We consider the numerical solution of large scale singular (continuous-time) Lyapunov equations of the form AX + XA^⊤ + BB^⊤ = 0, where A is semistable, that is, its spectrum is contained in the left half plane, with the exception of a few semisimple eigenvalues at zero. We also consider the case of a few semisimple eigenvalues on the imaginary axis. We assume that we know these few eigenvalues (zero or imaginary), and that we have or can compute the corresponding invariant subspaces. We use this information to build an appropriate newly proposed subspace on which to project the Lyapunov equations, and then compute a low-rank approximation to the least squares solution. Selected illustrative numerical examples are provided.

中文翻译：

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奇异李雅普诺夫方程的数值解

我们考虑形式为AX  +  XA ^⊤  +  BB ^⊤  = 0 的大规模奇异（连续时间）李雅普诺夫方程的数值解，其中A是半稳定的，也就是说，它的频谱包含在左半平面中，除了一些在零处的半简单特征值。我们还考虑了虚轴上一些半简单特征值的情况。我们假设我们知道这几个特征值（零或虚数），并且我们有或可以计算相应的不变子空间。我们使用这些信息来构建一个合适的新提出的子空间，在该子空间上投影 Lyapunov 方程，然后计算最小二乘解的低秩近似。提供了选定的说明性数值示例。