Finite iterative algorithms for solving generalized coupled Sylvester systems – Part I: One-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions

AbstractThe generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this two-part article, finite iterative methods are proposed for solving one-sided (or two-sided) and generalized coupled Sylvester matrix equations and the corresponding optimal approximation problem over generalized reflexive solutions (or reflexive solutions). In part I, an iterative algorithm is constructed to solve one-sided and coupled Sylvester matrix equations (AY−ZB,CY−ZD)=(E,F) over generalized reflexive matrices Y and Z. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [Y1,Z1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solution pair can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [Y^,Z^] to a given matrix pair [Y0,Z0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [Y∼∗,Z∼∗] of two new corresponding generalized coupled Sylvester matrix equations (AY∼-Z∼B,CY∼-Z∼D)=(E∼,F∼), where E∼=E-AY0+Z0B,F∼=F-CY0+Z0D. Several numerical examples are given to show the effectiveness of the presented iterative algorithm

Preconditioning techniques for generalized Sylvester matrix equations

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.Comment: 26 pages, 3 figures, 2 tables. Submitted manuscrip

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆-Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆-Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n3r) algorithm for computing the (unique) solution

Iterative Methods to Solve the Generalized Coupled Sylvester-Conjugate Matrix Equations for Obtaining the Centrally Symmetric (Centrally Antisymmetric) Matrix Solutions

The iterative method is presented for obtaining the centrally symmetric (centrally antisymmetric) matrix pair (X,Y) solutions of the generalized coupled Sylvester-conjugate matrix equations A1X+B1Y=D1X¯E1+F1, A2Y+B2X=D2Y¯E2+F2. On the condition that the coupled matrix equations are consistent, we show that the solution pair (X*,Y*) can be obtained within finite iterative steps in the absence of round-off error for any initial value given centrally symmetric (centrally antisymmetric) matrix. Moreover, by choosing appropriate initial value, we can get the least Frobenius norm solution for the new generalized coupled Sylvester-conjugate linear matrix equations. Finally, some numerical examples are given to illustrate that the proposed iterative method is quite efficient

Factorized solution of generalized stable Sylvester equations using many-core GPU accelerators

[EN] We investigate the factorized solution of generalized stable Sylvester equations such as those arising in model reduction, image restoration, and observer design. Our algorithms, based on the matrix sign function, take advantage of the current trend to integrate high performance graphics accelerators (also known as GPUs) in computer systems. As a result, our realisations provide a valuable tool to solve large-scale problems on a variety of platforms.We acknowledge support of the ANII - MPG Independent Research Group: "Efficient Hetergenous Computing" at UdelaR, a partner group of the Max Planck Institute in Magdeburg.Benner, P.; Dufrechou, E.; Ezzatti, P.; Gallardo, R.; Quintana-Ortí, ES. (2021). Factorized solution of generalized stable Sylvester equations using many-core GPU accelerators. The Journal of Supercomputing (Online). 77(9):10152-19164. https://doi.org/10.1007/s11227-021-03658-y101521916477

Least Squares Based Iterative Algorithm for the Coupled Sylvester Matrix Equations

By analyzing the eigenvalues of the related matrices, the convergence analysis of the least squares based iteration is given for solving the coupled Sylvester equations AX+YB=C and DX+YE=F in this paper. The analysis shows that the optimal convergence factor of this iterative algorithm is 1. In addition, the proposed iterative algorithm can solve the generalized Sylvester equation AXB+CXD=F. The analysis demonstrates that if the matrix equation has a unique solution then the least squares based iterative solution converges to the exact solution for any initial values. A numerical example illustrates the effectiveness of the proposed algorithm

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover "negative flows", leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation