gradient based iterative algorithm for solving coupled matrix equations abstract this paper is concerned with iterative methods for solving a class of coupled matrix equations including the well known coupled markovian jump lyapunov matrix equations as special cases the proposed method is developed from an optimization point of view and contains the well known jacobi iteration gauss seidel iteration and some recently reported iterative algorithms by using the hierarchical identification principle as special cases we have provided analytically the necessary and sufficient condition for the convergence of the proposed iterative algorithm simultaneously the optimal step size such that the convergence rate of the algorithm is maximized is also established in explicit form the proposed approach requires less computation and is numerically reliable as only matrix manipulation is required some other existing results require either matrix inversion or special matrix products numerical examples show the effectiveness of the proposed algorithm

iterative algorithms for solving a class of complex conjugate and transpose matrix equations this paper is concerned with iterative solutions to a class of complex matrix equations which include some previously investigated matrix equations as special cases by applying the hierarchical identification principle an iterative algorithm is constructed to solve this class of matrix equations a sufficient condition is presented to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix with a real representation of a complex matrix as tools by using some properties of the real representation a convergence condition that is easier to compute is also given in terms of original coefficient matrices a numerical example is employed to illustrate the effectiveness of the proposed methods

closed form solutions to the nonhomogeneous yakubovich conjugate matrix equation some closed form solutions are provided for the nonhomogeneous yakubovich conjugate matrix equation email protected f by r with x and y being unknown matrices the presented solutions can offer all the degrees of freedom which is represented by an arbitrarily chosen parameter matrix the primary feature of the solutions is that the matrices f and r are not restricted to be in any canonical form or may be even unknown a priori one of the solutions is neatly expressed in terms of controllability matrices and observability matrices

gradient based iterative solutions for general linear matrix equations in this paper we present a gradient based iterative algorithm for solving general linear matrix equations by extending the jacobi iteration and by applying the hierarchical identification principle convergence analysis indicates that the iterative solutions always converge fast to the exact solutions for any initial values and small condition numbers of the associated matrices two numerical examples are provided to show that the proposed algorithm is effective

gradient based iterative algorithms for generalized coupled sylvester conjugate matrix equations abstract by applying the hierarchical identification principle the gradient based iterative algorithm is suggested to solve a class of complex matrix equations with the real representation of a complex matrix as a tool the sufficient and necessary conditions for the convergence factor are determined to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices also we solve the problem which is proposed by wu et al 2010 finally some numerical examples are provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

on iterative solutions of general coupled matrix equations in this paper we study coupled matrix equations which are encountered in many systems and control applications first we extend the well known jacobi and gauss seidel iterations and present a large family of iterative methods which are then applied to develop iterative solutions to coupled sylvester matrix equations the basic idea is to regard the unknown matrices to be solved as parameters of a system to be identified and to obtain the iterative solutions by applying a hierarchical identification principle next we generalize the sylvester equations to general coupled matrix equations and propose a gradient based iterative algorithm for the solutions using a block matrix inner product the star star product we prove that the iterative algorithm always converges to the unique solutions for any initial values one advantage of the algorithms proposed is that they require less storage space in implementation than existing numerical methods finally we test the algorithms and show their effectiveness using numerical examples

consimilarity of quaternion matrices and complex matrices this paper discusses consimilarity of quaternion matrices obtains the jordan canonical form of a quaternion matrix under consimilarity and some necessary and sufficient conditions for consimilarity of quaternion matrices the consimilarity of quaternion matrices is a natural extension of the complex consimilarity of complex matrices has many good properties including some that are essentially different from those of complex consimilarity moreover using the consimilarity of quaternion matrices we obtain a new way to discuss complex consimilarity of complex matrices

a new general construction for generalized bent functions a simple yet general method of constructing bent functions is presented for certain applications it is of interest to minimize the size of the image of a bent function in this study a tight lower bound on the number of elements in the image is established under certain conditions

a shifted complex global lanczos method and the quasi minimal residual variant for the stein conjugate matrix equation x ax b c abstract in this paper based on a complex global lanczos process a shifted complex global lanczos method is proposed to solve the stein conjugate matrix equation x a x b c the new method adopts the shifted technique and is implemented by the original coefficient matrices the direct version of the proposed method is derived by exploiting the special structure of the l u factorization moreover the quasi minimal residual variant is developed to stabilize the convergence behavior finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

the mgpbicg method for solving the generalized coupled sylvester conjugate matrix equations in this paper we extend the generalized product type bi conjugate gradient gpbicg method for solving the generalized sylvester conjugate matrix equations a 1 x b 1 c 1 y d 1 s 1 a 2 x b 2 c 2 y d 2 s 2 by the real representation of the complex matrix and the properties of kronecker product and vectorization operator some numerical experiments demonstrate that the introduced iteration approach is efficient

an iterative algorithm for the reflexive solutions of the generalized coupled sylvester matrix equations and its optimal approximation abstract the generalized coupled sylvester matrix equations ay zb cy zd e f with unknown matrices y z are encountered in many systems and control applications also these matrix equations have several applications relating to the problem of computing stable eigendecompositions of matrix pencils in this work we construct an iterative algorithm to solve the generalized coupled sylvester matrix equations over reflexive matrices y z and when the matrix equations are consistent for any initial matrix pair y 0 z 0 a reflexive solution pair can be obtained within finite iteration steps in the absence of roundoff errors and the least frobenius norm reflexive solution pair can be obtained by choosing a special kind of initial matrix pair also we obtain the optimal approximation reflexive solution pair to a given matrix pair y z in the reflexive solution pair set of the generalized coupled sylvester matrix equations ay zb cy zd e f moreover several numerical examples are given to show the efficiency of the presented iterative algorithm

iterative solutions to coupled sylvester conjugate matrix equations this paper is concerned with iterative solutions to the coupled sylvester conjugate matrix equation with a unique solution by applying a hierarchical identification principle an iterative algorithm is established to solve this class of complex matrix equations with a real representation of a complex matrix as a tool a sufficient condition is given to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices in addition a sufficient convergence condition that is easier to compute is also given by the original coefficient matrices finally a numerical example is given to verify the effectiveness of the proposed algorithm

hierarchical gradient based identification of multivariable discrete time systems in this paper we use a hierarchical identification principle to study identification problems for multivariable discrete time systems we propose a hierarchical gradient iterative algorithm and a hierarchical stochastic gradient algorithm and prove that the parameter estimation errors given by the algorithms converge to zero for any initial values under persistent excitation the proposed algorithms can be applied to identification of systems involving non stationary signals and have significant computational advantage over existing identification algorithms finally we test the proposed algorithms by simulation and show their effectiveness

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

gradient based iterative algorithms for solving a class of matrix equations in this note we apply a hierarchical identification principle to study solving the sylvester and lyapunov matrix equations in our approach we regard the unknown matrix to be solved as system parameters to be identified and present a gradient iterative algorithm for solving the equations by minimizing certain criterion functions we prove that the iterative solution consistently converges to the true solution for any initial value and illustrate that the rate of convergence of the iterative solution can be enhanced by choosing the convergence factor or step size appropriately furthermore the iterative method is extended to solve general linear matrix equations the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

on solutions of the matrix equations xf ax c and xf ax c with the help of the concept of kronecker map an explicit solution for the matrix equation xf ax c is established this solution is neatly expressed by a symmetric operator matrix a controllability matrix and an observability matrix in addition the matrix equation xf ax c is also studied an explicit solution for this equation is also proposed by means of the real representation of a complex matrix this solution is neatly expressed by a symmetric operator matrix two controllability matrices and two observability matrices

kronecker maps and sylvester polynomial matrix equations the concepts of kronecker maps and sylvester polynomial matrix equations are proposed some properties of the kronecker maps are deduced it is shown that many kinds of matrix equations can be formulated into the so called sylvester polynomial matrix equations with the help of the kronecker maps general explicit solutions of the sylvester polynomial matrix equations are established when the polynomial matrices possess f primeness for the nonsquare case the proposed solutions can offer all the degrees of freedom represented by a free parameter matrix

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations abstract this paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled sylvester matrix equations j 1 q a i j x j b i j f i i 1 2 p we prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum norm solution

an efficient algorithm for the least squares reflexive solution of the matrix equation a1xb1 c1 a2xb2 c2 abstract a n n real matrix p is said to be a real generalized reflection matrix if pt p and p2 i a n n real matrix x is said to be a reflexive matrix with respect to the generalized reflection matrix p if x pxp in this paper the concept of gradient matrix f x is presented and an algorithm is constructed to solve the reflexive with respect to the generalized reflection matrix p solution of the minimum frobenius norm residual problem a 1 xb 1 a 2 xb 2 c 1 c 2 min by this algorithm for any initial reflexive matrix x1 a solution x can be obtained within finite iteration steps in the absence of roundoff errors and the solution x with least norm can be obtained by choosing a special kind of initial reflexive matrix in addition in the solution set of above problem the unique optimal approximation solution x to a given matrix x0 in frobenius norm can be obtained by finding the least norm reflexive solution x of the new minimum residual problem min a 1 x b 1 a 2 x b 2 c 1 c 2 where c 1 c 1 a 1 x 0 b 1 c 2 c 2 a 2 x 0 b 2 given numerical examples show that the iterative method is quite efficient

convergence of gradient based iterative solution of coupled markovian jump lyapunov equations the solution of coupled discrete time markovian jump lyapunov matrix equations cdmjlmes is important in stability analysis and controller design for markovian jump linear systems this paper presents a simple and effective iterative method to produce numerical solutions to this class of matrix equations the gradient based algorithm is developed from an optimization point of view a necessary and sufficient condition guaranteeing the convergence of the algorithm is established this condition shows that the algorithm always converges provided the cdmjlmes have unique solutions which is evidently different from the existing results that converge conditionally a simple sufficient condition which is easy to test is also provided the optimal step size in the algorithm such that the convergence rate of the algorithm is maximized is given explicitly it turns out that an upper bound of the convergence rate is bounded by a function of the condition number of the augmented coefficient matrix of the cdmjlmes some parameters are introduced to the algorithm that will potentially reduce the condition number and thus increase the convergence rate of the algorithm a numerical example is used to illustrate the efficiency of the proposed approach

on solutions of the matrix equations x axb c and x axb c this paper studies the solutions of complex matrix equations x axb c and x axb c and obtains explicit solutions of the equations by the method of characteristic polynomial and a method of real representation of a complex matrix respectively

synchronizing multiple chaotic maps with a randomized scalar coupling abstract synchronization between chaotic systems has recently been a topic of great interest among physicists and engineers in addition to theoretical results a number of applications in communications and control have also been proposed we have previously shown that identical chaotic maps can under certain conditions be synchronized by a common noise like input this raises the question whether the chaotic output from a map can synchronize other maps to itself in this paper we present results on the synchronization of two identical maps where the output of one map is used to drive both maps we then apply this method to synchronizing identical but internally inhomogeneous populations of chaotic oscillators using a randomly constructed scalar coupling signal which tends to white noise as the number of oscillators increases such a system may have applications in secure communication

an iterative algorithm for solving a pair of matrix equations ayb e cyd f over generalized centro symmetric matrices a matrix email protected r n x n is said to be a symmetric orthogonal matrix if p p t p 1 a matrix email protected r n x n is said to be generalized centro symmetric generalized central anti symmetric with respect to p if a pap a pap the generalized centro symmetric matrices have wide applications in information theory linear estimate theory and numerical analysis in this paper we propose a new iterative algorithm to compute a generalized centro symmetric solution of the linear matrix equations ayb e cyd f we show when the matrix equations are consistent over generalized centro symmetric matrix y for any initial generalized centro symmetric matrix y 1 the sequence y k generated by the introduced algorithm converges to a generalized centro symmetric solution of matrix equations ayb e cyd f the least frobenius norm generalized centro symmetric solution can be derived when a special initial generalized centro symmetric matrix is chosen furthermore the optimal approximation generalized centro symmetric solution to a given generalized centro symmetric matrix can be derived several numerical examples are given to show the efficiency of the presented method

an algebraic relation between consimilarity and similarity of complex matrices and its applications an antilinear operator in complex vector spaces is an important operator in the study of modern quantum theory quantum and semiclassical optics quantum electronics and quantum chemistry consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces and the theory of consimilarity of complex matrices plays an important role in the study of quantum theory this paper by means of a real representation of a complex matrix studies the relation between consimilarity and similarity of complex matrices sets up an algebraic bridge between consimilarity and similarity and turns the theory of consimilarity into that of ordinary similarity this paper also gives some applications of consimilarity of complex matrices