一种非递归、与维度无关的Schur分解算法用于N维Sylvester张量方程
A Non-Recursive, Dimension-Independent Schur-Decomposition Algorithm for $N$-Dimensional Sylvester Tensor Equations
AI总结：
本文提出一种非递归直接求解器，用于求解N维Sylvester张量方程，该方法基于Bartels-Stewart算法，通过Schur分解系数矩阵，实现与维度无关的计算，适用于高维问题，且在内存效率和实现复杂度上优于递归方法。
摘要：
本文提出了一种非递归直接求解器，基于Bartels-Stewart算法，用于求解N维Sylvester张量方程。该方法仅依赖于系数矩阵的Schur分解，并将计算简化为对张量条目的单次顺序扫描，使其完全独立于维度N。其主要优点是简单性、与维度无关的公式以及能够解决受限于可用内存的极高维问题。我们在标准笔记本电脑上成功求解了N=29的情况。与Chen和Kressner的递归阻塞方法相比，两种方法在准确性上相同。递归方法在大系数矩阵情况下更快，而我们的求解器在矩阵较小时更具竞争力，尤其是在大N时，递归方法无法有效利用BLAS-3内核。此外，该方法在内存使用上更高效：对于接近容量的问题（例如，阶数为19、N=7的矩阵，其中解和右端占用约28 GB），Chen-Kressner方法会超出可用内存，而我们的方法则能成功。该方法还显著更易于实现，完全独立于N，并正确处理单例维度。我们详细描述了该算法并推导了准确的成本估计。为了可重复性，我们提供了伪代码和完整的MATLAB实现。作为应用，我们计算了具有常系数的线性N维ODE系统的任意时间解，从而得到空间离散化后的进化PDE解，包括在R^N上的高精度解。
英文摘要：
In this paper we present a non-recursive direct solver, based on the Bartels-Stewart algorithm, for $N$-dimensional Sylvester tensor equations. The method relies only on Schur decompositions of the coefficient matrices and reduces the computation to a single sequential sweep over tensor entries, making it entirely independent of the dimension $N$. Its main advantages are simplicity, a dimension-independent formulation, and the ability to solve very high-dimensional problems limited only by available memory, which is used efficiently. We successfully solve cases up to $N=29$ on a standard laptop with $32$ GB RAM. Compared with the recursive blocked method of Chen and Kressner (state of the art), both approaches achieve identical accuracy. The recursive method is faster for large coefficient matrices, whereas our solver is competitive or superior when matrices are small, especially for large $N$, where recursive methods cannot effectively exploit BLAS-3 kernels. It also uses memory more efficiently: for near-capacity problems (e.g., matrices of order $19$ with $N=7$, where solution and right-hand side occupy $\approx 28$ GB), the Chen-Kressner method exceeds available memory, while ours succeeds. The method is also significantly simpler to implement, fully independent of $N$, and correctly handles singleton dimensions. We detail the algorithm and derive accurate cost estimates. For reproducibility, we provide pseudocode and complete MATLAB implementations. As an application, we compute solutions of linear $N$-dimensional ODE systems with constant coefficients at arbitrary times, and thus of evolutionary PDEs after spatial discretization, including highly accurate solutions of an advection-diffusion equation on $\mathbb{R}^N$.
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