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Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M -Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation A X + X B = C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

Projection methods based on Krylov subspaces are an effective tool for solving low-rank Sylvester matrix equations in the large scale cases. The initial problem is projected onto a sequence of nested subspaces. At each step of the process, a reduced Sylvester equation, whose coefficient matrices are restrictions of the original coefficient matrices to the projection subspaces, are solved by a direct method. In this paper, after recalling some important properties about the extended block Arnoldi process, we propose an alternative approach that enables us to determine approximate solutions by solving projected and reduced Sylvester equations whose coefficient matrices are restrictions of the inverse of the original matrices. Some numerical experiments are given in order to illustrate the performance of our approach.

In the first part of this paper, we investigate the use of Hessenberg-based methods for solving the Sylvester matrix equation A X + X B = C . To achieve this goal, the Sylvester form of the global generalized Hessenberg process is presented. Using this process, different methods based on a Petrov–Galerkin or on a minimal norm condition are derived. In the second part, we focus on the SGl-CMRH method which is based on the Sylvester form of the Hessenberg process with pivoting strategy combined with a minimal norm condition. In order to accelerate the SGl-CMRH method, a preconditioned framework of this method is also considered. It includes both fixed and flexible variants of the SGl-CMRH method. Moreover, the connection between the flexible preconditioned SGl-CMRH method and the fixed one is studied and some upper bounds for the residual norm are obtained. In particular, application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right-hand sides. Finally, some numerical experiments are given in order to evaluate the effectiveness of the proposed methods.

Many problems in control theory can be studied by obtaining the symmetric solution of linear matrix equations. In this investigation, we deal with the symmetric solutions X , Y and Z of the general Sylvester matrix equations A 1 X B 1 + C 1 Y D 1 + E 1 Z F 1 = G 1 , A 2 X B 2 + C 2 Y D 2 + E 2 Z F 2 = G 2 , ⋮ ⋮ ⋮ ⋮ A t X B t + C t Y D t + E t Z F t = G t . The Lanczos version of biconjugate residual (BCR) algorithm is generalized to compute the symmetric solutions of the general Sylvester matrix equations. The convergence properties of this algorithm are discussed and it is proven that it smoothly converges to the symmetric solutions of the general Sylvester matrix equations in a finite number of iterations in the absence of round-off errors. Finally, the comparative numerical results are carried out to support that the current algorithm may be more efficient than other iterative algorithms.

This paper introduces another version of biconjugate residual method (BCR) for solving the generalized coupled Sylvester matrix equations over centrosymmetric or anti-centrosymmetric matrix. We prove this version of BCR algorithm can find the centrosymmetric solution group of the generalized coupled matrix equations for any initial matrix group within finite steps in the absence of round-off errors. Furthermore, a method is provided for choosing the initial matrices to obtain the least norm solution of the problem. At last, some numerical examples are provided to illustrate the efficiency and validity of methods we have proposed.

In this paper, we first present a relaxed gradient based iterative (RGI) algorithm for solving matrix equations A 1 X B 1 = F 1 and A 2 X B 2 = F 2 . The idea is from (Niu et al., 2011; Xie and Ma, 2016) in which efficient algorithms were developed for solving the Sylvester matrix equations. Then the RGI algorithm is extended to solve the generalized linear matrix equations of the form A i X B i = F i , ( i = 1 , 2 , … , N ) . For any initial value, it is proved that the iterative solution converges to their true solution under some appropriate assumptions. Finally, a numerical example is given to illustrate that the introduced iterative algorithm is more efficient than gradient based maximal convergence rate iterative (GI) algorithm of Ding et al. (2010) in speed, elapsed time and iterative steps.

In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper.

Some explicit closed-form solutions of homogeneous and nonhomogeneous Sylvester-conjugate matrix equations are provided in this paper. One of the solutions is expressed in terms of controllability matrices and observability matrices. The proposed approach does not require all the coefficient matrices to be in any canonical forms and the solutions provide a significant degree of freedom which is represented by an arbitrarily chosen parameter matrix. By specifying the solutions of the homogeneous Sylvester-conjugate equation, some new expressions of the solutions of the normal Sylvester, normal Sylvester-conjugate and Sylvester equations are given. This fact reveals that the Sylveter-conjugate matrix equations are a more general class of some previously investigated matrix equations.

The alternate Sylvester sums are T"m(a,b)[email protected]?"n"@?"N"R(-1)^nn^m, where a and b are coprime, positive integers, and NR is the Frobenius set associated with a and b. In this note, we study the generating functions, recurrences and explicit expressions of the alternate Sylvester sums. It can be found that the results are closely related to the Bernoulli polynomials, the Euler polynomials, and the (alternate) power sums over the natural numbers.

Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but reversal of the elimination principle can be beneficial as well: competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task, defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms, that is their convergence from the start. A well known example is the dual linear and nonlinear programming, which enhances the power of optimization algorithms. We believe that this is just an ad hoc application of general Principle of Expansion with Independent Constraints; it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization. At the end of this paper we comment on other applications and extensions of this principle.Presently we show it at work for the approximation of a single zero of a univariate polynomial p of a degree n. Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros, such as Weierstrass–Durand–Kerner’s root-finder, which reduces its root-finding task to Viète’s (Vieta’s) system of n polynomial equations with n unknowns. We adjust this root-finder to the approximation of a single zero of p, preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear. Together with computing a zero of a polynomial p, the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of p. Alternatively by using m processors that exchange no data, one can concurrently approximate up to m zeros of p. Our tests confirm the efficiency of the proposed algorithms.Technically our root-finding boils down to computations with structured matrices, polynomials and partial fraction decompositions. Our study of these links can be of independent interest; e.g., as by-product we express the inverse of a Sylvester matrix via its last column, thus extending the celebrated result of Gohberg and Sementsul (1972) [22] from Toeplitz to Sylvester matrix inverses.

The classical Jacobi matrix polynomials only for commutative matrices were first studied by Defez et al. [E. Defez, L. Jodar, A. Law. Jacobi matrix differential equation, polynomial solutions and their properties, Comput. Math. Appl. 48 (2004) 789-803]. The main aim of this paper is to construct a multivariable extension with the help of the classical Jacobi matrix polynomials (JMPs). Generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived. Furthermore, general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented. (C) 2011 Elsevier Ltd. All rights reserved.

This note presents an adaptive control scheme for first-order continuous-time systems subject to bounded noise and unmodelled dynamics. The estimated plant model is controllable and then the adaptive scheme is free from singularities. The estimation scheme involves the use of a relative dead zone which freezes the estimation process when the size of the prediction error is small compared to the contribution of the unmodelled dynamics. The singularities are avoided through the use of a modification of the estimated plant parameter vector so that its associated Sylvester matrix is guaranteed to be nonsingular. That property is achieved by ensuring that the absolute value of its determinant does not lie below a prefixed positive threshold. In addition, the use of a hysteresis switching function is not used for a modification of the estimates while the absence of chattering is guaranteed in the eventual case when the Sylvester determinant tends to zero. The global stability of the closed-loop system is also guaranteed.