矩阵线性方程X=EXD+F的一种新算法

在离散控制系统设计中，经常遇到解矩阵方程X＝EXD＋F的问题，例如，CADSS软件包就是如此，本文给出了此方程的一种新算法，该方法适应范围广，数值稳定性高。

《线性代数》学习中三次方程求根方法探讨

在线性代数学习中,学生经常会遇到方程求根问题,如系数矩阵为方阵且含有参数的线性方程组解的判别,方阵的特征值的计算等均以方程的求根为基础.矩阵阶数越高对应方程的次数就越高,而高次方程的求解一直是个难点.本文以三元线性方程组解的判别和三阶方阵求特征值的问题为出发点总结归纳了三次方程的几种求根方法.

Applications of Lodrigues Matrix in 3D Coordinate Transformation

Three transformation models （Bursa-Wolf, Molodensky, and WTUSM） are generally used between two data systems transformation. The linear models are used when the rotation angles are small; however, when the rotation angles get bigger, model errors will be produced. In this paper, we present a method with three main terms：① the traditional rotation angles θ,φ,ψ are substituted with a,b,c which are three respective values in the anti-symmetrical or Lodrigues matrix; ② directly and accurately calculating the formula of seven parameters in any value of rotation angles; and ③ a corresponding adjustment model is established. This method does not use the triangle function. Instead it uses addition, subtraction, multiplication and division, and the complexity of the equation is reduced, making the calculation easy and quick.

Design of decentralized robust H_∞ state feedback controller

The design of decentralized robust H_∞ state feedback controller for large-scale interconnected systems with value bounded uncertainties existing in the state, control input and interconnected matrices was investigated. Based on the bounded real lemma a sufficient condition for the existence of a decentralized robust H_∞ state feedback controller was derived. This condition is expressed as the feasibility problem of a certain nonlinear matrix inequality. The controller, which makes the closed-loop large-scale system robust stable and satisfies the given H_∞ performance, is obtained by the offered homotopy iterative linear matrix inequality method. A numerical example is given to demonstrate the effectiveness of the proposed method.

Some new bound estimates of the Hermitian positive definite solutions of the nonlinear matrix equation X^s+ A~*X^(-t) A = Q

The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.

Computation of Jordan Chains by Generalized Characteristic Matrices

In this paper, we introduce a method to define generalized characteristic matrices of a defective matrix by the common form of Jordan chains. The generalized characteristic matrices can be obtained by solving a system of linear equations and they can be used to compute Jordan basis.

On a Matrix Equation AX+XB=C over a Skew Field

In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skew field,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).A convenient,simple and practical method of solving(I)is also given.As a particular case,we also give a simple method of finding a system of fundamental solutions of a homogeneous system of right linear equations over a skew field.

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p^([N ]), q^([N ]), r^([N ])and s^([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.

Loop Algebras and Bi-integrable Couplings

A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.

Variational algorithms for linear algebra

Quantum algorithms have been developed for efficiently solving linear algebra tasks.However,they generally require deep circuits and hence universal fault-tolerant quantum computers.In this work,we propose variational algorithms for linear algebra tasks that are compatible with noisy intermediate-scale quantum devices.We show that the solutions of linear systems of equations and matrix–vector multiplications can be translated as the ground states of the constructed Hamiltonians.Based on the variational quantum algorithms,we introduce Hamiltonian morphing together with an adaptive ans?tz for efficiently finding the ground state,and show the solution verification.Our algorithms are especially suitable for linear algebra problems with sparse matrices,and have wide applications in machine learning and optimisation problems.The algorithm for matrix multiplications can be also used for Hamiltonian simulation and open system simulation.We evaluate the cost and effectiveness of our algorithm through numerical simulations for solving linear systems of equations.We implement the algorithm on the IBM quantum cloud device with a high solution fidelity of 99.95%.

An orthogonally accumulated projection method for symmetric linear system of equations

A direct as well as iterative method(called the orthogonally accumulated projection method, or the OAP for short) for solving linear system of equations with symmetric coefficient matrix is introduced in this paper. With the Lanczos process the OAP creates a sequence of mutually orthogonal vectors, on the basis of which the projections of the unknown vectors are easily obtained, and thus the approximations to the unknown vectors can be simply constructed by a combination of these projections. This method is an application of the accumulated projection technique proposed recently by the authors of this paper, and can be regarded as a match of conjugate gradient method(CG) in its nature since both the CG and the OAP can be regarded as iterative methods, too. Unlike the CG method which can be only used to solve linear systems with symmetric positive definite coefficient matrices, the OAP can be used to handle systems with indefinite symmetric matrices. Unlike classical Krylov subspace methods which usually ignore the issue of loss of orthogonality, OAP uses an effective approach to detect the loss of orthogonality and a restart strategy is used to handle the loss of orthogonality.Numerical experiments are presented to demonstrate the efficiency of the OAP.