Differential equations over subalgebras of the full matrix algebra

We consider a general matrix differential equation whose parameters and solution are defined over subalgebras of the full matrix algebra. Some subalgebras are presented over which the equation splits into a set of equations over matrices of smaller orders. Multiplicative matrix equations of higher order in normal form are introduced and studied over some subalgebras of 2 × 2 matrices.     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

A note on the iterative solutions of general coupled matrix equation

Recently, Ding and Chen [F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim. 44 (2006) 2269-2284] developed a gradient-based iterative method for solving a class of coupled Sylvester matrix equations. The basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified, so that the iterative solutions are obtained by applying hierarchical identification principle. In this note, by considering the coupled Sylvester matrix equation as a linear operator equation we give a natural way to derive this algorithm. We also propose some faster algorithms and present some numerical results.     

On spectral decompositions of solutions to discrete Lyapunov equations

A new approach to solving discrete Lyapunov matrix algebraic equations is based on methods for spectral decomposition of their solutions. Assuming that all eigenvalues of the matrices on the left-hand side of the equation lie inside the unit disk, it is shown that the matrix of the solution to the equation can be calculated as a finite sum of matrix bilinear quadratic forms made up by products of Faddeev matrices obtained by decomposing the resolvents of the matrices of the Lyapunov equation. For a linear autonomous stochastic discrete dynamic system, analytical expressions are obtained for the decomposition of the asymptotic variance matrix of system’s states.     

Finite iterative solutions to coupled Sylvester-conjugate matrix equations

This paper is concerned with solutions to the so-called coupled Sylveter-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases. An iterative algorithm is constructed to solve this kind of matrix equations. By using the proposed algorithm, the existence of a solution to a coupled Sylvester-conjugate matrix equation can be determined automatically. When the considered matrix equation is consistent, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require to transform the coefficient matrices into any canonical forms. The algorithm is also generalized to solve a more general case. Two numerical examples are given to illustrate the effectiveness of the proposed methods.     

Numerical Solution of the Discrete BHH-Equation in the Normal Case

It is known that the solution of the semilinear matrix equation \(X - A\overline X B = C\) can be reduced to solving the classical Stein equation. The normal case means that the coefficients on the left-hand side of the resulting equation are normal matrices. We propose a method for solving the original semilinear equation in the normal case that permits to almost halve the execution time for equations of order n = 3000 compared to the library function dlyap, which solves Stein equations in Matlab.     

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.     

四元数矩阵的实表示与四元数矩阵方程

四元数矩阵与四元数矩阵方程在力学和工程问题的理论研究和实际数值计算中都起到重要的作用.该文借助四元数矩阵的实表示方法,研究了一般四元数矩阵方程AXB-CYD=E的解的问题,给出了一种求解四元数矩阵方程的算法技巧.该文还得到了四元数矩阵的Roth's定理.     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.     

On the Routh-Hurwitz-Fujiwara and the Schur-Cohn-Fujiwara theorems for the root-separation problem

In this paper, we give simple and elementary proofs of the two classical results of Fujiwara on the solution of the well-known Routh-Hurwitz and Schur-Cohn problems. We show that the Fujiwara matrix in each case satisfies a Lyapunov-type equation and then obtain Fujiwara's results by applying to this matrix equation some recent results on the inertia of matrices. These alternative proofs of Fujiwara's results thus establish a link between two apparently different approaches to the solution of the root-separation problem: the classical method of solution via quadratic forms, and the solution via matrix equations.     

Iterative solutions to the Kalman-Yakubovich-conjugate matrix equation

Two operations are introduced for complex matrices. In terms of these two operations an infinite series expression is obtained for the unique solution of the Kalman-Yakubovich-conjugate matrix equation. Based on the obtained explicit solution, some iterative algorithms are given for solving this class of matrix equations. Convergence properties of the proposed algorithms are also analyzed by using some properties of the proposed operations for complex matrices.     

On a relaxed SOR-method applied to nonsymmetric linear systems

Nonsymmetric linear systems are by far not as common as syemmtric ones but nevertheless systems with nonsymmetric matrices appear, e. g., in the numerical solution of the biharmonic equation, the computation of splines or the solution of some special integral equations. The SOR-method applied to linear systems X = BX + C with skew-symmetric matrix B is studied. Described is a region in the complex plane which contains the eigenvalues of the SOR-operator. Using this information a relaxed SOR-method is proposed; bounds for the spectral radius of the iteration operator are derived. The advantage is that the values of the corresponding iteration parameters can be directly calculated from the norm of the given matrix.     

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.     

The polynomial solution to the Sylvester matrix equation

For when the Sylvester matrix equation has a unique solution, this work provides a closed form solution, which is expressed as a polynomial of known matrices. In the case of non-uniqueness, the solution set of the Sylvester matrix equation is a subset of that of a deduced equation, which is a system of linear algebraic equations.     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

Hermite-广义反Hamilton矩阵的一类反问题

给定矩阵X和B,利用矩阵的广义奇异值分解,得到了矩阵方程X~HAX=B有Hermite-广义反Hamiton解的充分必要条件及有解时解的—般表达式.用S_E表示此矩阵方程的解集合,证明了S_E中存在唯一的矩阵(?),使得(?)与给定矩阵A的差的Frobenius范数最小,并且给出了矩阵(?)的表达式;同时也证明了S_E中存在唯一的矩阵A_o,使得A_o是此矩阵方程的极小Frobenius范数Hermite-广义反Hamilton解,并且给出了矩阵A_o的表达式.     

Lyapunov,Bézout,and Hankel

Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov-type equations of the form X?SXC=>W and X?CXS=W. We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.     

Classification of Discrete Symmetries of Ordinary Differential Equations

A simple method for determining all discrete point symmetries of a given differential equation has been developed recently. The method uses constant matrices that represent inequivalent automorphisms of the Lie algebra spanned by the Lie point symmetry generators. It may be difficult to obtain these matrices if there are three or more independent generators, because the matrix elements are determined by a large system of algebraic equations. This paper contains a classification of the automorphisms that can occur in the calculation of discrete symmetries of scalar ordinary differential equations, up to equivalence under real point transformations. (The results are also applicable to many partial differential equations.) Where these automorphisms can be realized as point transformations, we list all inequivalent realizations. By using this classification as a look-up table, readers can calculate the discrete point symmetries of a given ordinary differential equation with very little effort.     

一类新的矩阵型修正Korteweg-de Vries方程的Riemann-Hilbert方法与精确解北大核心CSCD

在本文中,一类新的矩阵型修正Korteweg-de Vries(简记为mmKdV)方程被首次通过RiemannHilbert方法研究,而且,这一方程可通过选取特殊的势矩阵来降阶为我们熟知的耦合型修正Kortewegde Vries方程.从方程对应的Lax对的谱分析入手,作者成功地建立了方程对应的Riemann-Hilbert问题.在无反射势的特殊条件下,mmKdV方程的精确解可由Riemann-Hilbert问题的解给出.而且,基于特殊势矩阵所对应的特殊对称性,作者可以对原有的孤子解进行分类,从而得到一些有趣的解的现象,比如呼吸孤子、钟形孤子等.     

Generating equations approach for quadratic matrix equations

We show how Van Loan's method for annulling the (2,1) block of skew‐Hamiltonian matrices by symplectic‐orthogonal similarity transformation generalizes to general matrices and provides a numerical algorithm for solving the general quadratic matrix equation: For skew‐Hamiltonian matrices we find their canonical form under a similarity transformation and find the class of all symplectic‐orthogonal similarity transformations for annulling the (2,1) block and simultaneously bringing the (1,1) block to Hessenberg form. We present a structure‐preserving algorithm for the solution of continuous‐time algebraic Riccati equation. Unlike other methods in the literature, the final transformed Hamiltonian matrix is not in Hamiltonian–Schur form. Three applications are presented: (a) for a special system of partial differential equations of second order for a single unknown function, we obtain the matrix of partial derivatives of second order of the unknown function by only algebraic operations and differentiation of functions; (b) for a similar transformation of a complex matrix into a symmetric (and three‐diagonal) one by applying only finite algebraic transformations; and (c) for finite‐step reduction of the eigenvalues–eigenvectors problem of a Hermitian matrix to the eigenvalues– eigenvectors problem of a real symmetric matrix of the same dimension. Copyright © 1999 John Wiley & Sons, Ltd.