Numerical methods for large-scale Lyapunov equations with symmetric banded data

The numerical solution of large-scale Lyapunov matrix equations with symmetric banded data has so far received little attention in the rich literature on Lyapunov equations. We aim to contribute to this open problem by introducing two efficient solution methods, which respectively address the cases of well conditioned and ill conditioned coefficient matrices. The proposed approaches conveniently exploit the possibly hidden structure of the solution matrix so as to deliver memory and computation saving approximate solutions. Numerical experiments are reported to illustrate the potential of the described methods

Stationary splitting iterative methods for the matrix equation AX B = C

Stationary splitting iterative methods for solving AXB = Care considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M −N by a matrix H such that (I −H) invertible. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications, our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer-aided geometric design (CAGD).The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, and the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

The Dynamical Functional Particle Method for Multi-Term Linear Matrix Equations

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dynamical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermitian positive definite or negative definite coefficients. In numerical experiments, our MATLAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels–Stewart algorithm, when A and B are well conditioned and have very different size