西尔维斯特方程解的条件数

中文摘要 AI 辅助

我们部分回答了Nick Higham提出的关于西尔维斯特（Sylvester）方程和李亚普诺夫（Lyapunov）方程解的条件数的公开问题，该问题源于对这些方程数值算法的向后稳定性分析。我们首先证明，即使矩阵A、B、C以及克罗内克和I⊗A−B^T⊗I均完全良态，西尔维斯特方程AX−XB=C的解仍可能具有任意差的条件数。随后，我们推导了解的条件数的一般先验界，以及当A和B可对角化时西尔维斯特方程的条件数界，还给出了涉及矩阵指数和佐洛塔廖夫数（Zolotarev numbers）的下界。对于李亚普诺夫方程AX+XA^T=−C，我们在两种情形下得到了上界：（i）当A为对称正定矩阵且C为对称负定矩阵时；（ii）当A为严格耗散矩阵且C为对称正定矩阵时。

英文摘要

We partially answer an open problem, posed by Nick Higham, concerning the conditioning of solutions to Sylvester and Lyapunov equations. The question arises in the backward stability analysis of numerical algorithms for these equations. We first show that the solution to the Sylvester equation $AX-XB = C$ can be arbitrarily ill-conditioned even if $A, B, C$ and the Kronecker sum $I \otimes A - B^T \otimes I$ are all perfectly conditioned. We then derive general a priori bounds on the condition number of the solution, as well as bounds for the Sylvester equation when $A$ and $B$ are diagonalizable. We also provide lower bounds involving matrix exponentials and Zolotarev numbers. For the Lyapunov equation $AX+XA^T = -C$, we obtain upper bounds in two settings: (i) when $A$ is symmetric positive definite while $C$ is symmetric negative definite, and (ii) when $A$ is strictly dissipative and $C$ is symmetric positive definite.

发表机构

• University of Leeds（利兹大学）
• University of Leicester（莱斯特大学）

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