’ Geometric Calculus, leads to Heaviside’s equations of the gravitomagnetic field. When derived from Einstein’s nonlinear field equations Heaviside’s “linearized” equations are known as the “weak field approximation”. When derived from the primordial field equation, there is no mention of field strength;the assumption that the primordial field was predominant at the big bang rather suggests that ultra-strong fields are governed by the equations. This aspect has physical significance, so we explore the assumption by formulating the gauge field version of Heaviside’s theory. We compare with recent linearized gravity formulations and discuss the significance of differences.

Use Elementary Column Operations to Calculate the Basis of the Null Space of a Matrix

This paper gives and proofs a theorem, for any matrix A, do elementary column operations, change it to a matrix which is partitioned to two blocks which left one is column full rank and right one is zero matrix. That is, use a invertible matrix P to let AP = （B,O）, O is zero matrix with n-r columns, r and n is rank and column number of A, so the P＇s right n-r columns is just the basis of the null space of the matrix A. On the basis of the theorem, lots of problems of linear algebra can be resolved and lots of theorems can be proofed by elementary column operations. Perhaps the textbooks used in universities will have a lot of change with the result of the paper. This result is first found by author in 2010.12.8 in http：//www.paper.edu.cn/index.php/default/releasepaper/content/201012-232, but is not formal published.

基于Mathematica的控制系统根轨迹探究性实验

为了使学生了解根轨迹法的思想并掌握其绘制与分析方法,设计了该实验教学方案。该方案从方程求根问题引入根轨迹概念,结合方程求根问题的发展历史,让学生领会求根问题对数系扩充和群论诞生等数学理论发展的推动作用。该方案设计了三种根轨迹绘制实验方法,利用Mathematica软件开发了根轨迹实验程序,能够通过交互式操作判断复平面上的点是否在根轨迹上。实验过程能够引导学生总结、发现根轨迹的绘制法则,并根据实验步骤掌握系统稳态和动态特性的分析方法。

ANALYSING THE EFFICIENCY OF SOLVING DENSE LINEAR EQUATIONS ON DWNING1000

In this paper, we consider solving dense linear equations on Dawning1000 byusing matrix partitioning technique. Based on this partitioning of matrix, we give aparallel block LU decomposition method. The efficiency of solving linear equationsby different ways is analysed. The numerical results are given on Dawning1000.By running our parallel program, the best speed up on 32 processors is over 25.

基于LU分解的安全外包求解线性代数方程组方法

由于现有协议的安全性为基于某种安全假设的计算安全,依赖于敌手的计算能力,因此,本文针对恶意敌手模型,使用矩阵伪装技术对方程的系数矩阵进行隐藏,结合矩阵的LU分解(lower-upper decomposition)算法,提出一种新的信息论安全外包求解线性代数方程组(information-theoretically secure outsourcing of linear algebraic equations,ITS-OutsLAE)方法 .与之前的研究相比,在保持计算和通信复杂度与现有最优方案保持一致的同时,首次将方程组唯一解的安全性提升至信息论安全(完美保密).给出了形式化的安全性证明,并通过理论分析和实验证明了所提方法的实用性.

从线性代数的角度看复变函数

文中从线性代数的角度详细解释了复数的合理性,并用向量值函数给出了复函数在一点可微充要条件的证明.

Boubaker多项式配置法求解混合线性积分方程

使用Boubaker多项式配置法求解混合线性积分方程的近似解。以Boubaker多项式为基函数,对未知函数用Boubaker多项式进行线性表示,将积分方程转化为一组线性代数方程,求解该方程组,得到积分方程的近似解。算例验证表明Boubaker多项式配置法求解混合线性积分方程具有可行性和有效性。该方法可以延展到求解二维的线性Volterra-Fredholm积分方程中。