用神经网络计算矩阵特征值与特征向量

该文研究用神经网格求解一般实对称矩阵的全部特征向量的问题。详细讨论了网络的平均态度合的结构并建立了平衡态集合的构造定理。通过求解简单的一维微分方程求出了网络的解析表达式。这一表达式是由对称矩阵的特征值与特征向量表达的、因而非常清晰利用解的解析表达式分析了网络的解的全局渐近行为。提出了用一些单位向量作为网络初始值计算对称矩阵的全部特征值与特征向量的具体算法。     

自洽场方法中广义本征值方程求解及其C++程序设计

本文研究自洽场方法中广义本征值方程求解的算法,并设计相应的C 程序来实现该算法。首先对重叠矩阵进行分解,并将广义本征值方程化为标准的本征值方程,再利用Householder变换将上一步变换所得的矩阵化为对称三对角矩阵,进而用QL方法求解这个三对角矩阵的本征值和本征矢量,从而得到自洽场方法中广义本征值方程的本征值和本征矢量。     

Finite iterative algorithms for the generalized Sylvester-conjugate matrix equation {AX+BY=E\overline{X}F+S}

This paper investigates the generalized Sylvester-conjugate matrix equation, which includes the normal Sylvester-conjugate, Kalman–Yakubovich-conjugate and generalized Sylvester matrix equations as its special cases. An iterative algorithm is presented for solving such a kind of matrix equations. This iterative method can give an exact solution within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it is implemented by original coefficient matrices. By specifying the proposed algorithm, iterative algorithms for some special matrix equations are also developed. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

On correction equations and domain decomposition for computing invariant subspaces

By considering the eigenvalue problem as a system of nonlinear equations, it is possible to develop a number of solution schemes which are related to the Newton iteration. For example, to compute eigenvalues and eigenvectors of an n × n matrix A, the Davidson and the Jacobi-Davidson techniques, construct ‘good’ basis vectors by approximately solving a “correction equation” which provides a correction to be added to the current approximation of the sought eigenvector. That equation is a linear system with the residual r of the approximated eigenvector as right-hand side.One of the goals of this paper is to extend this general technique to the “block” situation, i.e., the case where a set of p approximate eigenpairs is available, in which case the residual r becomes an n × p matrix. As will be seen, solving the correction equation in block form requires solving a Sylvester system of equations. The paper will define two algorithms based on this approach. For symmetric real matrices, the first algorithm converges quadratically and the second cubically. A second goal of the paper is to consider the class of substructuring methods such as the component mode synthesis (CMS) and the automatic multi-level substructuring (AMLS) methods, and to view them from the angle of the block correction equation. In particular this viewpoint allows us to define an iterative version of well-known one-level substructuring algorithms (CMS or one-level AMLS). Experiments are reported to illustrate the convergence behavior of these methods.     

Dampening controllers via a Riccati equation approach

An algorithm is presented which computes a state feedback for a standard linear system which not only stabilizes, but also dampens the closed-loop system dynamics. In other words, a feedback gain matrix is computed such that the eigenvalues of the closed-loop state matrix are within the region of the left half-plane where the magnitude of the real part of each eigenvalue is greater than that of the imaginary part, This may be accomplished by solving a damped algebraic Riccati equation and a degenerate Riccati equation. The solution to these equations are computed using numerically robust algorithms, Damped Riccati equations are unusual in that they may be formulated as an invariant subspace problem of a related periodic Hamiltonian system. This periodic Hamiltonian system induces two damped Riccati equations: one with a symmetric solution and another with a skew symmetric solution. These two solutions result in two different state feedbacks, both of which dampen the system dynamics, but produce different closed-loop eigenvalues, thus giving the controller designer greater freedom in choosing a desired feedback     

Construction of square root factor for solution of the Lyapunov matrix equation

The Lyapunov matrix equation is considered in this paper, where the solution is a nonnegative definite matrix, i.e. a matrix admitting decomposition in square root factors. An algorithm for findings the square root factor without preliminary finding the solution itself is given.     

Solving inverse eigenvalue problems by a projected Newton method

The inverse eigenvalue problem for symmetric matrices (IEP) can be formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. An iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method is proved to convergence locally quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so-called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here.     

分布时滞系统的反馈镇定

求解特征矩阵是镇定时滞系统的关键问题,本文给出了系统的特征根的代数重复度与几 何重复度均为一般值情况下特征矩阵的求法,即把它归结为求解一组线性代数方程的问题,并 得到了该方程组有解及对应于同一特征值的解向量组线性独立的充分条件.此外,还提出了 一种算法,用以处理系统对应于不同特征值的左特征向量线性相关情况下系统的镇定问题.     

求解矩阵特征值和特征向量的PSO算法

提出一种基于粒子群优化算法的求解方法,将线性方程组的求解转化为无约束优化问题加以解决,采用粒子群优化算法求解矩阵特征值和特征向量。仿真实验结果表明,该方法求解精度高、收敛速度快,能够在10代左右收敛,可以有效获得任意矩阵的特征值和特征向量。     

A guided genetic algorithm for diagonalization of symmetric and Hermitian matrices

The eigenvalues and eigenvectors of a matrix have many applications in engineering and science, such us studying and solving structural problems in both the treatment of signal or image processing, and the study of quantum mechanics. One of the most important aspects of an algorithm is the speed of execution, especially when it is used in large arrays. For this reason, in this paper the authors propose a new methodology using a genetic algorithm to compute all the eigenvectors and eigenvalues in real symmetric and Hermitian matrices. The algorithm uses a general-purpose library developed by the authors for genetic algorithms (GALGA). The speed of execution and the influence of population size have been studied. Moreover, the algorithm has been tested in different matrices and population sizes by comparing the speed of execution to the number of the eigenvectors. This new methodology is faster than the previous algorithm developed by the authors and all eigenvectors can be obtained with it. In addition, the performance using the Coope matrix has been tested contrasting the results with another technique published in the scientific literature.     

A novel finite element method for two-point boundary value problems

A finite element method is presented for solving boundary value problems for ordinary differential equations in which the general solution of the differential equation is computed first, followed by a selection procedure for the particular solution of the boundary value problem from the general solution. In this method, the discrete representation of the differential equation is a singular matrix equation, which is solved by using generalized matrix inversion. The technique is applied to both linear and nonlinear boundary value problems and to boundary value problems requiring eigenvalue evaluation. The solution of several examples involving different types of two-point boundary value problems is presented.     

求解矩阵特征值及特征向量的新方法

提出一种基于进化策略求解矩阵特征值及特征向量的新方法。该方法在进化过程中通过重组、突变、选择对个体进行训练学习,向最优解逼近。当达到预先给定的误差时,程序终止,得到最优解。实验结果表明,与传统方法相比,该方法的收敛速度较快,求解精度提高了10倍。该算法能够快速有效地获得任意矩阵对应的特征值及特征向量。     

Solution of progressively changing equilibrium equations for nonlinear structures

In the analysis of nonlinear structures by tangent stiffness methods, the equilibrium equations change progressively during the analysis. When direct methods of solving these equations are used, it may be possible to re-use a substantial part of the previously reduced coefficient matrix, and hence substantially reduce the equation solving effort. This paper examines procedures for re-solving equations when only selected parts of the reduced matrix need to be modified.The Crout and Cholesky algorithms are first reviewed for initial complete reduction and subsequent selective reduction of the coefficient matrix, and it is shown that the Cholesky algorithm is superior for selective reduction. A general procedure is then presented for identifying those parts of the coefficient matrix which remain unchanged as the structure changes. Finally, a general purpose in-core equation solver is presented, in which those parts of the previously reduced matrix which need to be modified are determined automatically during the solution process, and only these portions are changed. The equation solver is based on the Cholesky algorithm, and is applicable to both positive-definite and well conditioned non-positive-definite symmetrical systems of equations.     

Control theoretic analysis of a target search problem by a team of search vehicles

This article considers a target search problem by a set of unmanned aerial vehicles (UAVs). The problem is modelled as a discrete state, continuous-time Markov process. Convergence properties are investigated by using the eigenvalues and eigenvectors of a state transition rate matrix without explicitly solving differential equations or calculating matrix exponentials. The paper also studies the effect of cueing on convergence rate using eigenvalues analysis and optimal control theoretic perspective.     

一种计算矩阵特征值特征向量的神经网络方法

当把Oja学习规则描述的连续型全反馈神经网络(Oja-N)用于求解矩阵特征值特征向量时,网络初始向量需位于单位超球面上,这给应用带来不便.由此,提出一种求解矩阵特征值特征向量的神经网络(1yNN)方法.在lyNN解析解基础上得到了以下结果:初始向量属于任意特征值对应特征向量张成的子空间,则网络平衡向量也将属于该空间;分析了lyNN收敛于矩阵最大特征值对应特征向量的初始向量取值条件;明确了lyNN收敛于矩阵不同特征值的特征子空间时,网络初始向量的最大取值空间;网络初始向量与已知特征向量垂直,则lyNN平衡解向量将垂直于该特征向量;证明了平衡解向量位于由非零初始向量确定的超球面上的结论.基于以上分析,设计了用lyNN求矩阵特征值特征向量的具体算法,实例演算验证了该算法的有效性.1yNN不出现有限溢,而基于Oja-N的方法在矩阵负定、初始向量位于单位超球面外时必出现有限溢,算法失效.与基于优化的方法相比,lyNN实现容易,计算量较小.     

Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL)

In this article, we apply the method of lines (MOL) for solving the time-fractional diffusion equations (TFDEs). The use of MOL yields a system of fractional differential equations with the initial value. The solution of this system could be obtained in the form of Mittag–Leffler matrix function. A direct method which computes the Mittag–Leffler matrix by applying its eigenvalues and eigenvectors analytically has been discussed. The direct approach has been applied on one-, two-, and three-dimensional TFDEs with Dirichlet, Neumann, and periodic boundary conditions as well.

     

Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution

In this article, the Legendre wavelet operational matrix of integration is used to solve boundary ordinary differential equations with non-analytic solution. Although the standard Galerkin method using Legendre polynomials does not work well for solving ordinary differential equations in which at least one of the coefficient functions or solution function is not analytic, it is shown that the Legendre wavelet Galerkin method is very efficient and suitable for solving this kind of problems. Several numerical examples are given to illustrate the efficiency and performance of the presented method.     

DtiStudio: resource program for diffusion tensor computation and fiber bundle tracking

A versatile resource program was developed for diffusion tensor image (DTI) computation and fiber tracking. The software can read data formats from a variety of MR scanners. Tensor calculation is performed by solving an over-determined linear equation system using least square fitting. Various types of map data, such as tensor elements, eigenvalues, eigenvectors, diffusion anisotropy, diffusion constants, and color-coded orientations can be calculated. The results are visualized interactively in orthogonal views and in three-dimensional mode. Three-dimensional tract reconstruction is based on the Fiber Assignment by Continuous Tracking (FACT) algorithm and a brute-force reconstruction approach. To improve the time and memory efficiency, a rapid algorithm to perform the FACT is adopted. An index matrix for the fiber data is introduced to facilitate various types of fiber bundles selection based on approaches employing multiple regions of interest (ROIs). The program is developed using C++ and OpenGL on a Windows platform.     

Solution of the Lyapunov matrix differential equations by the frequency method

Methods for solving the Lyapunov matrix differential and algebraic equations in the time and frequency domains are considered. The solutions of these equations are finite and infinite Gramians of various forms. A feature of the proposed new approach to the calculation of Gramians is the expansion of the Gramians in a sum of matrix bilinear or quadratic forms that are formed using Faddeev’s matrices, where each form is a solution of the linear differential or algebraic equation corresponding to an eigenvalue of the matrix or to a combination of such eigenvalues. An example illustrating the calculation of finite and infinite Gramians is discussed.