线性代数中的线性方程组方法

本文从齐次线性方程组的同解理论、非零解的判定、解空间的维数公式、解空间与系数矩阵行空间的正交性等角度,阐述线性方程组方法在线性代数中的广泛应用．     

Legendre小波函数求解非线性Volterra积分微分方程组的数值解

Legendre小波函数被用于逼近非线性Volterra积分微分方程组的解,方法是基于Legendre小波的性质构建相应的积分算子矩阵,进而将原问题转化为关于未知解系数的线性方程组,通过求解该方程组,即得原问题的数值解.数值结果表明所述方法对于求解此类问题是行之有效的.     

由三个特殊次序向量对构造三对角对称矩阵

讨论利用给定的三个特殊次序向量对构造不可约三对角矩阵、Jacobi矩阵和负Jacobi矩阵的反问题.在求解方法中,将已知的一些关系式等价地转化为线性方程组,利用线性方程组有解的条件,得到了所研究问题有惟一解的充要条件,并给出了数值算法和例子.     

Moore-Penrose广义逆矩阵与线性方程组的解

线性方程组的逆矩阵求解方法只使用于系数矩阵为可逆方阵,对于一般线性方程组可以应用Moore-Penrose广义逆矩阵来研究并表示其通解,本文主要探讨Moore-Penrose广义逆矩阵及一般线性方程组通解和最小范数解.     

关于线性方程组同解的条件

讨论线性方程组同解的条件,得到了两个线性方程组同解的充分必要条件.     

三对角对称矩阵逆特征值问题存在唯一解的条件

讨论了利用给定的k(2≤k≤n)个特征对来构造相应的三对角对称矩阵的问题．在求解方法中,将已知的一些关系式等价转化成线性方程组,利用线性方程组的解存在唯一的条件,得到了所研究问题存在唯一解的充要条件,并给出了计算解的数值方法和数值实例．     

线性方程组的分块矩阵解法

本文利用分块矩阵证明线性方程组解的基本定理，并给出解线性方程组的一种改进方法．     

迭代求解复对称线性方程组的收敛性分析(英文)

本文提出求解系数矩阵不是埃尔米特但是对称复矩阵的线性方程组的一种分裂迭代法,详细讨论新方法的迭代矩阵的谱半径,最优参数选择,一些范数性质.证明在合理的假设下新方法是收敛的.最后以数值结果验证了新方法的有效性和可行性.     

解一类抛物方程反问题的数值方法

本文利用有限元方法建立了求解一类含有低阶未知系数的抛物方程反问题的数值公式,论证了近似解的收敛性和误差阶估计.     

范德蒙型方程组最小二乘解的快速算法

在用多项式进行曲线拟合等实际问题中,需要求解以范德蒙型矩阵VT为系数阵的线性方程组VTx=b的最小二乘解.     

解聯立方程的差分方程法

<正> 一. 引 聯立一次方程的求解早就不是一個理論問題,而是一個改進計算技術的問題.問題在如何組織計算使計算機械化從而節省工作量. 給定充分多始值後,線性差分方程是很容易解的.在本文內,我們把特種的及一般的聯立一次方程組的解看作線性差分方程滿足某種邊值的解,從而推求出求聯立一次方程組的準確解的一種簡單的機械的列表計算方法。     

A numerical approach to solve the model of an electromechanical system

In this paper, the model of an electromechanical system, which is a system of linear differential equations, is studied. Haar wavelet collocation method (HWCM) is applied for finding the approximate solution of the model. HWCM reduces the system of the model into a matrix‐vector form that contains the unknown Haar coefficients, and these coefficients are easily calculated. To demonstrate the validity and applicability of HWCM, numerical solutions of the system for different parameter values in the system are presented. The obtained results demonstrate the efficiency and accuracy of the method. All of the computations are performed via a program written in Mathematica.     

Determination of the source parameter in a heat equation with a non-local boundary condition

In this article we consider the inverse problem of identifying a time dependent unknown coefficient in a parabolic problem subject to initial and non-local boundary conditions along with an overspecified condition defined at a specific point in the spatial domain. Due to the non-local boundary condition, the system of linear equations resulting from the backward Euler approximation have a coefficient matrix that is a quasi-tridiagonal matrix. We consider an efficient method for solving the linear system and the predictor–corrector method for calculating the solution and updating the estimate of the unknown coefficient. Two model problems are solved to demonstrate the performance of the methods.     

常系数非齐次线性微分方程组的初等解法

利用初等变换将常系数非齐次线性微分方程组化为由若干个相互独立的高阶常系数非齐次线性微分方程组成的方程组,再利用高阶常系数齐次线性微分方程的特征根法和非齐次方程的待定系数法求该方程组的基本解组及特解,最后通过初等变换求原方程组的基本解组及特解,从而可求出其通解.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

Solution to fuzzy system of linear equations with crisp coefficients

This paper presents a new and simple method to solve fuzzy real system of linear equations by solving two n × n crisp systems of linear equations. In an original system, the coefficient matrix is considered as real crisp, whereas an unknown variable vector and right hand side vector are considered as fuzzy. The general system is initially solved by adding and subtracting the left and right bounds of the vectors respectively. Then obtained solutions are used to get a final solution of the original system. The proposed method is used to solve five example problems. The results obtained are also compared with the known solutions and found to be in good agreement with them.     

Divergence-free wavelet solution to the Stokes problem

In this paper, we use divergence-free wavelets to give an adaptive solution to the velocity field of the Stokes problem. We first use divergence-free wavelets to discretize the divergence-free weak formulation of the Stokes problem and obtain a discrete positive definite linear system of equations whose coefficient matrix is quasi-sparse; Secondly, an adaptive scheme is used to solve the discrete linear system of equations and the error estimation and complexity analysis are given.     

Construction of an asymptotically periodic solution to one nonstationary system with delay

In this paper we consider an ensemble of two subsystems of linear differential equations with constant delay. One of the subsystems contains an exponential multiplier in its right-hand side. We construct an asymptotically periodic solution of this system.     

Tikhonov solutions of approximately given systems of linear algebraic equations under finite perturbations of their matrices

The properties of a mathematical programming problem that arises in finding a stable (in the sense of Tikhonov) solution to a system of linear algebraic equations with an approximately given augmented coefficient matrix are examined. Conditions are obtained that determine whether this problem can be reduced to the minimization of a smoothing functional or to the minimal matrix correction of the underlying system of linear algebraic equations. A method for constructing (exact or approximately given) model systems of linear algebraic equations with known Tikhonov solutions is described. Sharp lower bounds are derived for the maximal error in the solution of an approximately given system of linear algebraic equations under finite perturbations of its coefficient matrix. Numerical examples are given.     

常系数线性分数阶微分方程组的解

本文研究了常系数线性分数阶微分方程组的求解问题.利用逆Laplace变换,Jordan标准矩阵和最小多项式,得到矩阵变量Mittag-Leffler函数的三种不同的计算方法,包含了常系数线性一阶微分方程组的解.