We discuss the use of a matrix-oriented approach for numerically solving the dense matrix equation AX + XA^T + M₁XN₁ + … + M_ℓXN_ℓ = F, with ℓ ≥ 1, and M_i, N_i, i = 1, … , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix-oriented method is reported. The application of the procedure as the core step in the solution of the large-scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block.

中文翻译：

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广义线性矩阵方程和应用的 Sherman-Morrison-Woodbury 公式

我们讨论了使用面向矩阵的方法对密集矩阵方程AX  +  XA ^T  +  M ₁ XN ₁  + … +  M _ℓ XN _ℓ  =  F进行数值求解，其中ℓ  ≥ 1，并且M _i ,  N _i , i  = 1, … ,  ℓ低等级的。该方法依赖于以问题的矢量化形式正式定义的 Sherman-Morrison-Woodbury 公式，但应用于矩阵设置。这使人们能够以比使用 Kronecker 公式显着降低的计算成本和内存要求来解决中等规模的密集问题。说明了导致这种形式的中等大小方程的应用问题，并报告了面向矩阵的方法的性能。还显示了该程序作为解决大规模问题的核心步骤的应用。此外，提出了一种新的线性张量方程显式方法，该方法使用所讨论的矩阵方程过程作为关键构建块。