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.

近年来，人们对多项线性矩阵方程重新产生了兴趣，因为它们已在许多重要应用中发挥作用。在这里，我们通过动态泛函粒子方法来考虑这些方程的解，这是一种依赖于阻尼二阶动态系统的数值积分的迭代技术。我们开发了一种新算法，用于求解一大类这些方程，其中包括所有具有 Hermitian 正定或负定系数的线性矩阵方程。在数值实验中，我们的 MATLAB 实现优于现有的求解多项 Sylvester 方程的方法。对于西尔维斯特方程 $\mathrm{一个}X+X乙=C$，特别是，它可以比 Bartels-Stewart 算法的内置实现更快、更准确，当$\mathrm{一个}$和$乙$条件良好，大小差异很大。