Preconditioning is essential in iterative methods for solving linear systems. It is also the implicit objective in updating approximations of Jacobians in optimization methods, e.g., in quasi-Newton methods. Motivated by the latter, we study a nonclassic matrix condition number, the $\omega$-condition number. We do this in the context of optimal conditioning for: (i) our application to low rank updating of generalized Jacobians; (ii) iterative methods for linear systems: (iia) clustering of eigenvalues and (iib) convergence rates. For a positive definite matrix, the $\omega$-condition measure is the ratio of the arithmetic and geometric means of the eigenvalues. In particular, our applications concentrate on linear systems with low rank updates of ill-conditioned positive definite matrices. These systems arise in the context of nonsmooth Newton methods using generalized Jacobians. We are able to use optimality conditions and derive explicit formulae for $\omega$-optimal preconditioners and preconditioned updates. Connections to partial Cholesky sparse preconditioners are made. Evaluating or estimating the classical condition number $\kappa$ can be expensive. We show that the $\omega$-condition number can be evaluated explicitly following a Cholesky or LU factorization. Moreover, the simplicity of $\omega$ allows for the derivation of formulae for optimal preconditioning in various scenarios, i.e., this avoids the need for expensive algorithmic calculations. Our empirics show that $\omega$ estimates the actual condition of a linear system significantly better. Moreover, our empirical results show a significant decrease in the number of iterations required for a requested accuracy in the residual during an iterative method, i.e., these results confirm the efficacy of using the $\omega$-condition number compared to the classical condition number.

翻译：预处理在求解线性系统的迭代方法中至关重要，它同时也是优化方法（如拟牛顿法）中更新雅可比近似矩阵的隐含目标。受后者启发，我们研究了一种非经典的矩阵条件数——ω-条件数。我们在以下最优条件化的背景下展开研究：(i) 广义雅可比矩阵低秩更新的应用；(ii) 线性系统的迭代方法：(iia) 特征值聚类与(iib) 收敛速率。对于正定矩阵，ω-条件测度是特征值算术平均与几何平均之比。特别地，我们的应用集中于具有病态正定矩阵低秩更新的线性系统。这类系统出现在使用广义雅可比矩阵的非光滑牛顿方法中。我们能够利用最优性条件，推导出ω-最优预处理矩阵及预处理更新的显式公式，并与部分Cholesky稀疏预处理方法建立联系。经典条件数κ的计算或估计可能代价高昂。我们证明ω-条件数可在Cholesky或LU分解后显式计算。此外，ω的简洁性使得能够在多种场景下推导最优预处理的显式公式，从而避免昂贵的算法计算。实验表明，ω能显著更准确地估计线性系统的实际条件。同时，实证结果显示，在迭代方法中为达到残差所需精度所需的迭代次数显著减少，这些结果证实了使用ω-条件数相较于经典条件数的有效性。